Optimal. Leaf size=167 \[ -\frac{2 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}+\frac{\sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}+\frac{2 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}-\frac{\sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}-\frac{2 x^3 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{16 x^4}{3 \sqrt{\sinh ^{-1}(a x)}} \]
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Rubi [A] time = 0.441375, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5667, 5774, 5669, 5448, 3308, 2180, 2204, 2205, 12} \[ -\frac{2 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}+\frac{\sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}+\frac{2 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}-\frac{\sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}-\frac{2 x^3 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{16 x^4}{3 \sqrt{\sinh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5669
Rule 5448
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 12
Rubi steps
\begin{align*} \int \frac{x^3}{\sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac{2 \int \frac{x^2}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx}{a}+\frac{1}{3} (8 a) \int \frac{x^4}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{16 x^4}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{64}{3} \int \frac{x^3}{\sqrt{\sinh ^{-1}(a x)}} \, dx+\frac{8 \int \frac{x}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac{2 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{16 x^4}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac{64 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac{2 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{16 x^4}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac{64 \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 )}{3 a^4}\\ &=-\frac{2 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{16 x^4}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}+\frac{4 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}-\frac{16 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac{2 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{16 x^4}{3 \sqrt{\sinh ^{-1}(a x)}}-\frac{4 \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}-\frac{2 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac{8 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}-\frac{8 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac{2 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{16 x^4}{3 \sqrt{\sinh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}+\frac{8 \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}-\frac{4 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{a^4}+\frac{4 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{a^4}+\frac{16 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}-\frac{16 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}\\ &=-\frac{2 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{16 x^4}{3 \sqrt{\sinh ^{-1}(a x)}}-\frac{2 \sqrt{\pi } \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}+\frac{\sqrt{2 \pi } \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}+\frac{2 \sqrt{\pi } \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}-\frac{\sqrt{2 \pi } \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a^4}\\ \end{align*}
Mathematica [A] time = 0.409386, size = 174, normalized size = 1.04 \[ \frac{4 \sinh ^{-1}(a x) \left (-\sqrt{2} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 \sinh ^{-1}(a x)\right )-\sqrt{2} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 \sinh ^{-1}(a x)\right )+e^{-2 \sinh ^{-1}(a x)}+e^{2 \sinh ^{-1}(a x)}\right )-4 \sinh ^{-1}(a x) \left (-2 \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 \sinh ^{-1}(a x)\right )-2 \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 \sinh ^{-1}(a x)\right )+e^{-4 \sinh ^{-1}(a x)}+e^{4 \sinh ^{-1}(a x)}\right )+2 \sinh \left (2 \sinh ^{-1}(a x)\right )-\sinh \left (4 \sinh ^{-1}(a x)\right )}{12 a^4 \sinh ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{5}{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 \frac{x^{3}}{\operatorname{arsinh}\left (a x\right )^{\frac{5}{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 \frac{x^{3}}{\operatorname{asinh}^{\frac{5}{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 \frac{x^{3}}{\operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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