Optimal. Leaf size=356 \[ \frac {3}{2} i x^2 \text {Li}_2\left (-e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-\frac {3}{2} i x^2 \text {Li}_2\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-\frac {3}{2} x \text {Li}_3\left (-e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {3}{2} x \text {Li}_3\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {3}{2} i \text {Li}_2\left (-e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-\frac {3}{4} i \text {Li}_4\left (-e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {3}{4} i \text {Li}_4\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \sin ^2(x) \sqrt {a \sec ^4(x)}-2 x^3 \cos ^2(x) \tanh ^{-1}\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} i x^2 \cos ^2(x) \sqrt {a \sec ^4(x)}-\frac {3}{2} x^2 \sin (x) \cos (x) \sqrt {a \sec ^4(x)}-3 x \log \left (1+e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)} \]
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Rubi [A] time = 0.64, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 17, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.944, Rules used = {6720, 2620, 14, 4420, 2551, 4419, 4183, 2531, 6609, 2282, 6589, 3720, 3719, 2190, 2279, 2391, 30} \[ \frac {3}{2} i x^2 \cos ^2(x) \text {PolyLog}\left (2,-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} i x^2 \cos ^2(x) \text {PolyLog}\left (2,e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x \cos ^2(x) \text {PolyLog}\left (3,-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} x \cos ^2(x) \text {PolyLog}\left (3,e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} i \cos ^2(x) \text {PolyLog}\left (2,-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{4} i \cos ^2(x) \text {PolyLog}\left (4,-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{4} i \cos ^2(x) \text {PolyLog}\left (4,e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {3}{2} i x^2 \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \sin ^2(x) \sqrt {a \sec ^4(x)}-2 x^3 \cos ^2(x) \tanh ^{-1}\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x^2 \sin (x) \cos (x) \sqrt {a \sec ^4(x)}-3 x \log \left (1+e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 2551
Rule 2620
Rule 3719
Rule 3720
Rule 4183
Rule 4419
Rule 4420
Rule 6589
Rule 6609
Rule 6720
Rubi steps
\begin {align*} \int x^3 \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx &=\left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x^3 \csc (x) \sec ^3(x) \, dx\\ &=x^3 \cos ^2(x) \log (\tan (x)) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)-\left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x^2 \left (\log (\tan (x))+\frac {\tan ^2(x)}{2}\right ) \, dx\\ &=x^3 \cos ^2(x) \log (\tan (x)) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)-\left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \left (x^2 \log (\tan (x))+\frac {1}{2} x^2 \tan ^2(x)\right ) \, dx\\ &=x^3 \cos ^2(x) \log (\tan (x)) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)-\frac {1}{2} \left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x^2 \tan ^2(x) \, dx-\left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x^2 \log (\tan (x)) \, dx\\ &=-\frac {3}{2} x^2 \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)+\left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x^3 \csc (x) \sec (x) \, dx+\frac {1}{2} \left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x^2 \, dx+\left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x \tan (x) \, dx\\ &=\frac {3}{2} i x^2 \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \cos ^2(x) \sqrt {a \sec ^4(x)}-\frac {3}{2} x^2 \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)-\left (6 i \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx+\left (2 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x^3 \csc (2 x) \, dx\\ &=\frac {3}{2} i x^2 \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \cos ^2(x) \sqrt {a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x^2 \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)-\left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x^2 \log \left (1-e^{2 i x}\right ) \, dx+\left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \log \left (1+e^{2 i x}\right ) \, dx+\left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x^2 \log \left (1+e^{2 i x}\right ) \, dx\\ &=\frac {3}{2} i x^2 \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \cos ^2(x) \sqrt {a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} i x^2 \cos ^2(x) \text {Li}_2\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} i x^2 \cos ^2(x) \text {Li}_2\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x^2 \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)-\frac {1}{2} \left (3 i \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i x}\right )-\left (3 i \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x \text {Li}_2\left (-e^{2 i x}\right ) \, dx+\left (3 i \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x \text {Li}_2\left (e^{2 i x}\right ) \, dx\\ &=\frac {3}{2} i x^2 \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \cos ^2(x) \sqrt {a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} i \cos ^2(x) \text {Li}_2\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} i x^2 \cos ^2(x) \text {Li}_2\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} i x^2 \cos ^2(x) \text {Li}_2\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x \cos ^2(x) \text {Li}_3\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} x \cos ^2(x) \text {Li}_3\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x^2 \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)+\frac {1}{2} \left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \text {Li}_3\left (-e^{2 i x}\right ) \, dx-\frac {1}{2} \left (3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \text {Li}_3\left (e^{2 i x}\right ) \, dx\\ &=\frac {3}{2} i x^2 \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \cos ^2(x) \sqrt {a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} i \cos ^2(x) \text {Li}_2\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} i x^2 \cos ^2(x) \text {Li}_2\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} i x^2 \cos ^2(x) \text {Li}_2\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x \cos ^2(x) \text {Li}_3\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} x \cos ^2(x) \text {Li}_3\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x^2 \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)-\frac {1}{4} \left (3 i \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 i x}\right )+\frac {1}{4} \left (3 i \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\frac {3}{2} i x^2 \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x^3 \cos ^2(x) \sqrt {a \sec ^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-3 x \cos ^2(x) \log \left (1+e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} i \cos ^2(x) \text {Li}_2\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} i x^2 \cos ^2(x) \text {Li}_2\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} i x^2 \cos ^2(x) \text {Li}_2\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x \cos ^2(x) \text {Li}_3\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{2} x \cos ^2(x) \text {Li}_3\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{4} i \cos ^2(x) \text {Li}_4\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {3}{4} i \cos ^2(x) \text {Li}_4\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {3}{2} x^2 \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x^3 \sqrt {a \sec ^4(x)} \sin ^2(x)\\ \end {align*}
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Mathematica [A] time = 1.07, size = 191, normalized size = 0.54 \[ \frac {1}{64} \cos ^2(x) \sqrt {a \sec ^4(x)} \left (96 i x^2 \text {Li}_2\left (e^{-2 i x}\right )+96 i \left (x^2+1\right ) \text {Li}_2\left (-e^{2 i x}\right )+96 x \text {Li}_3\left (e^{-2 i x}\right )-96 x \text {Li}_3\left (-e^{2 i x}\right )-48 i \text {Li}_4\left (e^{-2 i x}\right )-48 i \text {Li}_4\left (-e^{2 i x}\right )+32 i x^4+64 x^3 \log \left (1-e^{-2 i x}\right )-64 x^3 \log \left (1+e^{2 i x}\right )+32 x^3 \sec ^2(x)+96 i x^2-96 x^2 \tan (x)-192 x \log \left (1+e^{2 i x}\right )-i \pi ^4\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 1.08, size = 736, normalized size = 2.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sec \relax (x)^{4}} x^{3} \csc \relax (x) \sec \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 324, normalized size = 0.91 \[ \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, x^{2} \left (2 x -3 i-3 i {\mathrm e}^{-2 i x}\right )-2 i \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2} \left (-\frac {3 \,{\mathrm e}^{-2 i x} x^{2}}{2}-\frac {3 i {\mathrm e}^{-2 i x} x \ln \left ({\mathrm e}^{2 i x}+1\right )}{2}-\frac {3 \,{\mathrm e}^{-2 i x} \polylog \left (2, -{\mathrm e}^{2 i x}\right )}{4}+\frac {i {\mathrm e}^{-2 i x} x^{3} \ln \left (1+{\mathrm e}^{i x}\right )}{2}+\frac {3 \,{\mathrm e}^{-2 i x} x^{2} \polylog \left (2, -{\mathrm e}^{i x}\right )}{2}+3 i {\mathrm e}^{-2 i x} x \polylog \left (3, -{\mathrm e}^{i x}\right )-3 \,{\mathrm e}^{-2 i x} \polylog \left (4, -{\mathrm e}^{i x}\right )-\frac {i {\mathrm e}^{-2 i x} x^{3} \ln \left ({\mathrm e}^{2 i x}+1\right )}{2}-\frac {3 \,{\mathrm e}^{-2 i x} x^{2} \polylog \left (2, -{\mathrm e}^{2 i x}\right )}{4}-\frac {3 i {\mathrm e}^{-2 i x} x \polylog \left (3, -{\mathrm e}^{2 i x}\right )}{4}+\frac {3 \,{\mathrm e}^{-2 i x} \polylog \left (4, -{\mathrm e}^{2 i x}\right )}{8}+\frac {i {\mathrm e}^{-2 i x} x^{3} \ln \left (1-{\mathrm e}^{i x}\right )}{2}+\frac {3 \,{\mathrm e}^{-2 i x} x^{2} \polylog \left (2, {\mathrm e}^{i x}\right )}{2}+3 i {\mathrm e}^{-2 i x} x \polylog \left (3, {\mathrm e}^{i x}\right )-3 \,{\mathrm e}^{-2 i x} \polylog \left (4, {\mathrm e}^{i x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 870, normalized size = 2.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\sqrt {\frac {a}{{\cos \relax (x)}^4}}}{\cos \relax (x)\,\sin \relax (x)} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {a \sec ^{4}{\relax (x )}} \csc {\relax (x )} \sec {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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