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 = 0.18, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 8
Rule 2748
Rule 2792
Rule 3021
Rule 3767
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*}
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Mathematica [B] time = 6.20, size = 525, normalized size = 4.82 \[ \frac {\sin ^3(e+f x) \csc \left (\frac {1}{2} (e+f x)\right ) \left (-2 a^3 \cos \left (\frac {1}{2} (e+f x)\right )-9 a b^2 \cos \left (\frac {1}{2} (e+f x)\right )\right ) (a \csc (e+f x)+b)^3}{6 f (a+b \sin (e+f x))^3}+\frac {\sin ^3(e+f x) \sec \left (\frac {1}{2} (e+f x)\right ) \left (2 a^3 \sin \left (\frac {1}{2} (e+f x)\right )+9 a b^2 \sin \left (\frac {1}{2} (e+f x)\right )\right ) (a \csc (e+f x)+b)^3}{6 f (a+b \sin (e+f x))^3}-\frac {a^3 \sin ^3(e+f x) \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right ) (a \csc (e+f x)+b)^3}{24 f (a+b \sin (e+f x))^3}+\frac {a^3 \sin ^3(e+f x) \tan \left (\frac {1}{2} (e+f x)\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a \csc (e+f x)+b)^3}{24 f (a+b \sin (e+f x))^3}+\frac {\left (3 a^2 b+2 b^3\right ) \sin ^3(e+f x) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a \csc (e+f x)+b)^3}{2 f (a+b \sin (e+f x))^3}+\frac {\left (-3 a^2 b-2 b^3\right ) \sin ^3(e+f x) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right ) (a \csc (e+f x)+b)^3}{2 f (a+b \sin (e+f x))^3}-\frac {3 a^2 b \sin ^3(e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right ) (a \csc (e+f x)+b)^3}{8 f (a+b \sin (e+f x))^3}+\frac {3 a^2 b \sin ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a \csc (e+f x)+b)^3}{8 f (a+b \sin (e+f x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 191, normalized size = 1.75 \[ \frac {18 \, a^{2} b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 4 \, {\left (2 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3} - {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 3 \, {\left (3 \, a^{2} b + 2 \, b^{3} - {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) + 12 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (f x + e\right )}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 122, normalized size = 1.12 \[ -\frac {2 a^{3} \cot \left (f x +e \right )}{3 f}-\frac {a^{3} \cot \left (f x +e \right ) \left (\csc ^{2}\left (f x +e \right )\right )}{3 f}-\frac {3 a^{2} b \cot \left (f x +e \right ) \csc \left (f x +e \right )}{2 f}+\frac {3 a^{2} b \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2 f}-\frac {3 a \,b^{2} \cot \left (f x +e \right )}{f}+\frac {b^{3} \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 118, normalized size = 1.08 \[ \frac {9 \, a^{2} b {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - 6 \, b^{3} {\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {36 \, a b^{2}}{\tan \left (f x + e\right )} - \frac {4 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{3}}{\tan \left (f x + e\right )^{3}}}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.78, size = 150, normalized size = 1.38 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (\frac {3\,a^2\,b}{2}+b^3\right )}{f}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24\,f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (3\,a^3+12\,a\,b^2\right )+\frac {a^3}{3}+3\,a^2\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,a^3}{8}+\frac {3\,a\,b^2}{2}\right )}{f}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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