Optimal. Leaf size=123 \[ -\frac{2 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac{2 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac{4}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 x^2}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.482776, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {5668, 5775, 5670, 5448, 12, 3308, 2180, 2204, 2205, 5676} \[ -\frac{2 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac{2 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac{4}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 x^2}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5668
Rule 5775
Rule 5670
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5676
Rubi steps
\begin{align*} \int \frac{x}{\cosh ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac{2 \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx}{3 a}+\frac{1}{3} (4 a) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{4}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 x^2}{3 \sqrt{\cosh ^{-1}(a x)}}+\frac{16}{3} \int \frac{x}{\sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{4}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 x^2}{3 \sqrt{\cosh ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{4}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 x^2}{3 \sqrt{\cosh ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{4}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 x^2}{3 \sqrt{\cosh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{4}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 x^2}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^2}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{4}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 x^2}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac{8 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{4}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{8 x^2}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \sqrt{2 \pi } \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a^2}+\frac{2 \sqrt{2 \pi } \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a^2}\\ \end{align*}
Mathematica [A] time = 0.255095, size = 83, normalized size = 0.67 \[ -\frac{2 \sqrt{2 \pi } \left (\text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )-\text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )\right )+\frac{4 \cosh \left (2 \cosh ^{-1}(a x)\right )}{\sqrt{\cosh ^{-1}(a x)}}+\frac{\sinh \left (2 \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^{3/2}}}{3 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.113, size = 122, normalized size = 1. \begin{align*} -{\frac{\sqrt{2}}{3\,\sqrt{\pi }{a}^{2} \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}} \left ( 4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }{x}^{2}{a}^{2}+\sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}xa+2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) -2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\pi \,{\it erfi} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) -2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi } \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 \frac{x}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\operatorname{acosh}^{\frac{5}{2}}{\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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