Optimal. Leaf size=83 \[ \frac{a \sin ^3(c+d x)}{3 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{b \sin ^4(c+d x)}{4 d}-\frac{b \sin ^2(c+d x)}{d}+\frac{b \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0926954, antiderivative size = 83, 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, 766} \[ \frac{a \sin ^3(c+d x)}{3 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{b \sin ^4(c+d x)}{4 d}-\frac{b \sin ^2(c+d x)}{d}+\frac{b \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 766
Rubi steps
\begin{align*} \int \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 (a+x) \left (b^2-x^2\right )^2}{x^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+x) \left (b^2-x^2\right )^2}{x^2} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 a b^2+\frac{a b^4}{x^2}+\frac{b^4}{x}-2 b^2 x+a x^2+x^3\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac{a \csc (c+d x)}{d}+\frac{b \log (\sin (c+d x))}{d}-\frac{2 a \sin (c+d x)}{d}-\frac{b \sin ^2(c+d x)}{d}+\frac{a \sin ^3(c+d x)}{3 d}+\frac{b \sin ^4(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0301926, size = 83, normalized size = 1. \[ \frac{a \sin ^3(c+d x)}{3 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{b \sin ^4(c+d x)}{4 d}-\frac{b \sin ^2(c+d x)}{d}+\frac{b \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 116, normalized size = 1.4 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}-{\frac{8\,a\sin \left ( dx+c \right ) }{3\,d}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{d}}-{\frac{4\,a\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971597, size = 93, normalized size = 1.12 \begin{align*} \frac{3 \, b \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 12 \, b \sin \left (d x + c\right )^{2} + 12 \, b \log \left (\sin \left (d x + c\right )\right ) - 24 \, a \sin \left (d x + c\right ) - \frac{12 \, a}{\sin \left (d x + c\right )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75939, size = 250, normalized size = 3.01 \begin{align*} \frac{32 \, a \cos \left (d x + c\right )^{4} + 128 \, a \cos \left (d x + c\right )^{2} + 96 \, b \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 3 \,{\left (8 \, b \cos \left (d x + c\right )^{4} + 16 \, b \cos \left (d x + c\right )^{2} - 11 \, b\right )} \sin \left (d x + c\right ) - 256 \, a}{96 \, 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 = 1.29756, size = 107, normalized size = 1.29 \begin{align*} \frac{3 \, b \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 12 \, b \sin \left (d x + c\right )^{2} + 12 \, b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 24 \, a \sin \left (d x + c\right ) - \frac{12 \,{\left (b \sin \left (d x + c\right ) + a\right )}}{\sin \left (d x + c\right )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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