Optimal. Leaf size=268 \[ -2 i a^3 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )+\frac{2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac{4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\frac{\cosh ^{-1}(a x)^4}{3 x^3}+\frac{2 a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{3 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.80247, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {5662, 5748, 5761, 4180, 2531, 6609, 2282, 6589, 2279, 2391} \[ -2 i a^3 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )+\frac{2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac{4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\frac{\cosh ^{-1}(a x)^4}{3 x^3}+\frac{2 a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{3 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5662
Rule 5748
Rule 5761
Rule 4180
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^4}{x^4} \, dx &=-\frac{\cosh ^{-1}(a x)^4}{3 x^3}+\frac{1}{3} (4 a) \int \frac{\cosh ^{-1}(a x)^3}{x^3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac{\cosh ^{-1}(a x)^4}{3 x^3}-\left (2 a^2\right ) \int \frac{\cosh ^{-1}(a x)^2}{x^2} \, dx+\frac{1}{3} \left (2 a^3\right ) \int \frac{\cosh ^{-1}(a x)^3}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac{\cosh ^{-1}(a x)^4}{3 x^3}+\frac{1}{3} \left (2 a^3\right ) \operatorname{Subst}\left (\int x^3 \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 a^3\right ) \int \frac{\cosh ^{-1}(a x)}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac{\cosh ^{-1}(a x)^4}{3 x^3}+\frac{4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\left (2 i a^3\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (2 i a^3\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname{Subst}\left (\int x \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac{2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac{\cosh ^{-1}(a x)^4}{3 x^3}-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac{4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-2 i a^3 \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac{2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac{\cosh ^{-1}(a x)^4}{3 x^3}-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac{4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-2 i a^3 \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac{2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac{\cosh ^{-1}(a x)^4}{3 x^3}-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac{4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )-2 i a^3 \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )\\ &=\frac{2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac{\cosh ^{-1}(a x)^4}{3 x^3}-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac{4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )-2 i a^3 \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text{Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text{Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [B] time = 3.07534, size = 595, normalized size = 2.22 \[ a^3 \left (\frac{1}{2} i \left (-4 \cosh ^{-1}(a x)^2-4 i \pi \cosh ^{-1}(a x)+\pi ^2+8\right ) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-\frac{1}{96} i \left (192 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+384 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-384 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-192 i \pi \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{-\cosh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )-\frac{32 i \cosh ^{-1}(a x)^4}{a^3 x^3}+\frac{64 i \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^3}{a^2 x^2}-16 \cosh ^{-1}(a x)^4-32 i \pi \cosh ^{-1}(a x)^3+\frac{192 i \cosh ^{-1}(a x)^2}{a x}+24 \pi ^2 \cosh ^{-1}(a x)^2+8 i \pi ^3 \cosh ^{-1}(a x)-64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )-96 i \pi \cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+96 i \pi \cosh ^{-1}(a x)^2 \log \left (1-i e^{\cosh ^{-1}(a x)}\right )-384 \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )+48 \pi ^2 \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+384 \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \cosh ^{-1}(a x) \log \left (1-i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (\pi +2 i \cosh ^{-1}(a x)\right )\right )\right )+7 \pi ^4\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}}{{x}^{4}}}\, 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*} -\frac{\log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{4}}{3 \, x^{3}} + \int \frac{4 \,{\left (a^{3} x^{2} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{3}}{3 \,{\left (a^{3} x^{6} - a x^{4} +{\left (a^{2} x^{5} - x^{3}\right )} \sqrt{a x + 1} \sqrt{a x - 1}\right )}}\,{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{\operatorname{arcosh}\left (a x\right )^{4}}{x^{4}}, 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}^{4}{\left (a x \right )}}{x^{4}}\, 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 )^{4}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]