Optimal. Leaf size=53 \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{4 a}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{4 a}+x \sqrt{\cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.218892, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5654, 5781, 3307, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{4 a}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{4 a}+x \sqrt{\cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5654
Rule 5781
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sqrt{\cosh ^{-1}(a x)} \, dx &=x \sqrt{\cosh ^{-1}(a x)}-\frac{1}{2} a \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=x \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a}\\ &=x \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}\\ &=x \sqrt{\cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{2 a}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{2 a}\\ &=x \sqrt{\cosh ^{-1}(a x)}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{4 a}-\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0355784, size = 45, normalized size = 0.85 \[ \frac{\frac{\sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-\cosh ^{-1}(a x)\right )}{\sqrt{-\cosh ^{-1}(a x)}}+\text{Gamma}\left (\frac{3}{2},\cosh ^{-1}(a x)\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.074, size = 41, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,\sqrt{\pi }a} \left ( -4\,\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }xa+\pi \,{\it Erf} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) +\pi \,{\it erfi} \left ( \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 \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 \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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