Optimal. Leaf size=96 \[ \frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^2 b^3 d}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{a \sin (c+d x)}{b^2 d}-\frac{\csc (c+d x)}{a d}+\frac{\sin ^2(c+d x)}{2 b d} \]
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Rubi [A] time = 0.155076, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2837, 12, 894} \[ \frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^2 b^3 d}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{a \sin (c+d x)}{b^2 d}-\frac{\csc (c+d x)}{a d}+\frac{\sin ^2(c+d x)}{2 b d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^2 \left (b^2-x^2\right )^2}{x^2 (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^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a+\frac{b^4}{a x^2}-\frac{b^4}{a^2 x}+x+\frac{\left (a^2-b^2\right )^2}{a^2 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac{\csc (c+d x)}{a d}-\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^2 b^3 d}-\frac{a \sin (c+d x)}{b^2 d}+\frac{\sin ^2(c+d x)}{2 b d}\\ \end{align*}
Mathematica [A] time = 0.192026, size = 86, normalized size = 0.9 \[ \frac{\frac{2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^2 b^3}-\frac{2 b \log (\sin (c+d x))}{a^2}-\frac{2 a \sin (c+d x)}{b^2}-\frac{2 \csc (c+d x)}{a}+\frac{\sin ^2(c+d x)}{b}}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 124, normalized size = 1.3 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,bd}}-{\frac{a\sin \left ( dx+c \right ) }{{b}^{2}d}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{3}}}-2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{bd}}+{\frac{b\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{1}{da\sin \left ( dx+c \right ) }}-{\frac{b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982893, size = 123, normalized size = 1.28 \begin{align*} -\frac{\frac{2 \, b \log \left (\sin \left (d x + c\right )\right )}{a^{2}} - \frac{b \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )}{b^{2}} + \frac{2}{a \sin \left (d x + c\right )} - \frac{2 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2} b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16953, size = 317, normalized size = 3.3 \begin{align*} \frac{4 \, a^{3} b \cos \left (d x + c\right )^{2} - 4 \, b^{4} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 4 \, a^{3} b - 4 \, a b^{3} + 4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) -{\left (2 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} - a^{2} b^{2}\right )} \sin \left (d x + c\right )}{4 \, a^{2} b^{3} d \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 = 2.19449, size = 142, normalized size = 1.48 \begin{align*} -\frac{\frac{2 \, b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac{b \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )}{b^{2}} - \frac{2 \,{\left (b \sin \left (d x + c\right ) - a\right )}}{a^{2} \sin \left (d x + c\right )} - \frac{2 \,{\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^{2} b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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