Optimal. Leaf size=93 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^2}-\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^2}-\frac{\sqrt{\cosh ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.350346, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^2}-\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^2}-\frac{\sqrt{\cosh ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\cosh ^{-1}(a x)} \]
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 \sqrt{\cosh ^{-1}(a x)} \, dx &=\frac{1}{2} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{1}{4} a \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=\frac{1}{2} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^2}\\ &=\frac{1}{2} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^2}\\ &=-\frac{\sqrt{\cosh ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^2}\\ &=-\frac{\sqrt{\cosh ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^2}-\frac{\operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac{\sqrt{\cosh ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^2}-\frac{\operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^2}\\ &=-\frac{\sqrt{\cosh ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^2}-\frac{\sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^2}\\ \end{align*}
Mathematica [A] time = 0.0746745, size = 65, normalized size = 0.7 \[ \frac{8 \sqrt{\cosh ^{-1}(a x)} \cosh \left (2 \cosh ^{-1}(a x)\right )-\sqrt{2 \pi } \left (\text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+\text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )\right )}{32 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.089, size = 73, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}}{32\,\sqrt{\pi }{a}^{2}} \left ( -8\,\sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }{x}^{2}{a}^{2}+4\,\sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }+\pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) +\pi \,{\it erfi} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) \right ) } \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 \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 \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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