Optimal. Leaf size=157 \[ -\frac{15 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac{15 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac{\cosh ^{-1}(a x)^{5/2}}{4 a^2}-\frac{15 \sqrt{\cosh ^{-1}(a x)}}{64 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}+\frac{15}{32} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{8 a} \]
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Rubi [A] time = 0.708933, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {5664, 5759, 5676, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac{15 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac{15 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac{\cosh ^{-1}(a x)^{5/2}}{4 a^2}-\frac{15 \sqrt{\cosh ^{-1}(a x)}}{64 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}+\frac{15}{32} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{8 a} \]
Antiderivative was successfully verified.
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Rule 5664
Rule 5759
Rule 5676
Rule 5781
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x \cosh ^{-1}(a x)^{5/2} \, dx &=\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac{1}{4} (5 a) \int \frac{x^2 \cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}+\frac{15}{16} \int x \sqrt{\cosh ^{-1}(a x)} \, dx-\frac{5 \int \frac{\cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 a}\\ &=\frac{15}{32} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac{\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac{1}{64} (15 a) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=\frac{15}{32} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac{\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^2}\\ &=\frac{15}{32} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac{\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^2}\\ &=-\frac{15 \sqrt{\cosh ^{-1}(a x)}}{64 a^2}+\frac{15}{32} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac{\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a^2}\\ &=-\frac{15 \sqrt{\cosh ^{-1}(a x)}}{64 a^2}+\frac{15}{32} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac{\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a^2}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a^2}\\ &=-\frac{15 \sqrt{\cosh ^{-1}(a x)}}{64 a^2}+\frac{15}{32} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac{\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{128 a^2}-\frac{15 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{128 a^2}\\ &=-\frac{15 \sqrt{\cosh ^{-1}(a x)}}{64 a^2}+\frac{15}{32} x^2 \sqrt{\cosh ^{-1}(a x)}-\frac{5 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{8 a}-\frac{\cosh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{5/2}-\frac{15 \sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a^2}-\frac{15 \sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a^2}\\ \end{align*}
Mathematica [A] time = 0.164887, size = 92, normalized size = 0.59 \[ \frac{-15 \sqrt{2 \pi } \left (\text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+\text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )\right )+8 \left (16 \cosh ^{-1}(a x)^2+15\right ) \cosh \left (2 \cosh ^{-1}(a x)\right ) \sqrt{\cosh ^{-1}(a x)}-160 \cosh ^{-1}(a x)^{3/2} \sinh \left (2 \cosh ^{-1}(a x)\right )}{512 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.115, size = 139, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}}{512\,\sqrt{\pi }{a}^{2}} \left ( -128\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }{x}^{2}{a}^{2}+160\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}xa-120\,\sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }{x}^{2}{a}^{2}+64\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }+60\,\sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }+15\,\pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) +15\,\pi \,{\it erfi} \left ( \sqrt{2}\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 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(-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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