Optimal. Leaf size=237 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac{8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{a x-1} \sqrt{a x+1}}{15 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}-\frac{24 x^2 \sqrt{a x-1} \sqrt{a x+1}}{5 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
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Rubi [A] time = 0.847696, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5668, 5775, 5666, 3307, 2180, 2204, 2205, 5656, 5781} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac{8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{a x-1} \sqrt{a x+1}}{15 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}-\frac{24 x^2 \sqrt{a x-1} \sqrt{a x+1}}{5 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5668
Rule 5775
Rule 5666
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5656
Rule 5781
Rubi steps
\begin{align*} \int \frac{x^2}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac{1}{5} (6 a) \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac{12}{5} \int \frac{x^2}{\cosh ^{-1}(a x)^{3/2}} \, dx-\frac{8 \int \frac{1}{\cosh ^{-1}(a x)^{3/2}} \, dx}{15 a^2}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{-1+a x} \sqrt{1+a x}}{15 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \sqrt{\cosh ^{-1}(a x)}}-\frac{24 \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^3}-\frac{16 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx}{15 a}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{-1+a x} \sqrt{1+a x}}{15 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \sqrt{\cosh ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^3}+\frac{6 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}+\frac{18 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{-1+a x} \sqrt{1+a x}}{15 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \sqrt{\cosh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^3}-\frac{8 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}+\frac{9 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}+\frac{9 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{-1+a x} \sqrt{1+a x}}{15 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \sqrt{\cosh ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^3}-\frac{16 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac{6 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac{6 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac{18 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac{18 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{-1+a x} \sqrt{1+a x}}{15 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{24 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \sqrt{\cosh ^{-1}(a x)}}+\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac{3 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac{3 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{5 a^3}\\ \end{align*}
Mathematica [A] time = 0.795736, size = 286, normalized size = 1.21 \[ \frac{e^{-3 \cosh ^{-1}(a x)} \left (-e^{2 \cosh ^{-1}(a x)} \left (-2 e^{\cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )+2 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )+2 e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-2 \cosh ^{-1}(a x)^2+3 \sqrt{\frac{a x-1}{a x+1}} (a x+1) e^{\cosh ^{-1}(a x)}+e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)\right )-3 \left (-6 \sqrt{3} e^{3 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-3 \cosh ^{-1}(a x)\right )+6 \sqrt{3} e^{3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},3 \cosh ^{-1}(a x)\right )+6 e^{6 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-6 \cosh ^{-1}(a x)^2+e^{6 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)+e^{3 \cosh ^{-1}(a x)} \sinh \left (3 \cosh ^{-1}(a x)\right )\right )\right )}{30 a^3 \cosh ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \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^{2}}{\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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