Optimal. Leaf size=142 \[ \frac {1}{2} i \text {Li}_2\left (-e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x \sin ^2(x) \sqrt {a \sec ^4(x)}-2 x \cos ^2(x) \tanh ^{-1}\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {1}{2} \sin (x) \cos (x) \sqrt {a \sec ^4(x)} \]
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Rubi [A] time = 0.40, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6720, 2620, 14, 4420, 2548, 4419, 4183, 2279, 2391, 3473, 8} \[ \frac {1}{2} i \cos ^2(x) \text {PolyLog}\left (2,-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {1}{2} i \cos ^2(x) \text {PolyLog}\left (2,e^{2 i x}\right ) \sqrt {a \sec ^4(x)}+\frac {1}{2} x \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} x \sin ^2(x) \sqrt {a \sec ^4(x)}-2 x \cos ^2(x) \tanh ^{-1}\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {1}{2} \sin (x) \cos (x) \sqrt {a \sec ^4(x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 2279
Rule 2391
Rule 2548
Rule 2620
Rule 3473
Rule 4183
Rule 4419
Rule 4420
Rule 6720
Rubi steps
\begin {align*} \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx &=\left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x \csc (x) \sec ^3(x) \, dx\\ &=x \cos ^2(x) \log (\tan (x)) \sqrt {a \sec ^4(x)}+\frac {1}{2} x \sqrt {a \sec ^4(x)} \sin ^2(x)-\left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \left (\log (\tan (x))+\frac {\tan ^2(x)}{2}\right ) \, dx\\ &=x \cos ^2(x) \log (\tan (x)) \sqrt {a \sec ^4(x)}+\frac {1}{2} x \sqrt {a \sec ^4(x)} \sin ^2(x)-\frac {1}{2} \left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \tan ^2(x) \, dx-\left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \log (\tan (x)) \, dx\\ &=-\frac {1}{2} \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x \sqrt {a \sec ^4(x)} \sin ^2(x)+\frac {1}{2} \left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int 1 \, dx+\left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x \csc (x) \sec (x) \, dx\\ &=\frac {1}{2} x \cos ^2(x) \sqrt {a \sec ^4(x)}-\frac {1}{2} \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x \sqrt {a \sec ^4(x)} \sin ^2(x)+\left (2 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int x \csc (2 x) \, dx\\ &=\frac {1}{2} x \cos ^2(x) \sqrt {a \sec ^4(x)}-2 x \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-\frac {1}{2} \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x \sqrt {a \sec ^4(x)} \sin ^2(x)-\left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \log \left (1-e^{2 i x}\right ) \, dx+\left (\cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \log \left (1+e^{2 i x}\right ) \, dx\\ &=\frac {1}{2} x \cos ^2(x) \sqrt {a \sec ^4(x)}-2 x \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}-\frac {1}{2} \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x \sqrt {a \sec ^4(x)} \sin ^2(x)+\frac {1}{2} \left (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 )-\frac {1}{2} \left (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 )\\ &=\frac {1}{2} x \cos ^2(x) \sqrt {a \sec ^4(x)}-2 x \tanh ^{-1}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} i \cos ^2(x) \text {Li}_2\left (-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {1}{2} i \cos ^2(x) \text {Li}_2\left (e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {1}{2} \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x \sqrt {a \sec ^4(x)} \sin ^2(x)\\ \end {align*}
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Mathematica [A] time = 0.23, size = 85, normalized size = 0.60 \[ \frac {1}{2} \cos ^2(x) \sqrt {a \sec ^4(x)} \left (i \text {Li}_2\left (-e^{2 i x}\right )-i \text {Li}_2\left (e^{2 i x}\right )+2 x \log \left (1-e^{2 i x}\right )-2 x \log \left (1+e^{2 i x}\right )-\tan (x)+x \sec ^2(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 270, normalized size = 1.90 \[ \frac {1}{2} \, {\left (x \cos \relax (x)^{2} \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + x \cos \relax (x)^{2} \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - x \cos \relax (x)^{2} \log \left (i \, \cos \relax (x) + \sin \relax (x) + 1\right ) - x \cos \relax (x)^{2} \log \left (i \, \cos \relax (x) - \sin \relax (x) + 1\right ) - x \cos \relax (x)^{2} \log \left (-i \, \cos \relax (x) + \sin \relax (x) + 1\right ) - x \cos \relax (x)^{2} \log \left (-i \, \cos \relax (x) - \sin \relax (x) + 1\right ) + x \cos \relax (x)^{2} \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + x \cos \relax (x)^{2} \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - i \, \cos \relax (x)^{2} {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) + i \, \cos \relax (x)^{2} {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) - i \, \cos \relax (x)^{2} {\rm Li}_2\left (i \, \cos \relax (x) + \sin \relax (x)\right ) + i \, \cos \relax (x)^{2} {\rm Li}_2\left (i \, \cos \relax (x) - \sin \relax (x)\right ) + i \, \cos \relax (x)^{2} {\rm Li}_2\left (-i \, \cos \relax (x) + \sin \relax (x)\right ) - i \, \cos \relax (x)^{2} {\rm Li}_2\left (-i \, \cos \relax (x) - \sin \relax (x)\right ) + i \, \cos \relax (x)^{2} {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) - i \, \cos \relax (x)^{2} {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right ) - \cos \relax (x) \sin \relax (x) + x\right )} \sqrt {\frac {a}{\cos \relax (x)^{4}}} \]
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 \[ \int \sqrt {a \sec \relax (x)^{4}} x \csc \relax (x) \sec \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 165, normalized size = 1.16 \[ \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, \left (-i+2 x -i {\mathrm e}^{-2 i x}\right )-4 i \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2} \left (\frac {i {\mathrm e}^{-2 i x} x \ln \left (1+{\mathrm e}^{i x}\right )}{4}+\frac {{\mathrm e}^{-2 i x} \polylog \left (2, -{\mathrm e}^{i x}\right )}{4}-\frac {i {\mathrm e}^{-2 i x} x \ln \left ({\mathrm e}^{2 i x}+1\right )}{4}-\frac {{\mathrm e}^{-2 i x} \polylog \left (2, -{\mathrm e}^{2 i x}\right )}{8}+\frac {i {\mathrm e}^{-2 i x} x \ln \left (1-{\mathrm e}^{i x}\right )}{4}+\frac {{\mathrm e}^{-2 i x} \polylog \left (2, {\mathrm e}^{i x}\right )}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 432, normalized size = 3.04 \[ -\frac {{\left ({\left (2 \, x \cos \left (4 \, x\right ) + 4 \, x \cos \left (2 \, x\right ) + 2 i \, x \sin \left (4 \, x\right ) + 4 i \, x \sin \left (2 \, x\right ) + 2 \, x\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) - {\left (2 \, x \cos \left (4 \, x\right ) + 4 \, x \cos \left (2 \, x\right ) + 2 i \, x \sin \left (4 \, x\right ) + 4 i \, x \sin \left (2 \, x\right ) + 2 \, x\right )} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) + {\left (2 \, x \cos \left (4 \, x\right ) + 4 \, x \cos \left (2 \, x\right ) + 2 i \, x \sin \left (4 \, x\right ) + 4 i \, x \sin \left (2 \, x\right ) + 2 \, x\right )} \arctan \left (\sin \relax (x), -\cos \relax (x) + 1\right ) - 2 \, {\left (-2 i \, x - 1\right )} \cos \left (2 \, x\right ) - {\left (\cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + i \, \sin \left (4 \, x\right ) + 2 i \, \sin \left (2 \, x\right ) + 1\right )} {\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right ) + {\left (2 \, \cos \left (4 \, x\right ) + 4 \, \cos \left (2 \, x\right ) + 2 i \, \sin \left (4 \, x\right ) + 4 i \, \sin \left (2 \, x\right ) + 2\right )} {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + {\left (2 \, \cos \left (4 \, x\right ) + 4 \, \cos \left (2 \, x\right ) + 2 i \, \sin \left (4 \, x\right ) + 4 i \, \sin \left (2 \, x\right ) + 2\right )} {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + {\left (-i \, x \cos \left (4 \, x\right ) - 2 i \, x \cos \left (2 \, x\right ) + x \sin \left (4 \, x\right ) + 2 \, x \sin \left (2 \, x\right ) - i \, x\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + {\left (i \, x \cos \left (4 \, x\right ) + 2 i \, x \cos \left (2 \, x\right ) - x \sin \left (4 \, x\right ) - 2 \, x \sin \left (2 \, x\right ) + i \, x\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + {\left (i \, x \cos \left (4 \, x\right ) + 2 i \, x \cos \left (2 \, x\right ) - x \sin \left (4 \, x\right ) - 2 \, x \sin \left (2 \, x\right ) + i \, x\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - {\left (4 \, x - 2 i\right )} \sin \left (2 \, x\right ) + 2\right )} \sqrt {a}}{-2 i \, \cos \left (4 \, x\right ) - 4 i \, \cos \left (2 \, x\right ) + 2 \, \sin \left (4 \, x\right ) + 4 \, \sin \left (2 \, x\right ) - 2 i} \]
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 \[ \int \frac {x\,\sqrt {\frac {a}{{\cos \relax (x)}^4}}}{\cos \relax (x)\,\sin \relax (x)} \,d x \]
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 \[ \int x \sqrt {a \sec ^{4}{\relax (x )}} \csc {\relax (x )} \sec {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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