Optimal. Leaf size=36 \[ -\frac{\cos (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{2 x}{a^2} \]
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Rubi [A] time = 0.129302, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2869, 2746, 2735, 3770} \[ -\frac{\cos (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{2 x}{a^2} \]
Antiderivative was successfully verified.
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Rule 2869
Rule 2746
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \csc (c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=-\frac{\cos (c+d x)}{a^2 d}+\frac{\int \csc (c+d x) \left (a^2-2 a^2 \sin (c+d x)\right ) \, dx}{a^4}\\ &=-\frac{2 x}{a^2}-\frac{\cos (c+d x)}{a^2 d}+\frac{\int \csc (c+d x) \, dx}{a^2}\\ &=-\frac{2 x}{a^2}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cos (c+d x)}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.133405, size = 46, normalized size = 1.28 \[ -\frac{\cos (c+d x)-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+2 c+2 d x}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.121, size = 60, normalized size = 1.7 \begin{align*} -2\,{\frac{1}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-4\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}}+{\frac{1}{d{a}^{2}}\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 [B] time = 1.73861, size = 111, normalized size = 3.08 \begin{align*} -\frac{\frac{2}{a^{2} + \frac{a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}} + \frac{4 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15881, size = 135, normalized size = 3.75 \begin{align*} -\frac{4 \, d x + 2 \, \cos \left (d x + c\right ) + \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos ^{4}{\left (c + d x \right )} \csc{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31897, size = 70, normalized size = 1.94 \begin{align*} -\frac{\frac{2 \,{\left (d x + c\right )}}{a^{2}} - \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac{2}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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