Optimal. Leaf size=61 \[ -\frac{a^2 c \cot ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a^2 c \cot (e+f x) \csc (e+f x)}{2 f} \]
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Rubi [A] time = 0.162082, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2950, 2706, 2607, 30, 2611, 3770} \[ -\frac{a^2 c \cot ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a^2 c \cot (e+f x) \csc (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 2950
Rule 2706
Rule 2607
Rule 30
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx &=\left (a^2 c^2\right ) \int \frac{\cot ^4(e+f x)}{c-c \sin (e+f x)} \, dx\\ &=\left (a^2 c\right ) \int \cot ^2(e+f x) \csc (e+f x) \, dx+\left (a^2 c\right ) \int \cot ^2(e+f x) \csc ^2(e+f x) \, dx\\ &=-\frac{a^2 c \cot (e+f x) \csc (e+f x)}{2 f}-\frac{1}{2} \left (a^2 c\right ) \int \csc (e+f x) \, dx+\frac{\left (a^2 c\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,-\cot (e+f x)\right )}{f}\\ &=\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a^2 c \cot ^3(e+f x)}{3 f}-\frac{a^2 c \cot (e+f x) \csc (e+f x)}{2 f}\\ \end{align*}
Mathematica [B] time = 0.0741181, size = 172, normalized size = 2.82 \[ a^2 c \left (-\frac{\tan \left (\frac{1}{2} (e+f x)\right )}{6 f}+\frac{\cot \left (\frac{1}{2} (e+f x)\right )}{6 f}-\frac{\csc ^2\left (\frac{1}{2} (e+f x)\right )}{8 f}+\frac{\sec ^2\left (\frac{1}{2} (e+f x)\right )}{8 f}-\frac{\log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )}{2 f}+\frac{\log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )}{2 f}-\frac{\cot \left (\frac{1}{2} (e+f x)\right ) \csc ^2\left (\frac{1}{2} (e+f x)\right )}{24 f}+\frac{\tan \left (\frac{1}{2} (e+f x)\right ) \sec ^2\left (\frac{1}{2} (e+f x)\right )}{24 f}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 86, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}c\ln \left ( \csc \left ( fx+e \right ) -\cot \left ( fx+e \right ) \right ) }{2\,f}}+{\frac{{a}^{2}c\cot \left ( fx+e \right ) }{3\,f}}-{\frac{{a}^{2}c\cot \left ( fx+e \right ) \csc \left ( fx+e \right ) }{2\,f}}-{\frac{{a}^{2}c\cot \left ( fx+e \right ) \left ( \csc \left ( fx+e \right ) \right ) ^{2}}{3\,f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.973217, size = 162, normalized size = 2.66 \begin{align*} \frac{3 \, a^{2} c{\left (\frac{2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + 6 \, a^{2} c{\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + \frac{12 \, a^{2} c}{\tan \left (f x + e\right )} - \frac{4 \,{\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2} c}{\tan \left (f x + e\right )^{3}}}{12 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94318, size = 348, normalized size = 5.7 \begin{align*} \frac{4 \, a^{2} c \cos \left (f x + e\right )^{3} + 6 \, a^{2} c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \,{\left (a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right ) \sin \left (f x + e\right ) - 3 \,{\left (a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (-\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right ) \sin \left (f x + e\right )}{12 \,{\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\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 [B] time = 1.31485, size = 198, normalized size = 3.25 \begin{align*} \frac{a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 3 \, a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 12 \, a^{2} c \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) \right |}\right ) - 3 \, a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + \frac{22 \, a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 3 \, a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 3 \, a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - a^{2} c}{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3}}}{24 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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