Optimal. Leaf size=120 \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}+\frac{1}{3} x^3 \sqrt{\cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.397389, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}+\frac{1}{3} x^3 \sqrt{\cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5664
Rule 5781
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x^2 \sqrt{\cosh ^{-1}(a x)} \, dx &=\frac{1}{3} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{1}{6} a \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=\frac{1}{3} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cosh ^3(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{\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 )}{6 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{24 a^3}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{48 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^{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 )}{16 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{24 a^3}-\frac{\operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{24 a^3}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^3}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\cosh ^{-1}(a x)}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}-\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{48 a^3}\\ \end{align*}
Mathematica [A] time = 0.0830565, size = 100, normalized size = 0.83 \[ \frac{\sqrt{3} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-3 \cosh ^{-1}(a x)\right )+9 \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-\cosh ^{-1}(a x)\right )+\sqrt{-\cosh ^{-1}(a x)} \left (9 \text{Gamma}\left (\frac{3}{2},\cosh ^{-1}(a x)\right )+\sqrt{3} \text{Gamma}\left (\frac{3}{2},3 \cosh ^{-1}(a x)\right )\right )}{72 a^3 \sqrt{-\cosh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sqrt{{\rm arccosh} \left (ax\right )}\, 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} \sqrt{\operatorname{arcosh}\left (a x\right )}\,{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} \sqrt{\operatorname{acosh}{\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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