Optimal. Leaf size=56 \[ -\frac{3 \sin (c+d x)}{16 d (3 \cos (c+d x)+5)}-\frac{5 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{32 d}+\frac{5 x}{64} \]
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Rubi [A] time = 0.0294425, 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} \[ -\frac{3 \sin (c+d x)}{16 d (3 \cos (c+d x)+5)}-\frac{5 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{32 d}+\frac{5 x}{64} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 12
Rule 2658
Rubi steps
\begin{align*} \int \frac{1}{(-5-3 \cos (c+d x))^2} \, dx &=-\frac{3 \sin (c+d x)}{16 d (5+3 \cos (c+d x))}-\frac{1}{16} \int \frac{5}{-5-3 \cos (c+d x)} \, dx\\ &=-\frac{3 \sin (c+d x)}{16 d (5+3 \cos (c+d x))}-\frac{5}{16} \int \frac{1}{-5-3 \cos (c+d x)} \, dx\\ &=\frac{5 x}{64}-\frac{5 \tan ^{-1}\left (\frac{\sin (c+d x)}{3+\cos (c+d x)}\right )}{32 d}-\frac{3 \sin (c+d x)}{16 d (5+3 \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0710502, size = 43, normalized size = 0.77 \[ -\frac{\frac{6 \sin (c+d x)}{3 \cos (c+d x)+5}+5 \tan ^{-1}\left (2 \cot \left (\frac{1}{2} (c+d x)\right )\right )}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 48, normalized size = 0.9 \begin{align*} -{\frac{3}{16\,d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-1}}+{\frac{5}{32\,d}\arctan \left ({\frac{1}{2}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.97654, size = 92, normalized size = 1.64 \begin{align*} -\frac{\frac{6 \, \sin \left (d x + c\right )}{{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 4\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} - 5 \, \arctan \left (\frac{\sin \left (d x + c\right )}{2 \,{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64167, size = 163, normalized size = 2.91 \begin{align*} -\frac{5 \,{\left (3 \, \cos \left (d x + c\right ) + 5\right )} \arctan \left (\frac{5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right ) + 12 \, \sin \left (d x + c\right )}{64 \,{\left (3 \, d \cos \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 = 2.43728, size = 190, normalized size = 3.39 \begin{align*} \begin{cases} \frac{x}{\left (-5 - 3 \cosh{\left (2 \operatorname{atanh}{\left (2 \right )} \right )}\right )^{2}} & \text{for}\: c = - d x - 2 i \operatorname{atanh}{\left (2 \right )} \vee c = - d x + 2 i \operatorname{atanh}{\left (2 \right )} \\\frac{x}{\left (- 3 \cos{\left (c \right )} - 5\right )^{2}} & \text{for}\: d = 0 \\\frac{5 \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2} \right )} + \pi \left \lfloor{\frac{\frac{c}{2} + \frac{d x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{32 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 128 d} + \frac{20 \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2} \right )} + \pi \left \lfloor{\frac{\frac{c}{2} + \frac{d x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{32 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 128 d} - \frac{6 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{32 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 128 d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16973, size = 80, normalized size = 1.43 \begin{align*} \frac{5 \, d x + 5 \, c - \frac{12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4} - 10 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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