Optimal. Leaf size=220 \[ -\frac{15 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{64 a^3}-\frac{5 \sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{576 a^3}-\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{64 a^3}-\frac{5 \sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{576 a^3}+\frac{5 x \sqrt{\cosh ^{-1}(a x)}}{6 a^2}-\frac{5 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{9 a^3}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}+\frac{5}{36} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{18 a} \]
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Rubi [A] time = 1.05463, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5664, 5759, 5718, 5654, 5781, 3307, 2180, 2204, 2205, 3312} \[ -\frac{15 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{64 a^3}-\frac{5 \sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{576 a^3}-\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{64 a^3}-\frac{5 \sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{576 a^3}+\frac{5 x \sqrt{\cosh ^{-1}(a x)}}{6 a^2}-\frac{5 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{9 a^3}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}+\frac{5}{36} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{18 a} \]
Antiderivative was successfully verified.
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Rule 5664
Rule 5759
Rule 5718
Rule 5654
Rule 5781
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 3312
Rubi steps
\begin{align*} \int x^2 \cosh ^{-1}(a x)^{5/2} \, dx &=\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}-\frac{1}{6} (5 a) \int \frac{x^3 \cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{5 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}+\frac{5}{12} \int x^2 \sqrt{\cosh ^{-1}(a x)} \, dx-\frac{5 \int \frac{x \cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a}\\ &=\frac{5}{36} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}+\frac{5 \int \sqrt{\cosh ^{-1}(a x)} \, dx}{6 a^2}-\frac{1}{72} (5 a) \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=\frac{5 x \sqrt{\cosh ^{-1}(a x)}}{6 a^2}+\frac{5}{36} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}-\frac{5 \operatorname{Subst}\left (\int \frac{\cosh ^3(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{72 a^3}-\frac{5 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx}{12 a}\\ &=\frac{5 x \sqrt{\cosh ^{-1}(a x)}}{6 a^2}+\frac{5}{36} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}-\frac{5 \operatorname{Subst}\left (\int \left (\frac{3 \cosh (x)}{4 \sqrt{x}}+\frac{\cosh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{72 a^3}-\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^3}\\ &=\frac{5 x \sqrt{\cosh ^{-1}(a x)}}{6 a^2}+\frac{5}{36} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}-\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{288 a^3}-\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^3}-\frac{5 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{24 a^3}-\frac{5 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{24 a^3}\\ &=\frac{5 x \sqrt{\cosh ^{-1}(a x)}}{6 a^2}+\frac{5}{36} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}-\frac{5 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{576 a^3}-\frac{5 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{576 a^3}-\frac{5 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{192 a^3}-\frac{5 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{192 a^3}-\frac{5 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^3}-\frac{5 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^3}\\ &=\frac{5 x \sqrt{\cosh ^{-1}(a x)}}{6 a^2}+\frac{5}{36} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}-\frac{5 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{24 a^3}-\frac{5 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{24 a^3}-\frac{5 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{288 a^3}-\frac{5 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{288 a^3}-\frac{5 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{96 a^3}-\frac{5 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{96 a^3}\\ &=\frac{5 x \sqrt{\cosh ^{-1}(a x)}}{6 a^2}+\frac{5}{36} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)^{5/2}-\frac{15 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{64 a^3}-\frac{5 \sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{576 a^3}-\frac{15 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{64 a^3}-\frac{5 \sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{576 a^3}\\ \end{align*}
Mathematica [A] time = 0.0818516, size = 100, normalized size = 0.45 \[ \frac{\sqrt{3} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-3 \cosh ^{-1}(a x)\right )+81 \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-\cosh ^{-1}(a x)\right )+\sqrt{-\cosh ^{-1}(a x)} \left (81 \text{Gamma}\left (\frac{7}{2},\cosh ^{-1}(a x)\right )+\sqrt{3} \text{Gamma}\left (\frac{7}{2},3 \cosh ^{-1}(a x)\right )\right )}{648 a^3 \sqrt{-\cosh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\rm arccosh} \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 x^{2} \operatorname{arcosh}\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(-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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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