Optimal. Leaf size=45 \[ \frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{a^3} \]
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Rubi [A] time = 0.182899, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2875, 2872, 3770, 2648} \[ \frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2872
Rule 3770
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \csc (c+d x) \sec ^2(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (a+a \csc (c+d x)-\frac{4 a}{1+\sin (c+d x)}\right ) \, dx}{a^4}\\ &=\frac{x}{a^3}+\frac{\int \csc (c+d x) \, dx}{a^3}-\frac{4 \int \frac{1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=\frac{x}{a^3}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}\\ \end{align*}
Mathematica [B] time = 0.271051, size = 122, normalized size = 2.71 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5 \left (\cos \left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+c+d x\right )+\sin \left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+c+d x-8\right )\right )}{a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.145, size = 58, normalized size = 1.3 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}+8\,{\frac{1}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}+{\frac{1}{d{a}^{3}}\ln \left ( \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 = 1.52696, size = 105, normalized size = 2.33 \begin{align*} \frac{\frac{8}{a^{3} + \frac{a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}} + \frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.13666, size = 338, normalized size = 7.51 \begin{align*} \frac{2 \, d x + 2 \,{\left (d x + 4\right )} \cos \left (d x + c\right ) -{\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \,{\left (d x - 4\right )} \sin \left (d x + c\right ) + 8}{2 \,{\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36281, size = 63, normalized size = 1.4 \begin{align*} \frac{\frac{d x + c}{a^{3}} + \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac{8}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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