# Speed up evaluation of this integral?

Posted 6 months ago
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 I am trying to evaluate the integral $$\int_0^{2\pi}\frac{D_{11}^2+2D_{12}^2+D_{22}^2+D_{33}^2}{1+e\cos x(t)}dx$$Where $D_{ij}$ are third derivatives wrt $t$ as defined below. a[psi_] := k1/(1 + e*Cos[psi]); d[psi_] := p1/(1 + e*Cos[psi]); D11 = D[a[x[t]]*(Cos[x[t]]^2 - 1/3), {t, 3}]; D12 = D[a[x[t]]*Cos[x[t]]*Sin[x[t]], {t, 3}]; D22 = D[a[x[t]]*(Sin[x[t]]^2 - 1/3), {t, 3}]; D33 = D[-a[x[t]]/3]; To do this I have written the input Assuming[Element[k1, Reals] && Element[e, Reals] && x'[t] == p2/(d[x[t]]^2) && Element[p2, Reals] && Element[p2, Reals], Integrate[(D11^2 + 2*D12^2 + D22^2 + D33^2)/(1 + e*Cos[x[t]])^2, {x[ t], 0, 2*Pi}]] This is taking a very long time to evaluate (has been running for hours), is it OK to specify to Mathematica that I have the condition $\dot{x}=\frac{p_2}{d(x(t))^2}$ in this manner? How could this be sped up? This integral arose in the study of the loss of energy of an orbiting mass due to gravitational radiation.
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Posted 6 months ago
 It takes less than a minute. But first you need to simplify the expression a[psi_] := k1/(1 + e*Cos[psi]); d[psi_] := p1/(1 + e*Cos[psi]); D11 = D[a[x[t]]*(Cos[x[t]]^2 - 1/3), {t, 3}]; D12 = D[a[x[t]]*Cos[x[t]]*Sin[x[t]], {t, 3}]; D22 = D[a[x[t]]*(Sin[x[t]]^2 - 1/3), {t, 3}]; D33 = D[-a[x[t]]/3, {t,3}]; x'[t] = p2/d[x[t]]^2 x''[t] = D[x'[t], t] x'''[t] = D[x'[t], t, t] Et = (D11^2 + 2*D12^2 + D22^2 + D33^2)/(1 + e*Cos[x[t]])^2 //FullSimplify Integrate[Et, {x[t], 0, 2*Pi}] //AbsoluteTiming {2.57266, ((1536 + 20320 e^2 + 35616 e^4 + 10988 e^6 + 315 e^8) k1^2 p2^6 \[Pi])/(24 p1^12)} 
 Very insightful. There is a problem that the OP's numerator is not dimensionally consistent. The D33 term is the partial derivative with respect to nothing, which bothered me. Presuming that D33 should also be the third derivative with respect to t, your approach is even faster and contains no conditional expression. a[psi_] := k1/(1 + e*Cos[psi]); d[psi_] := p1/(1 + e*Cos[psi]); D11 = D[a[x[t]]*(Cos[x[t]]^2 - 1/3), {t, 3}]; D12 = D[a[x[t]]*Cos[x[t]]*Sin[x[t]], {t, 3}]; D22 = D[a[x[t]]*(Sin[x[t]]^2 - 1/3), {t, 3}]; (*D33=D[-a[x[t]]/3]*)(* From the OP *) D33 = D[-a[x[t]]/3, {t, 3}]; x'[t] = p2/d[x[t]]^2; x''[t] = D[x'[t], t]; x'''[t] = D[x'[t], t, t]; Et = (D11^2 + 2*D12^2 + D22^2 + D33^2)/(1 + e*Cos[x[t]])^2 // FullSimplify Integrate[Et, {x[t], 0, 2*Pi}] // AbsoluteTiming (* {2.819966958116478,((1536+20320 e^2+35616 e^4+10988 e^6+315 e^8) \ k1^2 p2^6 \[Pi])/(24 p1^12)} *) `