Optimal. Leaf size=73 \[ -i \cos ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(a x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{2 i \cos ^{-1}(a x)}\right )-\frac{1}{3} i \cos ^{-1}(a x)^3+\cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.0933503, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4626, 3719, 2190, 2531, 2282, 6589} \[ -i \cos ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(a x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{2 i \cos ^{-1}(a x)}\right )-\frac{1}{3} i \cos ^{-1}(a x)^3+\cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 4626
Rule 3719
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}(a x)^2}{x} \, dx &=-\operatorname{Subst}\left (\int x^2 \tan (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{1}{3} i \cos ^{-1}(a x)^3+2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} x^2}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{1}{3} i \cos ^{-1}(a x)^3+\cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-2 \operatorname{Subst}\left (\int x \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{1}{3} i \cos ^{-1}(a x)^3+\cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-i \cos ^{-1}(a x) \text{Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+i \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{1}{3} i \cos ^{-1}(a x)^3+\cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-i \cos ^{-1}(a x) \text{Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(a x)}\right )\\ &=-\frac{1}{3} i \cos ^{-1}(a x)^3+\cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-i \cos ^{-1}(a x) \text{Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+\frac{1}{2} \text{Li}_3\left (-e^{2 i \cos ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0196891, size = 73, normalized size = 1. \[ -i \cos ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(a x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{2 i \cos ^{-1}(a x)}\right )-\frac{1}{3} i \cos ^{-1}(a x)^3+\cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 101, normalized size = 1.4 \begin{align*} -{\frac{i}{3}} \left ( \arccos \left ( ax \right ) \right ) ^{3}+ \left ( \arccos \left ( ax \right ) \right ) ^{2}\ln \left ( 1+ \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) ^{2} \right ) -i\arccos \left ( ax \right ){\it polylog} \left ( 2,- \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) ^{2} \right ) +{\frac{1}{2}{\it polylog} \left ( 3,- \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 )^{2}}{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 )^{2}}{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}^{2}{\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 )^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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