Optimal. Leaf size=265 \[ \frac{3 b^2 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}+\frac{3 b^3 \text{PolyLog}\left (4,-e^{-2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right )}{4 c}-\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{4 b c}-\frac{\log \left (e^{-2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}+1\right ) \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.220762, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6681, 5660, 3718, 2190, 2531, 6609, 2282, 6589} \[ \frac{3 b^2 \text{PolyLog}\left (3,-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}-\frac{3 b \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}-\frac{3 b^3 \text{PolyLog}\left (4,-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right )}{4 c}+\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{4 b c}-\frac{\log \left (e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}+1\right ) \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 6681
Rule 5660
Rule 3718
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \tanh (x) \, dx,x,\cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^3}{1+e^{2 x}} \, dx,x,\cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1+e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1+e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{3 b \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1+e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{3 b \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}+\frac{3 b^2 \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{2 c}\\ &=\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1+e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{3 b \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}+\frac{3 b^2 \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{4 c}\\ &=\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1+e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{3 b \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}+\frac{3 b^2 \left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{3 b^3 \text{Li}_4\left (-e^{2 \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{4 c}\\ \end{align*}
Mathematica [F] time = 0.366583, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cosh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.942, size = 870, normalized size = 3.3 \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*} \text{result too large to display} \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^{3} \operatorname{arcosh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{3} + 3 \, a b^{2} \operatorname{arcosh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 3 \, a^{2} b \operatorname{arcosh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a^{3}}{c^{2} x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [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.
[In]
[Out]