Optimal. Leaf size=120 \[ \frac{\left (2 a^2-b^2\right ) \csc (c+d x)}{a^3 d}+\frac{b \left (2 a^2-b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^4 b d}+\frac{b \csc ^2(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.174388, antiderivative size = 120, 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 = {2837, 12, 894} \[ \frac{\left (2 a^2-b^2\right ) \csc (c+d x)}{a^3 d}+\frac{b \left (2 a^2-b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^4 b d}+\frac{b \csc ^2(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^4 \left (b^2-x^2\right )^2}{x^4 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{x^4 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{b^4}{a x^4}-\frac{b^4}{a^2 x^3}+\frac{-2 a^2 b^2+b^4}{a^3 x^2}+\frac{2 a^2 b^2-b^4}{a^4 x}+\frac{\left (a^2-b^2\right )^2}{a^4 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\left (2 a^2-b^2\right ) \csc (c+d x)}{a^3 d}+\frac{b \csc ^2(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{b \left (2 a^2-b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^4 b d}\\ \end{align*}
Mathematica [A] time = 0.283179, size = 110, normalized size = 0.92 \[ \frac{3 a^2 b^2 \csc ^2(c+d x)+6 a b \left (2 a^2-b^2\right ) \csc (c+d x)-6 b^2 \left (b^2-2 a^2\right ) \log (\sin (c+d x))+6 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-2 a^3 b \csc ^3(c+d x)}{6 a^4 b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 163, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{bd}}-2\,{\frac{b\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}+{\frac{{b}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{a}^{4}}}-{\frac{1}{3\,da \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+2\,{\frac{1}{da\sin \left ( dx+c \right ) }}-{\frac{{b}^{2}}{d{a}^{3}\sin \left ( dx+c \right ) }}+2\,{\frac{b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{{b}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{4}}}+{\frac{b}{2\,d{a}^{2} \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 = 0.974852, size = 153, normalized size = 1.27 \begin{align*} \frac{\frac{6 \,{\left (2 \, a^{2} b - b^{3}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{4}} + \frac{6 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} b} + \frac{3 \, a b \sin \left (d x + c\right ) + 6 \,{\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} - 2 \, a^{2}}{a^{3} \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 = 1.69746, size = 455, normalized size = 3.79 \begin{align*} -\frac{3 \, a^{2} b^{2} \sin \left (d x + c\right ) + 10 \, a^{3} b - 6 \, a b^{3} - 6 \,{\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{2} + 6 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} -{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) + 6 \,{\left (2 \, a^{2} b^{2} - b^{4} -{\left (2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{6 \,{\left (a^{4} b d \cos \left (d x + c\right )^{2} - a^{4} b 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 = 1.21953, size = 204, normalized size = 1.7 \begin{align*} \frac{\frac{6 \,{\left (2 \, a^{2} b - b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} + \frac{6 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b} - \frac{22 \, a^{2} b \sin \left (d x + c\right )^{3} - 11 \, b^{3} \sin \left (d x + c\right )^{3} - 12 \, a^{3} \sin \left (d x + c\right )^{2} + 6 \, a b^{2} \sin \left (d x + c\right )^{2} - 3 \, a^{2} b \sin \left (d x + c\right ) + 2 \, a^{3}}{a^{4} \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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