Optimal. Leaf size=294 \[ -\frac{3 i \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \sinh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{3 i \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \text{PolyLog}\left (4,-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{3 i \text{PolyLog}\left (4,i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left (a^2 x^2+1\right )}+\frac{3 \sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{a^2 x^2+1}}+\frac{\sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{6 \sinh ^{-1}(a x) \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2} \]
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Rubi [A] time = 0.306182, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {5690, 5693, 4180, 2531, 6609, 2282, 6589, 5717, 2279, 2391} \[ -\frac{3 i \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \sinh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{3 i \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \text{PolyLog}\left (4,-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{3 i \text{PolyLog}\left (4,i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left (a^2 x^2+1\right )}+\frac{3 \sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{a^2 x^2+1}}+\frac{\sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{6 \sinh ^{-1}(a x) \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 5690
Rule 5693
Rule 4180
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 5717
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac{(3 a) \int \frac{x \sinh ^{-1}(a x)^2}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{2 c^2}+\frac{\int \frac{\sinh ^{-1}(a x)^3}{c+a^2 c x^2} \, dx}{2 c}\\ &=\frac{3 \sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2}}+\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac{3 \int \frac{\sinh ^{-1}(a x)}{1+a^2 x^2} \, dx}{c^2}+\frac{\operatorname{Subst}\left (\int x^3 \text{sech}(x) \, dx,x,\sinh ^{-1}(a x)\right )}{2 a c^2}\\ &=\frac{3 \sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2}}+\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{(3 i) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{2 a c^2}+\frac{(3 i) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{2 a c^2}-\frac{3 \operatorname{Subst}\left (\int x \text{sech}(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2}\\ &=\frac{3 \sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2}}+\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac{6 \sinh ^{-1}(a x) \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{\sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{(3 i) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2}-\frac{(3 i) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2}+\frac{(3 i) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2}-\frac{(3 i) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2}\\ &=\frac{3 \sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2}}+\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac{6 \sinh ^{-1}(a x) \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{\sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x) \text{Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \sinh ^{-1}(a x) \text{Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{(3 i) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2}+\frac{(3 i) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2}\\ &=\frac{3 \sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2}}+\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac{6 \sinh ^{-1}(a x) \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{\sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{3 i \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}-\frac{3 i \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x) \text{Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \sinh ^{-1}(a x) \text{Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )}{a c^2}\\ &=\frac{3 \sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2}}+\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac{6 \sinh ^{-1}(a x) \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{\sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{3 i \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}-\frac{3 i \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x) \text{Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \sinh ^{-1}(a x) \text{Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c^2}-\frac{3 i \text{Li}_4\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c^2}+\frac{3 i \text{Li}_4\left (i e^{\sinh ^{-1}(a x)}\right )}{a c^2}\\ \end{align*}
Mathematica [A] time = 2.53661, size = 568, normalized size = 1.93 \[ -\frac{i \left (192 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(a x)}\right )+192 i \pi \sinh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )+384 \sinh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(a x)}\right )-384 \sinh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(a x)}\right )-48 \left (-4 \sinh ^{-1}(a x)^2-4 i \pi \sinh ^{-1}(a x)+\pi ^2+8\right ) \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(a x)}\right )-48 \pi ^2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )+192 i \pi \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(a x)}\right )-192 i \pi \text{PolyLog}\left (3,i e^{\sinh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{-\sinh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{\sinh ^{-1}(a x)}\right )+\frac{64 i a x \sinh ^{-1}(a x)^3}{a^2 x^2+1}+\frac{192 i \sinh ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-16 \sinh ^{-1}(a x)^4-32 i \pi \sinh ^{-1}(a x)^3+24 \pi ^2 \sinh ^{-1}(a x)^2+8 i \pi ^3 \sinh ^{-1}(a x)-64 \sinh ^{-1}(a x)^3 \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )+64 \sinh ^{-1}(a x)^3 \log \left (1+i e^{\sinh ^{-1}(a x)}\right )-96 i \pi \sinh ^{-1}(a x)^2 \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )+96 i \pi \sinh ^{-1}(a x)^2 \log \left (1-i e^{\sinh ^{-1}(a x)}\right )-384 \sinh ^{-1}(a x) \log \left (1-i e^{-\sinh ^{-1}(a x)}\right )+48 \pi ^2 \sinh ^{-1}(a x) \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )+384 \sinh ^{-1}(a x) \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )-48 \pi ^2 \sinh ^{-1}(a x) \log \left (1-i e^{\sinh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\sinh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (\pi +2 i \sinh ^{-1}(a x)\right )\right )\right )+7 \pi ^4\right )}{128 a c^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.126, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}}\, dx \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*} \int \frac{\operatorname{arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{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{\operatorname{arsinh}\left (a x\right )^{3}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{asinh}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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