Optimal. Leaf size=128 \[ \frac {2 i x \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 i x \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 \text {Li}_3\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {2 \text {Li}_3\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 x^2 \sec (x) \tanh ^{-1}\left (e^{i x}\right )}{\sqrt {a \sec ^2(x)}} \]
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Rubi [A] time = 0.59, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6720, 4183, 2531, 2282, 6589} \[ \frac {2 i x \sec (x) \text {PolyLog}\left (2,-e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {2 i x \sec (x) \text {PolyLog}\left (2,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {2 \sec (x) \text {PolyLog}\left (3,-e^{i x}\right )}{\sqrt {a \sec ^2(x)}}+\frac {2 \sec (x) \text {PolyLog}\left (3,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {2 x^2 \sec (x) \tanh ^{-1}\left (e^{i x}\right )}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4183
Rule 6589
Rule 6720
Rubi steps
\begin {align*} \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx &=\frac {\sec (x) \int x^2 \csc (x) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {(2 \sec (x)) \int x \log \left (1-e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}+\frac {(2 \sec (x)) \int x \log \left (1+e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {2 i x \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 i x \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {(2 i \sec (x)) \int \text {Li}_2\left (-e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}+\frac {(2 i \sec (x)) \int \text {Li}_2\left (e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {2 i x \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 i x \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {(2 \sec (x)) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}+\frac {(2 \sec (x)) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {2 i x \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 i x \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 \text {Li}_3\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {2 \text {Li}_3\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 99, normalized size = 0.77 \[ \frac {\sec (x) \left (2 i x \text {Li}_2\left (-e^{i x}\right )-2 i x \text {Li}_2\left (e^{i x}\right )-2 \text {Li}_3\left (-e^{i x}\right )+2 \text {Li}_3\left (e^{i x}\right )+x^2 \log \left (1-e^{i x}\right )-x^2 \log \left (1+e^{i x}\right )\right )}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.62, size = 227, normalized size = 1.77 \[ \frac {2 \, \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (3, \cos \relax (x) + i \, \sin \relax (x)\right ) + 2 \, \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (3, \cos \relax (x) - i \, \sin \relax (x)\right ) - 2 \, \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (3, -\cos \relax (x) + i \, \sin \relax (x)\right ) - 2 \, \sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) {\rm polylog}\left (3, -\cos \relax (x) - i \, \sin \relax (x)\right ) - {\left (x^{2} \cos \relax (x) \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + x^{2} \cos \relax (x) \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - x^{2} \cos \relax (x) \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) - x^{2} \cos \relax (x) \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + 2 i \, x \cos \relax (x) {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) - 2 i \, x \cos \relax (x) {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) + 2 i \, x \cos \relax (x) {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) - 2 i \, x \cos \relax (x) {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right )\right )} \sqrt {\frac {a}{\cos \relax (x)^{2}}}}{2 \, a} \]
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 \[ \int \frac {x^{2} \csc \relax (x) \sec \relax (x)}{\sqrt {a \sec \relax (x)^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 132, normalized size = 1.03 \[ -\frac {2 \left (\frac {{\mathrm e}^{i x} x^{2} \ln \left (1+{\mathrm e}^{i x}\right )}{2}-i {\mathrm e}^{i x} x \polylog \left (2, -{\mathrm e}^{i x}\right )+{\mathrm e}^{i x} \polylog \left (3, -{\mathrm e}^{i x}\right )-\frac {{\mathrm e}^{i x} x^{2} \ln \left (1-{\mathrm e}^{i x}\right )}{2}+i {\mathrm e}^{i x} x \polylog \left (2, {\mathrm e}^{i x}\right )-{\mathrm e}^{i x} \polylog \left (3, {\mathrm e}^{i x}\right )\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 107, normalized size = 0.84 \[ -\frac {2 i \, x^{2} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) + 2 i \, x^{2} \arctan \left (\sin \relax (x), -\cos \relax (x) + 1\right ) + x^{2} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) - x^{2} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - 4 i \, x {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + 4 i \, x {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + 4 \, {\rm Li}_{3}(-e^{\left (i \, x\right )}) - 4 \, {\rm Li}_{3}(e^{\left (i \, x\right )})}{2 \, \sqrt {a}} \]
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 \[ \int \frac {x^2}{\cos \relax (x)\,\sin \relax (x)\,\sqrt {\frac {a}{{\cos \relax (x)}^2}}} \,d x \]
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 \[ \int \frac {x^{2} \csc {\relax (x )} \sec {\relax (x )}}{\sqrt {a \sec ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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