Optimal. Leaf size=59 \[ -\frac {1}{2} i \text {Li}_2\left (e^{2 i \sin ^{-1}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \sin ^{-1}\left (\frac {x}{a}\right )^2+\sin ^{-1}\left (\frac {x}{a}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {x}{a}\right )}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5265, 4625, 3717, 2190, 2279, 2391} \[ -\frac {1}{2} i \text {PolyLog}\left (2,e^{2 i \sin ^{-1}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \sin ^{-1}\left (\frac {x}{a}\right )^2+\sin ^{-1}\left (\frac {x}{a}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {x}{a}\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 5265
Rubi steps
\begin {align*} \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x} \, dx &=\int \frac {\sin ^{-1}\left (\frac {x}{a}\right )}{x} \, dx\\ &=\operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )\\ &=-\frac {1}{2} i \sin ^{-1}\left (\frac {x}{a}\right )^2-2 i \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )\\ &=-\frac {1}{2} i \sin ^{-1}\left (\frac {x}{a}\right )^2+\sin ^{-1}\left (\frac {x}{a}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {x}{a}\right )}\right )-\operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )\\ &=-\frac {1}{2} i \sin ^{-1}\left (\frac {x}{a}\right )^2+\sin ^{-1}\left (\frac {x}{a}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {x}{a}\right )}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (\frac {x}{a}\right )}\right )\\ &=-\frac {1}{2} i \sin ^{-1}\left (\frac {x}{a}\right )^2+\sin ^{-1}\left (\frac {x}{a}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \text {Li}_2\left (e^{2 i \sin ^{-1}\left (\frac {x}{a}\right )}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 54, normalized size = 0.92 \[ \csc ^{-1}\left (\frac {a}{x}\right ) \log \left (1-e^{2 i \csc ^{-1}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} i \left (\csc ^{-1}\left (\frac {a}{x}\right )^2+\text {Li}_2\left (e^{2 i \csc ^{-1}\left (\frac {a}{x}\right )}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccsc}\left (\frac {a}{x}\right )}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arccsc}\left (\frac {a}{x}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 125, normalized size = 2.12 \[ -\frac {i \mathrm {arccsc}\left (\frac {a}{x}\right )^{2}}{2}+\mathrm {arccsc}\left (\frac {a}{x}\right ) \ln \left (1+\frac {i x}{a}+\sqrt {1-\frac {x^{2}}{a^{2}}}\right )+\mathrm {arccsc}\left (\frac {a}{x}\right ) \ln \left (1-\frac {i x}{a}-\sqrt {1-\frac {x^{2}}{a^{2}}}\right )-i \polylog \left (2, -\frac {i x}{a}-\sqrt {1-\frac {x^{2}}{a^{2}}}\right )-i \polylog \left (2, \frac {i x}{a}+\sqrt {1-\frac {x^{2}}{a^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccsc}\left (\frac {a}{x}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 49, normalized size = 0.83 \[ -\frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {x}{a}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {x}{a}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {x}{a}\right )-\frac {{\mathrm {asin}\left (\frac {x}{a}\right )}^2\,1{}\mathrm {i}}{2} \]
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 \[ \int \frac {\operatorname {acsc}{\left (\frac {a}{x} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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