Optimal. Leaf size=76 \[ -\frac{a^3 A \cos ^3(c+d x)}{3 d}+\frac{a^3 A \cos (c+d x)}{d}+\frac{a^3 A \sin (c+d x) \cos (c+d x)}{d}-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{d}+a^3 A x \]
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Rubi [A] time = 0.104153, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2966, 3770, 2635, 8, 2633} \[ -\frac{a^3 A \cos ^3(c+d x)}{3 d}+\frac{a^3 A \cos (c+d x)}{d}+\frac{a^3 A \sin (c+d x) \cos (c+d x)}{d}-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{d}+a^3 A x \]
Antiderivative was successfully verified.
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Rule 2966
Rule 3770
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \csc (c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx &=\int \left (2 a^3 A+a^3 A \csc (c+d x)-2 a^3 A \sin ^2(c+d x)-a^3 A \sin ^3(c+d x)\right ) \, dx\\ &=2 a^3 A x+\left (a^3 A\right ) \int \csc (c+d x) \, dx-\left (a^3 A\right ) \int \sin ^3(c+d x) \, dx-\left (2 a^3 A\right ) \int \sin ^2(c+d x) \, dx\\ &=2 a^3 A x-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 A \cos (c+d x) \sin (c+d x)}{d}-\left (a^3 A\right ) \int 1 \, dx+\frac{\left (a^3 A\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=a^3 A x-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 A \cos (c+d x)}{d}-\frac{a^3 A \cos ^3(c+d x)}{3 d}+\frac{a^3 A \cos (c+d x) \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.153143, size = 74, normalized size = 0.97 \[ \frac{a^3 A \left (9 \cos (c+d x)-\cos (3 (c+d x))+6 \left (\sin (2 (c+d x))+2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2 c+2 d x\right )\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 99, normalized size = 1.3 \begin{align*}{\frac{A\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{3}}{3\,d}}+{\frac{2\,{a}^{3}A\cos \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{3}A\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+{a}^{3}Ax+{\frac{{a}^{3}Ac}{d}}+{\frac{{a}^{3}A\ln \left ( \csc \left ( dx+c \right ) -\cot \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 = 0.969253, size = 115, normalized size = 1.51 \begin{align*} -\frac{2 \,{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} A a^{3} + 3 \,{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 12 \,{\left (d x + c\right )} A a^{3} + 6 \, A a^{3} \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0146, size = 247, normalized size = 3.25 \begin{align*} -\frac{2 \, A a^{3} \cos \left (d x + c\right )^{3} - 6 \, A a^{3} d x - 6 \, A a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, A a^{3} \cos \left (d x + c\right ) + 3 \, A a^{3} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 3 \, A a^{3} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{6 \, 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 = 1.15356, size = 144, normalized size = 1.89 \begin{align*} \frac{3 \,{\left (d x + c\right )} A a^{3} + 3 \, A a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \,{\left (3 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, A a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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