Optimal. Leaf size=51 \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(a x)}\right )-\frac{1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.0567818, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4626, 3719, 2190, 2279, 2391} \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(a x)}\right )-\frac{1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 4626
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}(a x)}{x} \, dx &=-\operatorname{Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{1}{2} i \cos ^{-1}(a x)^2+2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}(a x)}\right )\\ &=-\frac{1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\frac{1}{2} i \text{Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0169056, size = 51, normalized size = 1. \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(a x)}\right )-\frac{1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 68, normalized size = 1.3 \begin{align*} -{\frac{i}{2}} \left ( \arccos \left ( ax \right ) \right ) ^{2}+\arccos \left ( ax \right ) \ln \left ( 1+ \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) ^{2} \right ) -{\frac{i}{2}}{\it polylog} \left ( 2,- \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) ^{2} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arccos \left (a x\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arccos \left (a x\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acos}{\left (a x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arccos \left (a x\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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