Optimal. Leaf size=81 \[ -\frac{45 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)}-\frac{3 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}-\frac{59 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{1024 d}+\frac{59 x}{2048} \]
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Rubi [A] time = 0.0616755, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2664, 2754, 12, 2657} \[ -\frac{45 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)}-\frac{3 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}-\frac{59 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{1024 d}+\frac{59 x}{2048} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 2754
Rule 12
Rule 2657
Rubi steps
\begin{align*} \int \frac{1}{(5+3 \cos (c+d x))^3} \, dx &=-\frac{3 \sin (c+d x)}{32 d (5+3 \cos (c+d x))^2}-\frac{1}{32} \int \frac{-10+3 \cos (c+d x)}{(5+3 \cos (c+d x))^2} \, dx\\ &=-\frac{3 \sin (c+d x)}{32 d (5+3 \cos (c+d x))^2}-\frac{45 \sin (c+d x)}{512 d (5+3 \cos (c+d x))}+\frac{1}{512} \int \frac{59}{5+3 \cos (c+d x)} \, dx\\ &=-\frac{3 \sin (c+d x)}{32 d (5+3 \cos (c+d x))^2}-\frac{45 \sin (c+d x)}{512 d (5+3 \cos (c+d x))}+\frac{59}{512} \int \frac{1}{5+3 \cos (c+d x)} \, dx\\ &=\frac{59 x}{2048}-\frac{59 \tan ^{-1}\left (\frac{\sin (c+d x)}{3+\cos (c+d x)}\right )}{1024 d}-\frac{3 \sin (c+d x)}{32 d (5+3 \cos (c+d x))^2}-\frac{45 \sin (c+d x)}{512 d (5+3 \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.188741, size = 56, normalized size = 0.69 \[ -\frac{\frac{3 (182 \sin (c+d x)+45 \sin (2 (c+d x)))}{(3 \cos (c+d x)+5)^2}+59 \tan ^{-1}\left (2 \cot \left (\frac{1}{2} (c+d x)\right )\right )}{1024 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 79, normalized size = 1. \begin{align*} -{\frac{69}{512\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-2}}-{\frac{51}{128\,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 ) ^{-2}}+{\frac{59}{1024\,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 = 1.76045, size = 150, normalized size = 1.85 \begin{align*} -\frac{\frac{6 \,{\left (\frac{68 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{23 \, \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{\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 16} - 59 \, \arctan \left (\frac{\sin \left (d x + c\right )}{2 \,{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{1024 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66285, size = 258, normalized size = 3.19 \begin{align*} -\frac{59 \,{\left (9 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) + 25\right )} \arctan \left (\frac{5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right ) + 12 \,{\left (45 \, \cos \left (d x + c\right ) + 91\right )} \sin \left (d x + c\right )}{2048 \,{\left (9 \, d \cos \left (d x + c\right )^{2} + 30 \, d \cos \left (d x + c\right ) + 25 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.10102, size = 359, normalized size = 4.43 \begin{align*} \begin{cases} \frac{x}{\left (5 + 3 \cosh{\left (2 \operatorname{atanh}{\left (2 \right )} \right )}\right )^{3}} & \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 )^{3}} & \text{for}\: d = 0 \\\frac{59 \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 ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} + \frac{472 \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 )}}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} + \frac{944 \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 )}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} - \frac{138 \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} - \frac{408 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22912, size = 101, normalized size = 1.25 \begin{align*} \frac{59 \, d x + 59 \, c - \frac{12 \,{\left (23 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 68 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4\right )}^{2}} - 118 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{2048 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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