Optimal. Leaf size=107 \[ \frac{\left (a^2-2 b^2\right ) \sin (c+d x)}{b^3 d}-\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a b^4 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\log (\sin (c+d x))}{a d}+\frac{\sin ^3(c+d x)}{3 b d} \]
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Rubi [A] time = 0.14031, antiderivative size = 107, 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} \[ \frac{\left (a^2-2 b^2\right ) \sin (c+d x)}{b^3 d}-\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a b^4 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\log (\sin (c+d x))}{a d}+\frac{\sin ^3(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b \left (b^2-x^2\right )^2}{x (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 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 \left (1-\frac{2 b^2}{a^2}\right )+\frac{b^4}{a x}-a x+x^2-\frac{\left (a^2-b^2\right )^2}{a (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac{\log (\sin (c+d x))}{a d}-\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a b^4 d}+\frac{\left (a^2-2 b^2\right ) \sin (c+d x)}{b^3 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\sin ^3(c+d x)}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.139627, size = 101, normalized size = 0.94 \[ \frac{-3 a^2 b^2 \sin ^2(c+d x)+6 a b \left (a^2-2 b^2\right ) \sin (c+d x)+6 \left (b^4 \log (\sin (c+d x))-\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))\right )+2 a b^3 \sin ^3(c+d x)}{6 a b^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 140, normalized size = 1.3 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,bd}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{2\,{b}^{2}d}}+{\frac{{a}^{2}\sin \left ( dx+c \right ) }{{b}^{3}d}}-2\,{\frac{\sin \left ( dx+c \right ) }{bd}}-{\frac{{a}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{4}d}}+2\,{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{2}d}}-{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{da}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989278, size = 134, normalized size = 1.25 \begin{align*} \frac{\frac{6 \, \log \left (\sin \left (d x + c\right )\right )}{a} + \frac{2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \,{\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )}{b^{3}} - \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 b^{4}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75831, size = 250, normalized size = 2.34 \begin{align*} \frac{3 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} + 6 \, b^{4} \log \left (-\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 6 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \,{\left (a b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )}{6 \, a b^{4} d} \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 = 1.19394, size = 143, normalized size = 1.34 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac{2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right ) - 12 \, b^{2} \sin \left (d x + c\right )}{b^{3}} - \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 b^{4}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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