Optimal. Leaf size=199 \[ -\frac{3 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac{3 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{128 a^4}+\frac{3 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac{3 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{128 a^4}-\frac{3 x^3 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{9 x \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{64 a^3}-\frac{3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.487372, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5663, 5758, 5675, 5669, 5448, 12, 3308, 2180, 2204, 2205} \[ -\frac{3 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac{3 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{128 a^4}+\frac{3 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac{3 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{128 a^4}-\frac{3 x^3 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{9 x \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{64 a^3}-\frac{3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 5663
Rule 5758
Rule 5675
Rule 5669
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x^3 \sinh ^{-1}(a x)^{3/2} \, dx &=\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac{1}{8} (3 a) \int \frac{x^4 \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 x^3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac{3}{64} \int \frac{x^3}{\sqrt{\sinh ^{-1}(a x)}} \, dx+\frac{9 \int \frac{x^2 \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{32 a}\\ &=\frac{9 x \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{64 a^3}-\frac{3 x^3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^4}-\frac{9 \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{64 a^3}-\frac{9 \int \frac{x}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{128 a^2}\\ &=\frac{9 x \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{64 a^3}-\frac{3 x^3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \left (-\frac{\sinh (2 x)}{4 \sqrt{x}}+\frac{\sinh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a^4}\\ &=\frac{9 x \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{64 a^3}-\frac{3 x^3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a^4}\\ &=\frac{9 x \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{64 a^3}-\frac{3 x^3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a^4}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a^4}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a^4}\\ &=\frac{9 x \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{64 a^3}-\frac{3 x^3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac{3 \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac{3 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}-\frac{3 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}+\frac{9 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac{9 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}\\ &=\frac{9 x \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{64 a^3}-\frac{3 x^3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\pi } \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac{3 \sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac{3 \sqrt{\pi } \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac{3 \sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac{9 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}-\frac{9 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{256 a^4}\\ &=\frac{9 x \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{64 a^3}-\frac{3 x^3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\pi } \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac{3 \sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{128 a^4}+\frac{3 \sqrt{\pi } \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac{3 \sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{128 a^4}\\ \end{align*}
Mathematica [A] time = 0.0354882, size = 102, normalized size = 0.51 \[ \frac{-\sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-4 \sinh ^{-1}(a x)\right )+8 \sqrt{2} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt{-\sinh ^{-1}(a x)} \left (\text{Gamma}\left (\frac{5}{2},4 \sinh ^{-1}(a x)\right )-8 \sqrt{2} \text{Gamma}\left (\frac{5}{2},2 \sinh ^{-1}(a x)\right )\right )}{512 a^4 \sqrt{-\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.099, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{{\frac{3}{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 x^{3} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}\,{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} \operatorname{asinh}^{\frac{3}{2}}{\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} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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