Optimal. Leaf size=394 \[ -\frac{15 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac{\sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{Erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac{\sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{Erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{6400 a^5}+\frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{15 a^2}-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{15 a^3}+\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{5 a^4}-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}+\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{10 a} \]
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Rubi [A] time = 1.82197, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 44, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5664, 5759, 5718, 5654, 5781, 3307, 2180, 2204, 2205, 3312} \[ -\frac{15 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac{\sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{Erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac{\sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{Erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{6400 a^5}+\frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{15 a^2}-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{15 a^3}+\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{5 a^4}-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}+\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{10 a} \]
Antiderivative was successfully verified.
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Rule 5664
Rule 5759
Rule 5718
Rule 5654
Rule 5781
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 3312
Rubi steps
\begin{align*} \int x^4 \cosh ^{-1}(a x)^{5/2} \, dx &=\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac{1}{2} a \int \frac{x^5 \cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}+\frac{3}{20} \int x^4 \sqrt{\cosh ^{-1}(a x)} \, dx-\frac{2 \int \frac{x^3 \cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{5 a}\\ &=\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac{4 \int \frac{x \cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{15 a^3}+\frac{\int x^2 \sqrt{\cosh ^{-1}(a x)} \, dx}{5 a^2}-\frac{1}{200} (3 a) \int \frac{x^5}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=\frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh ^5(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{200 a^5}+\frac{2 \int \sqrt{\cosh ^{-1}(a x)} \, dx}{5 a^4}-\frac{\int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx}{30 a}\\ &=\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{5 a^4}+\frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac{3 \operatorname{Subst}\left (\int \left (\frac{5 \cosh (x)}{8 \sqrt{x}}+\frac{5 \cosh (3 x)}{16 \sqrt{x}}+\frac{\cosh (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{200 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cosh ^3(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{30 a^5}-\frac{\int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx}{5 a^3}\\ &=\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{5 a^4}+\frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3200 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{640 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{320 a^5}-\frac{\operatorname{Subst}\left (\int \left (\frac{3 \cosh (x)}{4 \sqrt{x}}+\frac{\cosh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{30 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^5}\\ &=\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{5 a^4}+\frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6400 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6400 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1280 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1280 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{640 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{640 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{120 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{40 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{10 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{10 a^5}\\ &=\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{5 a^4}+\frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac{3 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{3200 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{3200 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{240 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{240 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{640 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{640 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{320 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{320 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{80 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{80 a^5}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^5}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^5}\\ &=\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{5 a^4}+\frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac{67 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{640 a^5}-\frac{\sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac{67 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{640 a^5}-\frac{\sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac{\operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{120 a^5}-\frac{\operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{120 a^5}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{40 a^5}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{40 a^5}\\ &=\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{5 a^4}+\frac{x^3 \sqrt{\cosh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\cosh ^{-1}(a x)}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac{15 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac{\sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac{15 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac{\sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{6400 a^5}\\ \end{align*}
Mathematica [A] time = 0.0973618, size = 162, normalized size = 0.41 \[ \frac{27 \sqrt{5} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-5 \cosh ^{-1}(a x)\right )+625 \sqrt{3} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-3 \cosh ^{-1}(a x)\right )+33750 \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-\cosh ^{-1}(a x)\right )+33750 \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},\cosh ^{-1}(a x)\right )+625 \sqrt{3} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},3 \cosh ^{-1}(a x)\right )+27 \sqrt{5} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},5 \cosh ^{-1}(a x)\right )}{540000 a^5 \sqrt{-\cosh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ({\rm arccosh} \left (ax\right ) \right ) ^{{\frac{5}{2}}}\, dx \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^{4} \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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