Optimal. Leaf size=60 \[ \frac {1}{2} i \text {Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right )+\frac {1}{2} i \cos ^{-1}\left (\frac {a}{x}\right )^2-\cos ^{-1}\left (\frac {a}{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4831, 3719, 2190, 2279, 2391} \[ \frac {1}{2} i \text {PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right )+\frac {1}{2} i \cos ^{-1}\left (\frac {a}{x}\right )^2-\cos ^{-1}\left (\frac {a}{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4831
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}\left (\frac {a}{x}\right )}{x} \, dx &=\operatorname {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}\left (\frac {a}{x}\right )\right )\\ &=\frac {1}{2} i \cos ^{-1}\left (\frac {a}{x}\right )^2-2 i \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}\left (\frac {a}{x}\right )\right )\\ &=\frac {1}{2} i \cos ^{-1}\left (\frac {a}{x}\right )^2-\cos ^{-1}\left (\frac {a}{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right )+\operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (\frac {a}{x}\right )\right )\\ &=\frac {1}{2} i \cos ^{-1}\left (\frac {a}{x}\right )^2-\cos ^{-1}\left (\frac {a}{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right )\\ &=\frac {1}{2} i \cos ^{-1}\left (\frac {a}{x}\right )^2-\cos ^{-1}\left (\frac {a}{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right )+\frac {1}{2} i \text {Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 1.00 \[ \frac {1}{2} i \text {Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right )+\frac {1}{2} i \cos ^{-1}\left (\frac {a}{x}\right )^2-\cos ^{-1}\left (\frac {a}{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {a}{x}\right )}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arccos \left (\frac {a}{x}\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 77, normalized size = 1.28 \[ \frac {i \arccos \left (\frac {a}{x}\right )^{2}}{2}-\arccos \left (\frac {a}{x}\right ) \ln \left (1+\left (\frac {a}{x}+i \sqrt {1-\frac {a^{2}}{x^{2}}}\right )^{2}\right )+\frac {i \polylog \left (2, -\left (\frac {a}{x}+i \sqrt {1-\frac {a^{2}}{x^{2}}}\right )^{2}\right )}{2} \]
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 \[ -i \, a^{2} \int -\frac {\log \relax (x)}{a^{2} x - x^{3}}\,{d x} - a \int -\frac {\sqrt {a + x} \sqrt {-a + x} \log \relax (x)}{a^{2} x - x^{3}}\,{d x} + \arctan \left (\frac {\sqrt {a + x} \sqrt {-a + x}}{a}\right ) \log \relax (x) - \frac {1}{2} i \, \log \left (x^{2}\right ) \log \relax (x) + \frac {1}{2} i \, \log \relax (x)^{2} + \frac {1}{2} i \, \log \relax (x) \log \left (\frac {a + x}{a}\right ) + \frac {1}{2} i \, \log \relax (x) \log \left (\frac {a - x}{a}\right ) + \frac {1}{2} i \, {\rm Li}_2\left (\frac {x}{a}\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-\frac {x}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acos}\left (\frac {a}{x}\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {acos}{\left (\frac {a}{x} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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