Optimal. Leaf size=109 \[ -\frac {i x \text {Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}+\frac {\text {Li}_3\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}-\frac {i x^3 \sec ^2(x)}{3 \sqrt {a \sec ^4(x)}}+\frac {x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}} \]
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Rubi [A] time = 0.57, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6720, 3717, 2190, 2531, 2282, 6589} \[ -\frac {i x \sec ^2(x) \text {PolyLog}\left (2,e^{2 i x}\right )}{\sqrt {a \sec ^4(x)}}+\frac {\sec ^2(x) \text {PolyLog}\left (3,e^{2 i x}\right )}{2 \sqrt {a \sec ^4(x)}}-\frac {i x^3 \sec ^2(x)}{3 \sqrt {a \sec ^4(x)}}+\frac {x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 6589
Rule 6720
Rubi steps
\begin {align*} \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx &=\frac {\sec ^2(x) \int x^2 \cot (x) \, dx}{\sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^3 \sec ^2(x)}{3 \sqrt {a \sec ^4(x)}}-\frac {\left (2 i \sec ^2(x)\right ) \int \frac {e^{2 i x} x^2}{1-e^{2 i x}} \, dx}{\sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^3 \sec ^2(x)}{3 \sqrt {a \sec ^4(x)}}+\frac {x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {\left (2 \sec ^2(x)\right ) \int x \log \left (1-e^{2 i x}\right ) \, dx}{\sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^3 \sec ^2(x)}{3 \sqrt {a \sec ^4(x)}}+\frac {x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {i x \text {Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}+\frac {\left (i \sec ^2(x)\right ) \int \text {Li}_2\left (e^{2 i x}\right ) \, dx}{\sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^3 \sec ^2(x)}{3 \sqrt {a \sec ^4(x)}}+\frac {x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {i x \text {Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}+\frac {\sec ^2(x) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i x}\right )}{2 \sqrt {a \sec ^4(x)}}\\ &=-\frac {i x^3 \sec ^2(x)}{3 \sqrt {a \sec ^4(x)}}+\frac {x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {i x \text {Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}+\frac {\text {Li}_3\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 0.69 \[ \frac {\sec ^2(x) \left (24 i x \text {Li}_2\left (e^{-2 i x}\right )+12 \text {Li}_3\left (e^{-2 i x}\right )+8 i x^3+24 x^2 \log \left (1-e^{-2 i x}\right )-i \pi ^3\right )}{24 \sqrt {a \sec ^4(x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 2.50, size = 248, normalized size = 2.28 \[ \frac {2 \, \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (3, \cos \relax (x) + i \, \sin \relax (x)\right ) + 2 \, \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (3, \cos \relax (x) - i \, \sin \relax (x)\right ) + 2 \, \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (3, -\cos \relax (x) + i \, \sin \relax (x)\right ) + 2 \, \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x)^{2} {\rm polylog}\left (3, -\cos \relax (x) - i \, \sin \relax (x)\right ) + {\left (x^{2} \cos \relax (x)^{2} \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + x^{2} \cos \relax (x)^{2} \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + x^{2} \cos \relax (x)^{2} \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + x^{2} \cos \relax (x)^{2} \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - 2 i \, x \cos \relax (x)^{2} {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) + 2 i \, x \cos \relax (x)^{2} {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) + 2 i \, x \cos \relax (x)^{2} {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) - 2 i \, x \cos \relax (x)^{2} {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right )\right )} \sqrt {\frac {a}{\cos \relax (x)^{4}}}}{2 \, a} \]
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 \[ \int \frac {x^{2} \csc \relax (x) \sec \relax (x)}{\sqrt {a \sec \relax (x)^{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 183, normalized size = 1.68 \[ \frac {i {\mathrm e}^{2 i x} x^{3}}{3 \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2}}-\frac {2 \left (\frac {i {\mathrm e}^{2 i x} x^{3}}{3}-\frac {{\mathrm e}^{2 i x} x^{2} \ln \left (1+{\mathrm e}^{i x}\right )}{2}+i {\mathrm e}^{2 i x} x \polylog \left (2, -{\mathrm e}^{i x}\right )-{\mathrm e}^{2 i x} \polylog \left (3, -{\mathrm e}^{i x}\right )-\frac {{\mathrm e}^{2 i x} x^{2} \ln \left (1-{\mathrm e}^{i x}\right )}{2}+i {\mathrm e}^{2 i x} x \polylog \left (2, {\mathrm e}^{i x}\right )-{\mathrm e}^{2 i x} \polylog \left (3, {\mathrm e}^{i x}\right )\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 113, normalized size = 1.04 \[ \frac {-2 i \, x^{3} + 6 i \, x^{2} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - 6 i \, x^{2} \arctan \left (\sin \relax (x), -\cos \relax (x) + 1\right ) + 3 \, x^{2} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + 3 \, x^{2} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - 12 i \, x {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - 12 i \, x {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + 12 \, {\rm Li}_{3}(-e^{\left (i \, x\right )}) + 12 \, {\rm Li}_{3}(e^{\left (i \, x\right )})}{6 \, \sqrt {a}} \]
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 \[ \int \frac {x^2}{\cos \relax (x)\,\sin \relax (x)\,\sqrt {\frac {a}{{\cos \relax (x)}^4}}} \,d x \]
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 \[ \int \frac {x^{2} \csc {\relax (x )} \sec {\relax (x )}}{\sqrt {a \sec ^{4}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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