Optimal. Leaf size=222 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a^3}-\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a^3}+\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}-\frac{24 x^2 \sqrt{a^2 x^2+1}}{5 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 x^2 \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 \sqrt{a^2 x^2+1}}{15 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{8 x}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \sinh ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.433931, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5667, 5774, 5665, 3308, 2180, 2204, 2205, 5655, 5779} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a^3}-\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a^3}+\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}-\frac{24 x^2 \sqrt{a^2 x^2+1}}{5 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 x^2 \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 \sqrt{a^2 x^2+1}}{15 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{8 x}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \sinh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5665
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5655
Rule 5779
Rubi steps
\begin{align*} \int \frac{x^2}{\sinh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}+\frac{4 \int \frac{x}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac{1}{5} (6 a) \int \frac{x^3}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \sinh ^{-1}(a x)^{3/2}}+\frac{12}{5} \int \frac{x^2}{\sinh ^{-1}(a x)^{3/2}} \, dx+\frac{8 \int \frac{1}{\sinh ^{-1}(a x)^{3/2}} \, dx}{15 a^2}\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \sinh ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1+a^2 x^2}}{15 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{1+a^2 x^2}}{5 a \sqrt{\sinh ^{-1}(a x)}}+\frac{24 \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 \sqrt{x}}+\frac{3 \sinh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^3}+\frac{16 \int \frac{x}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx}{15 a}\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \sinh ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1+a^2 x^2}}{15 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{1+a^2 x^2}}{5 a \sqrt{\sinh ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^3}-\frac{6 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^3}+\frac{18 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \sinh ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1+a^2 x^2}}{15 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{1+a^2 x^2}}{5 a \sqrt{\sinh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^3}+\frac{8 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^3}-\frac{9 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^3}+\frac{9 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \sinh ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1+a^2 x^2}}{15 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{1+a^2 x^2}}{5 a \sqrt{\sinh ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{15 a^3}+\frac{16 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{15 a^3}+\frac{6 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}-\frac{6 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}-\frac{18 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}+\frac{18 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \sinh ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1+a^2 x^2}}{15 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{1+a^2 x^2}}{5 a \sqrt{\sinh ^{-1}(a x)}}+\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a^3}-\frac{3 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}-\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a^3}+\frac{3 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{5 a^3}\\ \end{align*}
Mathematica [A] time = 0.352793, size = 221, normalized size = 1. \[ \frac{36 \sqrt{3} \left (-\sinh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-3 \sinh ^{-1}(a x)\right )-4 \left (-\sinh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )+e^{-\sinh ^{-1}(a x)} \left (-4 e^{\sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )+4 \sinh ^{-1}(a x)^2-2 \sinh ^{-1}(a x)+3\right )+e^{-3 \sinh ^{-1}(a x)} \left (36 \sqrt{3} e^{3 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},3 \sinh ^{-1}(a x)\right )-36 \sinh ^{-1}(a x)^2+6 \sinh ^{-1}(a x)-3\right )+e^{\sinh ^{-1}(a x)} \left (4 \sinh ^{-1}(a x)^2+2 \sinh ^{-1}(a x)+3\right )-3 e^{3 \sinh ^{-1}(a x)} \left (12 \sinh ^{-1}(a x)^2+2 \sinh ^{-1}(a x)+1\right )}{60 a^3 \sinh ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{7}{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^{2}}{\operatorname{arsinh}\left (a x\right )^{\frac{7}{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2}}{\operatorname{arsinh}\left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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