Optimal. Leaf size=207 \[ \frac{i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{b^2 \text{PolyLog}\left (3,-e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right )}{2 c}+\frac{i \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{3 b c}-\frac{\log \left (1+e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{c} \]
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Rubi [A] time = 0.174358, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {6681, 4626, 3719, 2190, 2531, 2282, 6589} \[ \frac{i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{b^2 \text{PolyLog}\left (3,-e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right )}{2 c}+\frac{i \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{3 b c}-\frac{\log \left (1+e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{c} \]
Antiderivative was successfully verified.
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Rule 6681
Rule 4626
Rule 3719
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cos ^{-1}(x)\right )^2}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^2 \tan (x) \, dx,x,\cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{i \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{3 b c}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)^2}{1+e^{2 i x}} \, dx,x,\cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{i \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{3 b c}-\frac{\left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \log \left (1+e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{i \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{3 b c}-\frac{\left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \log \left (1+e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{i b \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{i \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{3 b c}-\frac{\left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \log \left (1+e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{i b \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}\\ &=\frac{i \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{3 b c}-\frac{\left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \log \left (1+e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{i b \left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{b^2 \text{Li}_3\left (-e^{2 i \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}\\ \end{align*}
Mathematica [F] time = 0.695027, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cos ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.007, size = 401, normalized size = 1.9 \begin{align*}{\frac{{a}^{2}\ln \left ( cx+1 \right ) }{2\,c}}-{\frac{{a}^{2}\ln \left ( cx-1 \right ) }{2\,c}}+{\frac{{\frac{i}{3}}{b}^{2}}{c} \left ( \arccos \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) \right ) ^{3}}-{\frac{{b}^{2}}{c} \left ( \arccos \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) \right ) ^{2}\ln \left ( 1+ \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}}+i\sqrt{1-{\frac{-cx+1}{cx+1}}} \right ) ^{2} \right ) }+{\frac{i{b}^{2}}{c}\arccos \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ){\it polylog} \left ( 2,- \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}}+i\sqrt{1-{\frac{-cx+1}{cx+1}}} \right ) ^{2} \right ) }-{\frac{{b}^{2}}{2\,c}{\it polylog} \left ( 3,- \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}}+i\sqrt{1-{\frac{-cx+1}{cx+1}}} \right ) ^{2} \right ) }+{\frac{iab}{c} \left ( \arccos \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) \right ) ^{2}}-2\,{\frac{ab}{c}\arccos \left ({\frac{\sqrt{-cx+1}}{\sqrt{cx+1}}} \right ) \ln \left ( 1+ \left ({\frac{\sqrt{-cx+1}}{\sqrt{cx+1}}}+i\sqrt{1-{\frac{-cx+1}{cx+1}}} \right ) ^{2} \right ) }+{\frac{iab}{c}{\it polylog} \left ( 2,- \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}}+i\sqrt{1-{\frac{-cx+1}{cx+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*} \frac{1}{2} \, a^{2}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} - \int \frac{b^{2} \arctan \left (\sqrt{2} \sqrt{c} \sqrt{x}, \sqrt{-c x + 1}\right )^{2} + 2 \, a b \arctan \left (\sqrt{2} \sqrt{c} \sqrt{x}, \sqrt{-c x + 1}\right )}{c^{2} x^{2} - 1}\,{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 (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 2 \, a b \arccos \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a^{2}}{c^{2} x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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