Optimal. Leaf size=143 \[ -\frac {3 i x^2 \text {Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {3 x \text {Li}_3\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {3 i \text {Li}_4\left (e^{2 i x}\right ) \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}-\frac {i x^4 \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}+\frac {x^3 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}} \]
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Rubi [A] time = 0.61, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6720, 3717, 2190, 2531, 6609, 2282, 6589} \[ -\frac {3 i x^2 \sec ^2(x) \text {PolyLog}\left (2,e^{2 i x}\right )}{2 \sqrt {a \sec ^4(x)}}+\frac {3 x \sec ^2(x) \text {PolyLog}\left (3,e^{2 i x}\right )}{2 \sqrt {a \sec ^4(x)}}+\frac {3 i \sec ^2(x) \text {PolyLog}\left (4,e^{2 i x}\right )}{4 \sqrt {a \sec ^4(x)}}-\frac {i x^4 \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}+\frac {x^3 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 6589
Rule 6609
Rule 6720
Rubi steps
\begin {align*} \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx &=\frac {\sec ^2(x) \int x^3 \cot (x) \, dx}{\sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^4 \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}-\frac {\left (2 i \sec ^2(x)\right ) \int \frac {e^{2 i x} x^3}{1-e^{2 i x}} \, dx}{\sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^4 \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}+\frac {x^3 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {\left (3 \sec ^2(x)\right ) \int x^2 \log \left (1-e^{2 i x}\right ) \, dx}{\sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^4 \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}+\frac {x^3 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {3 i x^2 \text {Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {\left (3 i \sec ^2(x)\right ) \int x \text {Li}_2\left (e^{2 i x}\right ) \, dx}{\sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^4 \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}+\frac {x^3 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {3 i x^2 \text {Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {3 x \text {Li}_3\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}-\frac {\left (3 \sec ^2(x)\right ) \int \text {Li}_3\left (e^{2 i x}\right ) \, dx}{2 \sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^4 \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}+\frac {x^3 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {3 i x^2 \text {Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {3 x \text {Li}_3\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {\left (3 i \sec ^2(x)\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i x}\right )}{4 \sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^4 \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}+\frac {x^3 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {3 i x^2 \text {Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {3 x \text {Li}_3\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {3 i \text {Li}_4\left (e^{2 i x}\right ) \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 87, normalized size = 0.61 \[ -\frac {i \sec ^2(x) \left (-96 x^2 \text {Li}_2\left (e^{-2 i x}\right )+96 i x \text {Li}_3\left (e^{-2 i x}\right )+48 \text {Li}_4\left (e^{-2 i x}\right )-16 x^4+64 i x^3 \log \left (1-e^{-2 i x}\right )+\pi ^4\right )}{64 \sqrt {a \sec ^4(x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 2.05, size = 356, normalized size = 2.49 \[ \frac {6 \, x \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (3, \cos \relax (x) + i \, \sin \relax (x)\right ) + 6 \, x \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (3, \cos \relax (x) - i \, \sin \relax (x)\right ) + 6 \, x \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (3, -\cos \relax (x) + i \, \sin \relax (x)\right ) + 6 \, x \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (3, -\cos \relax (x) - i \, \sin \relax (x)\right ) + 6 i \, \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (4, \cos \relax (x) + i \, \sin \relax (x)\right ) - 6 i \, \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (4, \cos \relax (x) - i \, \sin \relax (x)\right ) - 6 i \, \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (4, -\cos \relax (x) + i \, \sin \relax (x)\right ) + 6 i \, \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (4, -\cos \relax (x) - i \, \sin \relax (x)\right ) + {\left (x^{3} \cos \relax (x)^{2} \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + x^{3} \cos \relax (x)^{2} \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + x^{3} \cos \relax (x)^{2} \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + x^{3} \cos \relax (x)^{2} \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - 3 i \, x^{2} \cos \relax (x)^{2} {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) + 3 i \, x^{2} \cos \relax (x)^{2} {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) + 3 i \, x^{2} \cos \relax (x)^{2} {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) - 3 i \, x^{2} \cos \relax (x)^{2} {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right )\right )} \sqrt {\frac {a}{\cos \relax (x)^{4}}}}{2 \, a} \]
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 \[ \int \frac {x^{3} \csc \relax (x) \sec \relax (x)}{\sqrt {a \sec \relax (x)^{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 221, normalized size = 1.55 \[ \frac {i {\mathrm e}^{2 i x} x^{4}}{4 \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {2 i \left (-\frac {{\mathrm e}^{2 i x} x^{4}}{4}-\frac {i {\mathrm e}^{2 i x} x^{3} \ln \left (1+{\mathrm e}^{i x}\right )}{2}-\frac {3 \,{\mathrm e}^{2 i x} x^{2} \polylog \left (2, -{\mathrm e}^{i x}\right )}{2}-3 i {\mathrm e}^{2 i x} x \polylog \left (3, -{\mathrm e}^{i x}\right )+3 \,{\mathrm e}^{2 i x} \polylog \left (4, -{\mathrm e}^{i x}\right )-\frac {i {\mathrm e}^{2 i x} x^{3} \ln \left (1-{\mathrm e}^{i x}\right )}{2}-\frac {3 \,{\mathrm e}^{2 i x} x^{2} \polylog \left (2, {\mathrm e}^{i x}\right )}{2}-3 i {\mathrm e}^{2 i x} x \polylog \left (3, {\mathrm e}^{i x}\right )+3 \,{\mathrm e}^{2 i x} \polylog \left (4, {\mathrm e}^{i x}\right )\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 137, normalized size = 0.96 \[ \frac {-i \, x^{4} + 4 i \, x^{3} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - 4 i \, x^{3} \arctan \left (\sin \relax (x), -\cos \relax (x) + 1\right ) + 2 \, x^{3} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + 2 \, x^{3} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - 12 i \, x^{2} {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - 12 i \, x^{2} {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + 24 \, x {\rm Li}_{3}(-e^{\left (i \, x\right )}) + 24 \, x {\rm Li}_{3}(e^{\left (i \, x\right )}) + 24 i \, {\rm Li}_{4}(-e^{\left (i \, x\right )}) + 24 i \, {\rm Li}_{4}(e^{\left (i \, x\right )})}{4 \, \sqrt {a}} \]
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 \[ \int \frac {x^3}{\cos \relax (x)\,\sin \relax (x)\,\sqrt {\frac {a}{{\cos \relax (x)}^4}}} \,d x \]
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 \[ \int \frac {x^{3} \csc {\relax (x )} \sec {\relax (x )}}{\sqrt {a \sec ^{4}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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