Optimal. Leaf size=107 \[ \frac{\sqrt{\frac{\pi }{2}} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^2}-\frac{\sqrt{\frac{\pi }{2}} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^2} \]
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Rubi [A] time = 0.184953, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5670, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^2}-\frac{\sqrt{\frac{\pi }{2}} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^2} \]
Antiderivative was successfully verified.
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Rule 5670
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^2}+\frac{\operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^2}\\ &=-\frac{\operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{2 b c^2}+\frac{\operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{2 b c^2}\\ &=-\frac{e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^2}+\frac{e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^2}\\ \end{align*}
Mathematica [A] time = 0.219912, size = 104, normalized size = 0.97 \[ -\frac{\sqrt{\frac{\pi }{2}} \left (\left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )+\left (\sinh \left (\frac{2 a}{b}\right )-\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )\right )}{4 \sqrt{b} c^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt{a+b{\rm arccosh} \left (cx\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 \frac{x}{\sqrt{b \operatorname{arcosh}\left (c x\right ) + a}}\,{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 \frac{x}{\sqrt{a + b \operatorname{acosh}{\left (c 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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