Optimal. Leaf size=189 \[ -\frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{32 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{96 a^3}+\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{32 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{96 a^3}-\frac{\sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{3 a^3}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{6 a} \]
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Rubi [A] time = 0.639637, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5664, 5759, 5718, 5658, 3308, 2180, 2204, 2205, 5670, 5448} \[ -\frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{32 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{96 a^3}+\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{32 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{96 a^3}-\frac{\sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{3 a^3}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{6 a} \]
Antiderivative was successfully verified.
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Rule 5664
Rule 5759
Rule 5718
Rule 5658
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5670
Rule 5448
Rubi steps
\begin{align*} \int x^2 \cosh ^{-1}(a x)^{3/2} \, dx &=\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac{1}{2} a \int \frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}+\frac{1}{12} \int \frac{x^2}{\sqrt{\cosh ^{-1}(a x)}} \, dx-\frac{\int \frac{x \sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 a}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}+\frac{\operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^3}+\frac{\int \frac{1}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{6 a^2}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}+\frac{\operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 \sqrt{x}}+\frac{\sinh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^3}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{48 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{48 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^3}+\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^3}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac{\operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^3}+\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^3}+\frac{\operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^3}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^3}+\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^3}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^3}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^3}-\frac{\operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}+\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}+\frac{\operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{32 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{96 a^3}+\frac{3 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{32 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{96 a^3}\\ \end{align*}
Mathematica [A] time = 0.0873498, size = 100, normalized size = 0.53 \[ \frac{\sqrt{3} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-3 \cosh ^{-1}(a x)\right )+27 \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-\cosh ^{-1}(a x)\right )+\sqrt{\cosh ^{-1}(a x)} \left (27 \text{Gamma}\left (\frac{5}{2},\cosh ^{-1}(a x)\right )+\sqrt{3} \text{Gamma}\left (\frac{5}{2},3 \cosh ^{-1}(a x)\right )\right )}{216 a^3 \sqrt{\cosh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.091, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\rm arccosh} \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^{2} \operatorname{arcosh}\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^{2} \operatorname{acosh}^{\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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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