Optimal. Leaf size=88 \[ -\frac{5 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}+\frac{3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}-\frac{3 \log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{64 d} \]
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Rubi [A] time = 0.0433838, antiderivative size = 88, 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} \[ -\frac{5 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}+\frac{3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}-\frac{3 \log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{64 d} \]
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}{-9-3 x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}+\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{3 \log \left (3+\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{5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.150623, size = 126, normalized size = 1.43 \[ \frac{20 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\frac{3}{3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+\frac{1}{\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )}\right )+9 \left (\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{192 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 76, normalized size = 0.9 \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 = 0.964615, size = 155, normalized size = 1.76 \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 = 2.12007, size = 247, normalized size = 2.81 \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 = 2.28905, size = 468, normalized size = 5.32 \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 = 1.17508, size = 109, normalized size = 1.24 \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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