Optimal. Leaf size=265 \[ -\frac{3 i \sqrt{a x-1} \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{3 i \sqrt{a x-1} \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{6 i \sqrt{a x-1} \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{6 i \sqrt{a x-1} \cosh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{6 i \sqrt{a x-1} \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{6 i \sqrt{a x-1} \text{PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{2 \sqrt{a x-1} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.478743, antiderivative size = 356, normalized size of antiderivative = 1.34, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {5798, 5761, 4180, 2531, 6609, 2282, 6589} \[ -\frac{3 i \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{6 i \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{6 i \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{6 i \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{6 i \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5761
Rule 4180
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^3 \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (3 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (3 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (6 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (6 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{6 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{6 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (6 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (6 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{6 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{6 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (6 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (6 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{6 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{6 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{6 i \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{6 i \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.647126, size = 488, normalized size = 1.84 \[ \frac{i \sqrt{-(a x-1) (a x+1)} \left (192 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+384 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-384 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-48 \left (\pi -2 i \cosh ^{-1}(a x)\right )^2 \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-192 i \pi \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{-\cosh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )-16 \cosh ^{-1}(a x)^4-32 i \pi \cosh ^{-1}(a x)^3+24 \pi ^2 \cosh ^{-1}(a x)^2+8 i \pi ^3 \cosh ^{-1}(a x)-64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )-96 i \pi \cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+96 i \pi \cosh ^{-1}(a x)^2 \log \left (1-i e^{\cosh ^{-1}(a x)}\right )+48 \pi ^2 \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \cosh ^{-1}(a x) \log \left (1-i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (\pi +2 i \cosh ^{-1}(a x)\right )\right )\right )+7 \pi ^4\right )}{64 \sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.149, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{x}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \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*} \int \frac{\operatorname{arcosh}\left (a x\right )^{3}}{\sqrt{-a^{2} x^{2} + 1} x}\,{d x} \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{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )^{3}}{a^{2} x^{3} - x}, 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{\operatorname{acosh}^{3}{\left (a x \right )}}{x \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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{\operatorname{arcosh}\left (a x\right )^{3}}{\sqrt{-a^{2} x^{2} + 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]