Optimal. Leaf size=99 \[ \frac{1}{3} a^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-\frac{1}{3} a^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 x^2}-\frac{a^2}{3 x}+\frac{2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac{\sinh ^{-1}(a x)^2}{3 x^3} \]
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Rubi [A] time = 0.162589, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5661, 5747, 5760, 4182, 2279, 2391, 30} \[ \frac{1}{3} a^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-\frac{1}{3} a^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 x^2}-\frac{a^2}{3 x}+\frac{2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac{\sinh ^{-1}(a x)^2}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 5747
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 30
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^2}{x^4} \, dx &=-\frac{\sinh ^{-1}(a x)^2}{3 x^3}+\frac{1}{3} (2 a) \int \frac{\sinh ^{-1}(a x)}{x^3 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac{\sinh ^{-1}(a x)^2}{3 x^3}+\frac{1}{3} a^2 \int \frac{1}{x^2} \, dx-\frac{1}{3} a^3 \int \frac{\sinh ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a^2}{3 x}-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac{\sinh ^{-1}(a x)^2}{3 x^3}-\frac{1}{3} a^3 \operatorname{Subst}\left (\int x \text{csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{a^2}{3 x}-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac{\sinh ^{-1}(a x)^2}{3 x^3}+\frac{2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac{1}{3} a^3 \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\frac{1}{3} a^3 \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{a^2}{3 x}-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac{\sinh ^{-1}(a x)^2}{3 x^3}+\frac{2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac{1}{3} a^3 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-\frac{1}{3} a^3 \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac{a^2}{3 x}-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 x^2}-\frac{\sinh ^{-1}(a x)^2}{3 x^3}+\frac{2}{3} a^3 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac{1}{3} a^3 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-\frac{1}{3} a^3 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.489779, size = 125, normalized size = 1.26 \[ -\frac{a^3 x^3 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )-a^3 x^3 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )+a^2 x^2+a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+a^3 x^3 \sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-a^3 x^3 \sinh ^{-1}(a x) \log \left (e^{-\sinh ^{-1}(a x)}+1\right )+\sinh ^{-1}(a x)^2}{3 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.092, size = 144, normalized size = 1.5 \begin{align*} -{\frac{a{\it Arcsinh} \left ( ax \right ) }{3\,{x}^{2}}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}}{3\,x}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{3\,{x}^{3}}}+{\frac{{a}^{3}{\it Arcsinh} \left ( ax \right ) }{3}\ln \left ( 1+ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) }+{\frac{{a}^{3}}{3}{\it polylog} \left ( 2,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) }-{\frac{{a}^{3}{\it Arcsinh} \left ( ax \right ) }{3}\ln \left ( 1-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) }-{\frac{{a}^{3}}{3}{\it polylog} \left ( 2,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) } \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*} -\frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{3 \, x^{3}} + \int \frac{2 \,{\left (a^{3} x^{2} + \sqrt{a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{3 \,{\left (a^{3} x^{6} + a x^{4} +{\left (a^{2} x^{5} + x^{3}\right )} \sqrt{a^{2} x^{2} + 1}\right )}}\,{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 )^{2}}{x^{4}}, 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*} \int \frac{\operatorname{asinh}^{2}{\left (a x \right )}}{x^{4}}\, dx \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 )^{2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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