Optimal. Leaf size=92 \[ -i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} b^2 \text{PolyLog}\left (3,-e^{2 i \cos ^{-1}(c x)}\right )-\frac{i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+\log \left (1+e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )^2 \]
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Rubi [A] time = 0.118095, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4626, 3719, 2190, 2531, 2282, 6589} \[ -i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} b^2 \text{PolyLog}\left (3,-e^{2 i \cos ^{-1}(c x)}\right )-\frac{i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+\log \left (1+e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )^2 \]
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{\left (a+b \cos ^{-1}(c x)\right )^2}{x} \, dx &=-\operatorname{Subst}\left (\int (a+b x)^2 \tan (x) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)^2}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-i b \left (a+b \cos ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-i b \left (a+b \cos ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )\\ &=-\frac{i \left (a+b \cos ^{-1}(c x)\right )^3}{3 b}+\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-i b \left (a+b \cos ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac{1}{2} b^2 \text{Li}_3\left (-e^{2 i \cos ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.116828, size = 128, normalized size = 1.39 \[ -i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} b^2 \text{PolyLog}\left (3,-e^{2 i \cos ^{-1}(c x)}\right )+a^2 \log (c x)-i a b \cos ^{-1}(c x)^2+2 a b \cos ^{-1}(c x) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac{1}{3} i b^2 \cos ^{-1}(c x)^3+b^2 \cos ^{-1}(c x)^2 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.058, size = 194, normalized size = 2.1 \begin{align*}{a}^{2}\ln \left ( cx \right ) -{\frac{i}{3}}{b}^{2} \left ( \arccos \left ( cx \right ) \right ) ^{3}+{b}^{2} \left ( \arccos \left ( cx \right ) \right ) ^{2}\ln \left ( 1+ \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) -i{b}^{2}\arccos \left ( cx \right ){\it polylog} \left ( 2,- \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) +{\frac{{b}^{2}}{2}{\it polylog} \left ( 3,- \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }-iab \left ( \arccos \left ( cx \right ) \right ) ^{2}+2\,ab\arccos \left ( cx \right ) \ln \left ( 1+ \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) -iab{\it polylog} \left ( 2,- \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \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*} a^{2} \log \left (x\right ) + \int \frac{b^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right )^{2} + 2 \, a b \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c 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{b^{2} \arccos \left (c x\right )^{2} + 2 \, a b \arccos \left (c x\right ) + a^{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{\left (a + b \operatorname{acos}{\left (c x \right )}\right )^{2}}{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{{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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