Optimal. Leaf size=74 \[ \frac{1}{d \left (a^3 \sin (c+d x)+a^3\right )}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.0582604, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 44} \[ \frac{1}{d \left (a^3 \sin (c+d x)+a^3\right )}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 44
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x}-\frac{1}{a (a+x)^3}-\frac{1}{a^2 (a+x)^2}-\frac{1}{a^3 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (1+\sin (c+d x))}{a^3 d}+\frac{1}{2 a d (a+a \sin (c+d x))^2}+\frac{1}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.175268, size = 52, normalized size = 0.7 \[ \frac{\frac{2 \sin (c+d x)+3}{(\sin (c+d x)+1)^2}+2 \log (\sin (c+d x))-2 \log (\sin (c+d x)+1)}{2 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 68, normalized size = 0.9 \begin{align*}{\frac{1}{2\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{3}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76862, size = 97, normalized size = 1.31 \begin{align*} \frac{\frac{2 \, \sin \left (d x + c\right ) + 3}{a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}} - \frac{2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6075, size = 284, normalized size = 3.84 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 2 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, \sin \left (d x + c\right ) - 3}{2 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \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{\cot{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.9801, size = 80, normalized size = 1.08 \begin{align*} -\frac{\frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{2 \, \sin \left (d x + c\right ) + 3}{a^{3}{\left (\sin \left (d x + c\right ) + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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