Optimal. Leaf size=113 \[ -\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}+\frac{43 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d} \]
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Rubi [A] time = 0.0719102, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2664, 2754, 12, 2659, 206} \[ -\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}+\frac{43 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 2754
Rule 12
Rule 2659
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(-3+5 \cos (c+d x))^3} \, dx &=\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac{1}{32} \int \frac{6+5 \cos (c+d x)}{(-3+5 \cos (c+d x))^2} \, dx\\ &=\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac{1}{512} \int \frac{43}{-3+5 \cos (c+d x)} \, dx\\ &=\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac{43}{512} \int \frac{1}{-3+5 \cos (c+d x)} \, dx\\ &=\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac{43 \operatorname{Subst}\left (\int \frac{1}{2-8 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{256 d}\\ &=-\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}+\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}+\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.131317, size = 211, normalized size = 1.87 \[ \frac{45 \sin \left (\frac{1}{2} (c+d x)\right )}{1024 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{45 \sin \left (\frac{1}{2} (c+d x)\right )}{1024 d \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{5}{512 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{5}{512 d \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}+\frac{43 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 120, normalized size = 1.1 \begin{align*}{\frac{25}{2048\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-2}}-{\frac{35}{2048\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}}-{\frac{43}{2048\,d}\ln \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }-{\frac{25}{2048\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-2}}-{\frac{35}{2048\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}}+{\frac{43}{2048\,d}\ln \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52223, size = 185, normalized size = 1.64 \begin{align*} -\frac{\frac{20 \,{\left (\frac{17 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{28 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{\frac{8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{16 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1} - 43 \, \log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + 43 \, \log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{2048 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63687, size = 382, normalized size = 3.38 \begin{align*} \frac{43 \,{\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (-\frac{3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) - 43 \,{\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (-\frac{3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) + 40 \,{\left (45 \, \cos \left (d x + c\right ) - 11\right )} \sin \left (d x + c\right )}{4096 \,{\left (25 \, d \cos \left (d x + c\right )^{2} - 30 \, d \cos \left (d x + c\right ) + 9 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.17821, size = 490, normalized size = 4.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34633, size = 113, normalized size = 1. \begin{align*} -\frac{\frac{20 \,{\left (28 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 17 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2}} - 43 \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 43 \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{2048 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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