Optimal. Leaf size=90 \[ \frac{5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}+\frac{3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-3 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}-\frac{3 \log \left (3 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{64 d} \]
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Rubi [A] time = 0.0476817, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2664, 12, 2660, 616, 31} \[ \frac{5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}+\frac{3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-3 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}-\frac{3 \log \left (3 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{64 d} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 12
Rule 2660
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(-3+5 \sin (c+d x))^2} \, dx &=\frac{5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}+\frac{1}{16} \int \frac{3}{-3+5 \sin (c+d x)} \, dx\\ &=\frac{5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}+\frac{3}{16} \int \frac{1}{-3+5 \sin (c+d x)} \, dx\\ &=\frac{5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3+10 x-3 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}\\ &=\frac{5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{1-3 x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{9-3 x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}\\ &=\frac{3 \log \left (1-3 \tan \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}-\frac{3 \log \left (3-\tan \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}+\frac{5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0192063, size = 130, normalized size = 1.44 \[ \frac{20 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\frac{1}{3 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{3}{\cos \left (\frac{1}{2} (c+d x)\right )-3 \sin \left (\frac{1}{2} (c+d x)\right )}\right )+9 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-3 \sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (3 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{192 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 76, normalized size = 0.8 \begin{align*} -{\frac{5}{48\,d} \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}}+{\frac{3}{64\,d}\ln \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }-{\frac{5}{16\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -3 \right ) ^{-1}}-{\frac{3}{64\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -3 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966043, size = 155, normalized size = 1.72 \begin{align*} \frac{\frac{40 \,{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 3\right )}}{\frac{10 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 3} + 9 \, \log \left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right ) - 9 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 3\right )}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71167, size = 248, normalized size = 2.76 \begin{align*} -\frac{3 \,{\left (5 \, \sin \left (d x + c\right ) - 3\right )} \log \left (4 \, \cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) + 5\right ) - 3 \,{\left (5 \, \sin \left (d x + c\right ) - 3\right )} \log \left (-4 \, \cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) + 5\right ) + 40 \, \cos \left (d x + c\right )}{128 \,{\left (5 \, d \sin \left (d x + c\right ) - 3 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.17596, size = 462, normalized size = 5.13 \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 = 1.17406, size = 109, normalized size = 1.21 \begin{align*} -\frac{\frac{40 \,{\left (5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3\right )}}{3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 10 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3} - 9 \, \log \left ({\left | 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 9 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \right |}\right )}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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