Optimal. Leaf size=321 \[ \frac{i b \text{PolyLog}\left (2,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right ) \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{i b \text{PolyLog}\left (2,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right )}\right ) \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right )}{2 c}-\frac{b^2 \text{PolyLog}\left (3,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right )}\right )}{2 c}-\frac{2 \coth ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.319821, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6681, 4851, 4989, 4885, 4993, 6610} \[ \frac{i b \text{PolyLog}\left (2,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right ) \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{i b \text{PolyLog}\left (2,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right )}\right ) \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right )}{2 c}-\frac{b^2 \text{PolyLog}\left (3,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right )}\right )}{2 c}-\frac{2 \coth ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6681
Rule 4851
Rule 4989
Rule 4885
Rule 4993
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (a+b \cot ^{-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 \cot ^{-1}(x)\right )^2}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cot ^{-1}(x)\right ) \coth ^{-1}\left (1-\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cot ^{-1}(x)\right ) \log \left (\frac{2 i}{i+x}\right )}{1+x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cot ^{-1}(x)\right ) \log \left (\frac{2 x}{i+x}\right )}{1+x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{i b \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (1-\frac{2 i}{i+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{i b \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (1-\frac{2 \sqrt{1-c x}}{\sqrt{1+c x} \left (i+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (1-\frac{2 i}{i+x}\right )}{1+x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (1-\frac{2 x}{i+x}\right )}{1+x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{i b \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (1-\frac{2 i}{i+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{i b \left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (1-\frac{2 \sqrt{1-c x}}{\sqrt{1+c x} \left (i+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{b^2 \text{Li}_3\left (1-\frac{2 i}{i+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}-\frac{b^2 \text{Li}_3\left (1-\frac{2 \sqrt{1-c x}}{\sqrt{1+c x} \left (i+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}\\ \end{align*}
Mathematica [F] time = 0.535587, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 1.422, size = 931, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )} - \frac{b^{2} \log \left (2\right )^{2} \log \left (c x + 1\right ) - b^{2} \log \left (2\right )^{2} \log \left (-c x + 1\right ) - 4 \,{\left (b^{2} \log \left (c x + 1\right ) - b^{2} \log \left (-c x + 1\right )\right )} \arctan \left (\sqrt{c x + 1}, \sqrt{-c x + 1}\right )^{2} -{\left (12 \, b^{2}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} \arctan \left (\frac{\sqrt{c x + 1}}{\sqrt{-c x + 1}}\right )^{2} + b^{2}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} \log \left (2\right )^{2} + 4 \, b^{2} \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (\frac{\sqrt{c x + 1}}{\sqrt{-c x + 1}}\right ) \log \left (c x + 1\right )}{c^{2} x^{2} - 1}\,{d x} - 4 \, b^{2} \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (\frac{\sqrt{c x + 1}}{\sqrt{-c x + 1}}\right ) \log \left (-c x + 1\right )}{c^{2} x^{2} - 1}\,{d x} + \frac{32 \,{\left ({\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )} \arctan \left (\sqrt{c x + 1}, \sqrt{-c x + 1}\right ) + c \int \frac{e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) - e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right )}{{\left (c^{2} x^{2} - 1\right )}{\left (c x + 1\right )} -{\left (c^{2} x^{2} - 1\right )}{\left (c x - 1\right )}}\,{d x}\right )} a b}{c}\right )} c}{32 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b^{2} \operatorname{arccot}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 2 \, a b \operatorname{arccot}\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a^{2}}{c^{2} x^{2} - 1}\, dx - \int \frac{b^{2} \operatorname{acot}^{2}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac{2 a b \operatorname{acot}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b \operatorname{arccot}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]