Optimal. Leaf size=76 \[ \frac {i \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {i \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 x \sec (x) \tanh ^{-1}\left (e^{i x}\right )}{\sqrt {a \sec ^2(x)}} \]
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Rubi [A] time = 0.53, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6720, 4183, 2279, 2391} \[ \frac {i \sec (x) \text {PolyLog}\left (2,-e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {i \sec (x) \text {PolyLog}\left (2,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {2 x \sec (x) \tanh ^{-1}\left (e^{i x}\right )}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4183
Rule 6720
Rubi steps
\begin {align*} \int \frac {x \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx &=\frac {\sec (x) \int x \csc (x) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {\sec (x) \int \log \left (1-e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}+\frac {\sec (x) \int \log \left (1+e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {(i \sec (x)) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}-\frac {(i \sec (x)) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {2 x \tanh ^{-1}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {i \text {Li}_2\left (-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {i \text {Li}_2\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 69, normalized size = 0.91 \[ \frac {\sec (x) \left (i \text {Li}_2\left (-e^{i x}\right )-i \text {Li}_2\left (e^{i x}\right )+x \left (\log \left (1-e^{i x}\right )-\log \left (1+e^{i x}\right )\right )\right )}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.27, size = 124, normalized size = 1.63 \[ -\frac {{\left (x \cos \relax (x) \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + x \cos \relax (x) \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - x \cos \relax (x) \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) - x \cos \relax (x) \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + i \, \cos \relax (x) {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) - i \, \cos \relax (x) {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) + i \, \cos \relax (x) {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) - i \, \cos \relax (x) {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right )\right )} \sqrt {\frac {a}{\cos \relax (x)^{2}}}}{2 \, a} \]
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 \[ \int \frac {x \csc \relax (x) \sec \relax (x)}{\sqrt {a \sec \relax (x)^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 98, normalized size = 1.29 \[ -\frac {2 i \left (-\frac {i {\mathrm e}^{i x} x \ln \left (1+{\mathrm e}^{i x}\right )}{2}-\frac {{\mathrm e}^{i x} \polylog \left (2, -{\mathrm e}^{i x}\right )}{2}+\frac {i {\mathrm e}^{i x} x \ln \left (1-{\mathrm e}^{i x}\right )}{2}+\frac {{\mathrm e}^{i x} \polylog \left (2, {\mathrm e}^{i x}\right )}{2}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 79, normalized size = 1.04 \[ -\frac {2 i \, x \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) + 2 i \, x \arctan \left (\sin \relax (x), -\cos \relax (x) + 1\right ) + x \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) - x \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - 2 i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + 2 i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right )}{2 \, \sqrt {a}} \]
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 \[ \int \frac {x}{\cos \relax (x)\,\sin \relax (x)\,\sqrt {\frac {a}{{\cos \relax (x)}^2}}} \,d x \]
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 \[ \int \frac {x \csc {\relax (x )} \sec {\relax (x )}}{\sqrt {a \sec ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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