Optimal. Leaf size=64 \[ -\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.092183, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 75} \[ -\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 75
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4 (a-x)^2 (a+x)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^4}-\frac{a^2}{x^3}-\frac{a}{x^2}+\frac{1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\csc (c+d x)}{a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\log (\sin (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.0778566, size = 48, normalized size = 0.75 \[ \frac{-2 \csc ^3(c+d x)+3 \csc ^2(c+d x)+6 \csc (c+d x)+6 \log (\sin (c+d x))}{6 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 63, normalized size = 1. \begin{align*}{\frac{1}{da\sin \left ( dx+c \right ) }}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}-{\frac{1}{3\,da \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0733, size = 68, normalized size = 1.06 \begin{align*} \frac{\frac{6 \, \log \left (\sin \left (d x + c\right )\right )}{a} + \frac{6 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11885, size = 198, normalized size = 3.09 \begin{align*} \frac{6 \,{\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) - 4}{6 \,{\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\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 = 1.24, size = 84, normalized size = 1.31 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{11 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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