Optimal. Leaf size=56 \[ \frac{3 \cos (c+d x)}{16 d (3 \sin (c+d x)+5)}+\frac{5 \tan ^{-1}\left (\frac{\cos (c+d x)}{\sin (c+d x)+3}\right )}{32 d}+\frac{5 x}{64} \]
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Rubi [A] time = 0.0307029, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2664, 12, 2658} \[ \frac{3 \cos (c+d x)}{16 d (3 \sin (c+d x)+5)}+\frac{5 \tan ^{-1}\left (\frac{\cos (c+d x)}{\sin (c+d x)+3}\right )}{32 d}+\frac{5 x}{64} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 12
Rule 2658
Rubi steps
\begin{align*} \int \frac{1}{(-5-3 \sin (c+d x))^2} \, dx &=\frac{3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))}-\frac{1}{16} \int \frac{5}{-5-3 \sin (c+d x)} \, dx\\ &=\frac{3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))}-\frac{5}{16} \int \frac{1}{-5-3 \sin (c+d x)} \, dx\\ &=\frac{5 x}{64}+\frac{5 \tan ^{-1}\left (\frac{\cos (c+d x)}{3+\sin (c+d x)}\right )}{32 d}+\frac{3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0991729, size = 91, normalized size = 1.62 \[ \frac{\frac{6 (-3 \sin (c+d x)+5 \cos (c+d x)-5)}{3 \sin (c+d x)+5}+25 \tan ^{-1}\left (\frac{2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}\right )}{160 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 92, normalized size = 1.6 \begin{align*}{\frac{9}{200\,d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+{\frac{6}{5}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+1 \right ) ^{-1}}+{\frac{3}{40\,d} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+{\frac{6}{5}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+1 \right ) ^{-1}}+{\frac{5}{32\,d}\arctan \left ({\frac{5}{4}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{3}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44171, size = 126, normalized size = 2.25 \begin{align*} \frac{\frac{12 \,{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 5\right )}}{\frac{6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 5} + 25 \, \arctan \left (\frac{5 \, \sin \left (d x + c\right )}{4 \,{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{3}{4}\right )}{160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93719, size = 162, normalized size = 2.89 \begin{align*} \frac{5 \,{\left (3 \, \sin \left (d x + c\right ) + 5\right )} \arctan \left (\frac{5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right ) + 12 \, \cos \left (d x + c\right )}{64 \,{\left (3 \, d \sin \left (d x + c\right ) + 5 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.07124, size = 389, normalized size = 6.95 \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 = 1.18743, size = 128, normalized size = 2.29 \begin{align*} \frac{25 \, d x + 25 \, c + \frac{24 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5\right )}}{5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 6 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5} + 50 \, \arctan \left (-\frac{3 \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) - 9}\right )}{320 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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