Optimal. Leaf size=99 \[ -\frac{15 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a}-\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a}+x \cosh ^{-1}(a x)^{5/2}-\frac{5 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{2 a}+\frac{15}{4} x \sqrt{\cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.395465, antiderivative size = 99, 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 = {5654, 5718, 5781, 3307, 2180, 2204, 2205} \[ -\frac{15 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a}-\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a}+x \cosh ^{-1}(a x)^{5/2}-\frac{5 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{2 a}+\frac{15}{4} x \sqrt{\cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5654
Rule 5718
Rule 5781
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \cosh ^{-1}(a x)^{5/2} \, dx &=x \cosh ^{-1}(a x)^{5/2}-\frac{1}{2} (5 a) \int \frac{x \cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{2 a}+x \cosh ^{-1}(a x)^{5/2}+\frac{15}{4} \int \sqrt{\cosh ^{-1}(a x)} \, dx\\ &=\frac{15}{4} x \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{2 a}+x \cosh ^{-1}(a x)^{5/2}-\frac{1}{8} (15 a) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=\frac{15}{4} x \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{2 a}+x \cosh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a}\\ &=\frac{15}{4} x \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{2 a}+x \cosh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a}-\frac{15 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a}\\ &=\frac{15}{4} x \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{2 a}+x \cosh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a}-\frac{15 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a}\\ &=\frac{15}{4} x \sqrt{\cosh ^{-1}(a x)}-\frac{5 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{2 a}+x \cosh ^{-1}(a x)^{5/2}-\frac{15 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a}-\frac{15 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{16 a}\\ \end{align*}
Mathematica [A] time = 0.035386, size = 45, normalized size = 0.45 \[ \frac{\frac{\sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-\cosh ^{-1}(a x)\right )}{\sqrt{-\cosh ^{-1}(a x)}}+\text{Gamma}\left (\frac{7}{2},\cosh ^{-1}(a x)\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.098, size = 81, normalized size = 0.8 \begin{align*} -{\frac{1}{16\,\sqrt{\pi }a} \left ( -16\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{5/2}\sqrt{\pi }xa+40\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}-60\,\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }xa+15\,\pi \,{\it Erf} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) +15\,\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 \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(-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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