Optimal. Leaf size=139 \[ -\frac{\sqrt{\pi } \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{32 a^4}-\frac{\sqrt{\pi } \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{32 a^4}-\frac{3 \sqrt{\sinh ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.261595, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5663, 5779, 3312, 3307, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{32 a^4}-\frac{\sqrt{\pi } \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{32 a^4}-\frac{3 \sqrt{\sinh ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5663
Rule 5779
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x^3 \sqrt{\sinh ^{-1}(a x)} \, dx &=\frac{1}{4} x^4 \sqrt{\sinh ^{-1}(a x)}-\frac{1}{8} a \int \frac{x^4}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx\\ &=\frac{1}{4} x^4 \sqrt{\sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^4(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}\\ &=\frac{1}{4} x^4 \sqrt{\sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}-\frac{\cosh (2 x)}{2 \sqrt{x}}+\frac{\cosh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}\\ &=-\frac{3 \sqrt{\sinh ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^4}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^4}\\ &=-\frac{3 \sqrt{\sinh ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a^4}-\frac{\operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a^4}+\frac{\operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^4}+\frac{\operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^4}\\ &=-\frac{3 \sqrt{\sinh ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{64 a^4}-\frac{\operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{64 a^4}+\frac{\operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^4}+\frac{\operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^4}\\ &=-\frac{3 \sqrt{\sinh ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sinh ^{-1}(a x)}-\frac{\sqrt{\pi } \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{32 a^4}-\frac{\sqrt{\pi } \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{32 a^4}\\ \end{align*}
Mathematica [A] time = 0.0357477, size = 101, normalized size = 0.73 \[ \frac{\sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-4 \sinh ^{-1}(a x)\right )-4 \sqrt{2} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt{-\sinh ^{-1}(a x)} \left (\text{Gamma}\left (\frac{3}{2},4 \sinh ^{-1}(a x)\right )-4 \sqrt{2} \text{Gamma}\left (\frac{3}{2},2 \sinh ^{-1}(a x)\right )\right )}{128 a^4 \sqrt{-\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\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^{3} \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^{3} \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^{3} \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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