Optimal. Leaf size=155 \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{48 a^5}-\frac{27 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Shi}\left (5 \sinh ^{-1}(a x)\right )}{96 a^5}-\frac{25 x^4 \sqrt{a^2 x^2+1}}{6 a \sinh ^{-1}(a x)}-\frac{x^4 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{a^2 x^2+1}}{a^3 \sinh ^{-1}(a x)}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.319825, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5667, 5774, 5665, 3298} \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{48 a^5}-\frac{27 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Shi}\left (5 \sinh ^{-1}(a x)\right )}{96 a^5}-\frac{25 x^4 \sqrt{a^2 x^2+1}}{6 a \sinh ^{-1}(a x)}-\frac{x^4 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{a^2 x^2+1}}{a^3 \sinh ^{-1}(a x)}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5665
Rule 3298
Rubi steps
\begin{align*} \int \frac{x^4}{\sinh ^{-1}(a x)^4} \, dx &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}+\frac{4 \int \frac{x^3}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx}{3 a}+\frac{1}{3} (5 a) \int \frac{x^5}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx\\ &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2}+\frac{25}{6} \int \frac{x^4}{\sinh ^{-1}(a x)^2} \, dx+\frac{2 \int \frac{x^2}{\sinh ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac{25 x^4 \sqrt{1+a^2 x^2}}{6 a \sinh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 x}+\frac{3 \sinh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}+\frac{25 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 x}-\frac{9 \sinh (3 x)}{16 x}+\frac{5 \sinh (5 x)}{16 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^5}\\ &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac{25 x^4 \sqrt{1+a^2 x^2}}{6 a \sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{48 a^5}+\frac{125 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^5}-\frac{75 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}\\ &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac{25 x^4 \sqrt{1+a^2 x^2}}{6 a \sinh ^{-1}(a x)}+\frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{48 a^5}-\frac{27 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Shi}\left (5 \sinh ^{-1}(a x)\right )}{96 a^5}\\ \end{align*}
Mathematica [A] time = 0.328406, size = 156, normalized size = 1.01 \[ -\frac{32 a^4 x^4 \sqrt{a^2 x^2+1}+80 a^5 x^5 \sinh ^{-1}(a x)+400 a^4 x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2+64 a^3 x^3 \sinh ^{-1}(a x)+192 a^2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2-2 \sinh ^{-1}(a x)^3 \text{Shi}\left (\sinh ^{-1}(a x)\right )+81 \sinh ^{-1}(a x)^3 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )-125 \sinh ^{-1}(a x)^3 \text{Shi}\left (5 \sinh ^{-1}(a x)\right )}{96 a^5 \sinh ^{-1}(a x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 169, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{5}} \left ( -{\frac{1}{24\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{ax}{48\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}-{\frac{1}{48\,{\it Arcsinh} \left ( ax \right ) }\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{{\it Shi} \left ({\it Arcsinh} \left ( ax \right ) \right ) }{48}}+{\frac{\cosh \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{16\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}}+{\frac{3\,\sinh \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{32\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}+{\frac{9\,\cosh \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{32\,{\it Arcsinh} \left ( ax \right ) }}-{\frac{27\,{\it Shi} \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{32}}-{\frac{\cosh \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{48\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}}-{\frac{5\,\sinh \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{96\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}-{\frac{25\,\cosh \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{96\,{\it Arcsinh} \left ( ax \right ) }}+{\frac{125\,{\it Shi} \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{96}} \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*} \text{result too large to display} \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{x^{4}}{\operatorname{arsinh}\left (a x\right )^{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{x^{4}}{\operatorname{asinh}^{4}{\left (a x \right )}}\, 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{x^{4}}{\operatorname{arsinh}\left (a x\right )^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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