Optimal. Leaf size=109 \[ -\frac{a \left (2 a^2+9 b^2\right ) \cot (e+f x)}{3 f}-\frac{b \left (3 a^2+2 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f} \]
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Rubi [A] time = 0.181392, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2792, 3021, 2748, 3767, 8, 3770} \[ -\frac{a \left (2 a^2+9 b^2\right ) \cot (e+f x)}{3 f}-\frac{b \left (3 a^2+2 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3021
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx &=-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}+\frac{1}{3} \int \csc ^3(e+f x) \left (7 a^2 b+a \left (2 a^2+9 b^2\right ) \sin (e+f x)+b \left (a^2+3 b^2\right ) \sin ^2(e+f x)\right ) \, dx\\ &=-\frac{7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}+\frac{1}{6} \int \csc ^2(e+f x) \left (2 a \left (2 a^2+9 b^2\right )+3 b \left (3 a^2+2 b^2\right ) \sin (e+f x)\right ) \, dx\\ &=-\frac{7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}+\frac{1}{2} \left (b \left (3 a^2+2 b^2\right )\right ) \int \csc (e+f x) \, dx+\frac{1}{3} \left (a \left (2 a^2+9 b^2\right )\right ) \int \csc ^2(e+f x) \, dx\\ &=-\frac{b \left (3 a^2+2 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}-\frac{\left (a \left (2 a^2+9 b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (e+f x))}{3 f}\\ &=-\frac{b \left (3 a^2+2 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a \left (2 a^2+9 b^2\right ) \cot (e+f x)}{3 f}-\frac{7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}\\ \end{align*}
Mathematica [B] time = 6.183, size = 525, normalized size = 4.82 \[ \frac{\sin ^3(e+f x) \csc \left (\frac{1}{2} (e+f x)\right ) \left (-2 a^3 \cos \left (\frac{1}{2} (e+f x)\right )-9 a b^2 \cos \left (\frac{1}{2} (e+f x)\right )\right ) (a \csc (e+f x)+b)^3}{6 f (a+b \sin (e+f x))^3}+\frac{\left (3 a^2 b+2 b^3\right ) \sin ^3(e+f x) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right ) (a \csc (e+f x)+b)^3}{2 f (a+b \sin (e+f x))^3}+\frac{\sin ^3(e+f x) \sec \left (\frac{1}{2} (e+f x)\right ) \left (2 a^3 \sin \left (\frac{1}{2} (e+f x)\right )+9 a b^2 \sin \left (\frac{1}{2} (e+f x)\right )\right ) (a \csc (e+f x)+b)^3}{6 f (a+b \sin (e+f x))^3}+\frac{\left (-3 a^2 b-2 b^3\right ) \sin ^3(e+f x) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right ) (a \csc (e+f x)+b)^3}{2 f (a+b \sin (e+f x))^3}-\frac{3 a^2 b \sin ^3(e+f x) \csc ^2\left (\frac{1}{2} (e+f x)\right ) (a \csc (e+f x)+b)^3}{8 f (a+b \sin (e+f x))^3}-\frac{a^3 \sin ^3(e+f x) \cot \left (\frac{1}{2} (e+f x)\right ) \csc ^2\left (\frac{1}{2} (e+f x)\right ) (a \csc (e+f x)+b)^3}{24 f (a+b \sin (e+f x))^3}+\frac{3 a^2 b \sin ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a \csc (e+f x)+b)^3}{8 f (a+b \sin (e+f x))^3}+\frac{a^3 \sin ^3(e+f x) \tan \left (\frac{1}{2} (e+f x)\right ) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a \csc (e+f x)+b)^3}{24 f (a+b \sin (e+f x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 122, normalized size = 1.1 \begin{align*} -{\frac{2\,{a}^{3}\cot \left ( fx+e \right ) }{3\,f}}-{\frac{{a}^{3}\cot \left ( fx+e \right ) \left ( \csc \left ( fx+e \right ) \right ) ^{2}}{3\,f}}-{\frac{3\,{a}^{2}b\cot \left ( fx+e \right ) \csc \left ( fx+e \right ) }{2\,f}}+{\frac{3\,{a}^{2}b\ln \left ( \csc \left ( fx+e \right ) -\cot \left ( fx+e \right ) \right ) }{2\,f}}-3\,{\frac{a{b}^{2}\cot \left ( fx+e \right ) }{f}}+{\frac{{b}^{3}\ln \left ( \csc \left ( fx+e \right ) -\cot \left ( fx+e \right ) \right ) }{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.64138, size = 159, normalized size = 1.46 \begin{align*} \frac{9 \, a^{2} b{\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 \, b^{3}{\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac{36 \, a b^{2}}{\tan \left (f x + e\right )} - \frac{4 \,{\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{3}}{\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 [A] time = 1.72114, size = 471, normalized size = 4.32 \begin{align*} \frac{18 \, a^{2} b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 4 \,{\left (2 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (3 \, a^{2} b + 2 \, b^{3} -{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right ) \sin \left (f x + e\right ) - 3 \,{\left (3 \, a^{2} b + 2 \, b^{3} -{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right ) \sin \left (f x + e\right ) + 12 \,{\left (a^{3} + 3 \, a b^{2}\right )} \cos \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 [A] time = 2.11076, size = 271, normalized size = 2.49 \begin{align*} \frac{a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 9 \, a^{2} b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 9 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 36 \, a b^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 12 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) \right |}\right ) - \frac{66 \, a^{2} b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 44 \, b^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 9 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 36 \, a b^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 9 \, a^{2} b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + a^{3}}{\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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