Optimal. Leaf size=122 \[ \frac{4 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}+\frac{4 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
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Rubi [A] time = 0.443552, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {5656, 5775, 5781, 3307, 2180, 2204, 2205} \[ \frac{4 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}+\frac{4 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5656
Rule 5775
Rule 5781
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{1}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{1}{5} (2 a) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{4}{15} \int \frac{1}{\cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{1}{15} (8 a) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a}\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a}+\frac{4 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a}\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}+\frac{8 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{4 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}+\frac{4 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}\\ \end{align*}
Mathematica [A] time = 0.181454, size = 147, normalized size = 1.2 \[ -\frac{2 e^{-\cosh ^{-1}(a x)} \left (-2 e^{\cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )+2 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )+2 e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-2 \cosh ^{-1}(a x)^2+3 \sqrt{\frac{a x-1}{a x+1}} (a x+1) e^{\cosh ^{-1}(a x)}+e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)\right )}{15 a \cosh ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.101, size = 111, normalized size = 0.9 \begin{align*}{\frac{2}{15\,\sqrt{\pi }a \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}} \left ( -4\,\sqrt{ax-1} \left ({\rm arccosh} \left (ax\right ) \right ) ^{5/2}\sqrt{ax+1}\sqrt{\pi }+2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\pi \,{\it Erf} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) +2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\pi \,{\it erfi} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) -2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{\pi }xa-3\,\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1} \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{1}{\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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