Optimal. Leaf size=186 \[ \frac {3 i x^2 \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 x \text {Li}_3\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 x \text {Li}_3\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 i \text {Li}_4\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 i \text {Li}_4\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 x^3 \sec (x) \tanh ^{-1}\left (e^{i x}\right )}{\sqrt {a \sec ^2(x)}} \]
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Rubi [A] time = 0.57, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6720, 4183, 2531, 6609, 2282, 6589} \[ \frac {3 i x^2 \sec (x) \text {PolyLog}\left (2,-e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \sec (x) \text {PolyLog}\left (2,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {6 x \sec (x) \text {PolyLog}\left (3,-e^{i x}\right )}{\sqrt {a \sec ^2(x)}}+\frac {6 x \sec (x) \text {PolyLog}\left (3,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {6 i \sec (x) \text {PolyLog}\left (4,-e^{i x}\right )}{\sqrt {a \sec ^2(x)}}+\frac {6 i \sec (x) \text {PolyLog}\left (4,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {2 x^3 \sec (x) \tanh ^{-1}\left (e^{i x}\right )}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4183
Rule 6589
Rule 6609
Rule 6720
Rubi steps
\begin {align*} \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx &=\frac {\sec (x) \int x^3 \csc (x) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {(3 \sec (x)) \int x^2 \log \left (1-e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}+\frac {(3 \sec (x)) \int x^2 \log \left (1+e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {(6 i \sec (x)) \int x \text {Li}_2\left (-e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}+\frac {(6 i \sec (x)) \int x \text {Li}_2\left (e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 x \text {Li}_3\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 x \text {Li}_3\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {(6 \sec (x)) \int \text {Li}_3\left (-e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}-\frac {(6 \sec (x)) \int \text {Li}_3\left (e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 x \text {Li}_3\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 x \text {Li}_3\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {(6 i \sec (x)) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}+\frac {(6 i \sec (x)) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 x \text {Li}_3\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 x \text {Li}_3\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 i \text {Li}_4\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 i \text {Li}_4\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 147, normalized size = 0.79 \[ -\frac {i \sec (x) \left (-24 x^2 \text {Li}_2\left (e^{-i x}\right )-24 x^2 \text {Li}_2\left (-e^{i x}\right )+48 i x \text {Li}_3\left (e^{-i x}\right )-48 i x \text {Li}_3\left (-e^{i x}\right )+48 \text {Li}_4\left (e^{-i x}\right )+48 \text {Li}_4\left (-e^{i x}\right )-2 x^4+8 i x^3 \log \left (1-e^{-i x}\right )-8 i x^3 \log \left (1+e^{i x}\right )+\pi ^4\right )}{8 \sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.31, size = 327, normalized size = 1.76 \[ \frac {6 \, x \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (3, \cos \relax (x) + i \, \sin \relax (x)\right ) + 6 \, x \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (3, \cos \relax (x) - i \, \sin \relax (x)\right ) - 6 \, x \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (3, -\cos \relax (x) + i \, \sin \relax (x)\right ) - 6 \, x \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (3, -\cos \relax (x) - i \, \sin \relax (x)\right ) + 6 i \, \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (4, \cos \relax (x) + i \, \sin \relax (x)\right ) - 6 i \, \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (4, \cos \relax (x) - i \, \sin \relax (x)\right ) + 6 i \, \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (4, -\cos \relax (x) + i \, \sin \relax (x)\right ) - 6 i \, \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (4, -\cos \relax (x) - i \, \sin \relax (x)\right ) - {\left (x^{3} \cos \relax (x) \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + x^{3} \cos \relax (x) \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - x^{3} \cos \relax (x) \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) - x^{3} \cos \relax (x) \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + 3 i \, x^{2} \cos \relax (x) {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) - 3 i \, x^{2} \cos \relax (x) {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) + 3 i \, x^{2} \cos \relax (x) {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) - 3 i \, x^{2} \cos \relax (x) {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right )\right )} \sqrt {\frac {a}{\cos \relax (x)^{2}}}}{2 \, a} \]
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 \[ \int \frac {x^{3} \csc \relax (x) \sec \relax (x)}{\sqrt {a \sec \relax (x)^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 172, normalized size = 0.92 \[ \frac {2 i \left (\frac {i {\mathrm e}^{i x} x^{3} \ln \left (1+{\mathrm e}^{i x}\right )}{2}+\frac {3 \,{\mathrm e}^{i x} x^{2} \polylog \left (2, -{\mathrm e}^{i x}\right )}{2}+3 i {\mathrm e}^{i x} x \polylog \left (3, -{\mathrm e}^{i x}\right )-3 \,{\mathrm e}^{i x} \polylog \left (4, -{\mathrm e}^{i x}\right )-\frac {i {\mathrm e}^{i x} x^{3} \ln \left (1-{\mathrm e}^{i x}\right )}{2}-\frac {3 \,{\mathrm e}^{i x} x^{2} \polylog \left (2, {\mathrm e}^{i x}\right )}{2}-3 i {\mathrm e}^{i x} x \polylog \left (3, {\mathrm e}^{i x}\right )+3 \,{\mathrm e}^{i x} \polylog \left (4, {\mathrm e}^{i x}\right )\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 131, normalized size = 0.70 \[ -\frac {2 i \, x^{3} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) + 2 i \, x^{3} \arctan \left (\sin \relax (x), -\cos \relax (x) + 1\right ) + x^{3} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) - x^{3} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - 6 i \, x^{2} {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + 6 i \, x^{2} {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + 12 \, x {\rm Li}_{3}(-e^{\left (i \, x\right )}) - 12 \, x {\rm Li}_{3}(e^{\left (i \, x\right )}) + 12 i \, {\rm Li}_{4}(-e^{\left (i \, x\right )}) - 12 i \, {\rm Li}_{4}(e^{\left (i \, x\right )})}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{\cos \relax (x)\,\sin \relax (x)\,\sqrt {\frac {a}{{\cos \relax (x)}^2}}} \,d x \]
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 \[ \int \frac {x^{3} \csc {\relax (x )} \sec {\relax (x )}}{\sqrt {a \sec ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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