Optimal. Leaf size=157 \[ \frac{8 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{8 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
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Rubi [A] time = 0.495435, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5668, 5775, 5666, 3307, 2180, 2204, 2205, 5676} \[ \frac{8 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{8 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5668
Rule 5775
Rule 5666
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5676
Rubi steps
\begin{align*} \int \frac{x}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{2 \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac{1}{5} (4 a) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{16}{15} \int \frac{x}{\cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^2}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{32 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{8 \sqrt{2 \pi } \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{8 \sqrt{2 \pi } \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}\\ \end{align*}
Mathematica [A] time = 0.285803, size = 91, normalized size = 0.58 \[ -\frac{-8 \sqrt{2 \pi } \left (\text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+\text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )\right )+\frac{4 \cosh \left (2 \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^{3/2}}+\frac{\left (16 \cosh ^{-1}(a x)^2+3\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^{5/2}}}{15 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.118, size = 153, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{15\,\sqrt{\pi }{a}^{2} \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}} \left ( -16\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}xa-4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }{x}^{2}{a}^{2}-3\,\sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}xa+8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) +8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\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{7}{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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