Optimal. Leaf size=182 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}+\frac{\sqrt{\frac{\pi }{5}} \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{320 a^5}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac{\sqrt{\frac{\pi }{5}} \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{320 a^5}+\frac{1}{5} x^5 \sqrt{\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.324456, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5663, 5779, 3312, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}+\frac{\sqrt{\frac{\pi }{5}} \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{320 a^5}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac{\sqrt{\frac{\pi }{5}} \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{320 a^5}+\frac{1}{5} x^5 \sqrt{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5663
Rule 5779
Rule 3312
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x^4 \sqrt{\sinh ^{-1}(a x)} \, dx &=\frac{1}{5} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{1}{10} a \int \frac{x^5}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx\\ &=\frac{1}{5} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^5(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{10 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\sinh ^{-1}(a x)}+\frac{i \operatorname{Subst}\left (\int \left (\frac{5 i \sinh (x)}{8 \sqrt{x}}-\frac{5 i \sinh (3 x)}{16 \sqrt{x}}+\frac{i \sinh (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{10 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{160 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{320 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{320 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{160 a^5}-\frac{\operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{160 a^5}-\frac{\operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac{\operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\sinh ^{-1}(a x)}+\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}+\frac{\sqrt{\frac{\pi }{5}} \text{erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{320 a^5}-\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac{\sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac{\sqrt{\frac{\pi }{5}} \text{erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{320 a^5}\\ \end{align*}
Mathematica [A] time = 0.0446512, size = 161, normalized size = 0.88 \[ \frac{\frac{\sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-5 \sinh ^{-1}(a x)\right )}{160 \sqrt{5} \sqrt{-\sinh ^{-1}(a x)}}-\frac{\sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-3 \sinh ^{-1}(a x)\right )}{32 \sqrt{3} \sqrt{-\sinh ^{-1}(a x)}}+\frac{\sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-\sinh ^{-1}(a x)\right )}{16 \sqrt{-\sinh ^{-1}(a x)}}-\frac{1}{16} \text{Gamma}\left (\frac{3}{2},\sinh ^{-1}(a x)\right )+\frac{\text{Gamma}\left (\frac{3}{2},3 \sinh ^{-1}(a x)\right )}{32 \sqrt{3}}-\frac{\text{Gamma}\left (\frac{3}{2},5 \sinh ^{-1}(a x)\right )}{160 \sqrt{5}}}{a^5} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.241, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}\sqrt{{\it Arcsinh} \left ( ax \right ) }\, 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 x^{4} \sqrt{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 x^{4} \sqrt{\operatorname{asinh}{\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 x^{4} \sqrt{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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