Optimal. Leaf size=300 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{16 x^3}{15 a^2 \cosh ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{a x-1} \sqrt{a x+1}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^5}{3 \cosh ^{-1}(a x)^{3/2}}-\frac{40 x^4 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^4 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
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Rubi [A] time = 0.931954, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5668, 5775, 5666, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{16 x^3}{15 a^2 \cosh ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{a x-1} \sqrt{a x+1}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^5}{3 \cosh ^{-1}(a x)^{3/2}}-\frac{40 x^4 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^4 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5668
Rule 5775
Rule 5666
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x^4}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{8 \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{5 a}+(2 a) \int \frac{x^5}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{16 x^3}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \cosh ^{-1}(a x)^{3/2}}+\frac{20}{3} \int \frac{x^4}{\cosh ^{-1}(a x)^{3/2}} \, dx-\frac{16 \int \frac{x^2}{\cosh ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{16 x^3}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \cosh ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \sqrt{\cosh ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{x}}-\frac{3 \cosh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^5}-\frac{40 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{8 \sqrt{x}}-\frac{9 \cosh (3 x)}{16 \sqrt{x}}-\frac{5 \cosh (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{16 x^3}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \cosh ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \sqrt{\cosh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^5}-\frac{24 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^5}+\frac{15 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{16 x^3}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \cosh ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \sqrt{\cosh ^{-1}(a x)}}-\frac{4 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}-\frac{12 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^5}-\frac{12 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^5}+\frac{15 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^5}+\frac{15 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{16 x^3}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \cosh ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \sqrt{\cosh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^5}-\frac{8 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^5}+\frac{5 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^5}+\frac{5 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^5}-\frac{24 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^5}-\frac{24 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^5}+\frac{15 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{2 a^5}+\frac{15 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{2 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{16 x^3}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \cosh ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \sqrt{\cosh ^{-1}(a x)}}+\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}\\ \end{align*}
Mathematica [A] time = 1.98824, size = 374, normalized size = 1.25 \[ \frac{4 \left (-\cosh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )-4 \cosh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )-5 \cosh ^{-1}(a x) \left (10 \sqrt{5} \left (-\cosh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-5 \cosh ^{-1}(a x)\right )+10 \sqrt{5} \cosh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},5 \cosh ^{-1}(a x)\right )+e^{-5 \cosh ^{-1}(a x)} \left (1-10 \cosh ^{-1}(a x)\right )+e^{5 \cosh ^{-1}(a x)} \left (10 \cosh ^{-1}(a x)+1\right )\right )-9 e^{-3 \cosh ^{-1}(a x)} \left (-6 \sqrt{3} e^{3 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-3 \cosh ^{-1}(a x)\right )+6 \sqrt{3} e^{3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},3 \cosh ^{-1}(a x)\right )+6 e^{6 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-6 \cosh ^{-1}(a x)^2+e^{6 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)+e^{3 \cosh ^{-1}(a x)} \sinh \left (3 \cosh ^{-1}(a x)\right )\right )-6 \sqrt{\frac{a x-1}{a x+1}} (a x+1)+4 e^{-\cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-4 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-2 e^{-\cosh ^{-1}(a x)} \cosh ^{-1}(a x)-2 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)-3 \sinh \left (5 \cosh ^{-1}(a x)\right )}{120 a^5 \cosh ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.172, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{7}{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 \frac{x^{4}}{\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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