Optimal. Leaf size=244 \[ \frac{16 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{4 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{16 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{4 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{a x-1} \sqrt{a x+1}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{128 x^3 \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^3 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
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Rubi [A] time = 0.756945, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 21, 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} \[ \frac{16 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{4 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{16 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{4 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{a x-1} \sqrt{a x+1}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{128 x^3 \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^3 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
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^3}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{6 \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac{1}{5} (8 a) \int \frac{x^4}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{64}{15} \int \frac{x^3}{\cosh ^{-1}(a x)^{3/2}} \, dx-\frac{8 \int \frac{x}{\cosh ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{128 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^4}-\frac{128 \operatorname{Subst}\left (\int \left (-\frac{\cosh (2 x)}{2 \sqrt{x}}-\frac{\cosh (4 x)}{2 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{128 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^4}-\frac{8 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^4}+\frac{64 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac{64 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{128 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac{32 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac{32 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac{32 \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}-\frac{16 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^4}-\frac{16 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{5 a^4}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{128 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{4 \sqrt{2 \pi } \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{5 a^4}-\frac{4 \sqrt{2 \pi } \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{5 a^4}+\frac{64 \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{64 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{64 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{64 \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a^3 \sqrt{\cosh ^{-1}(a x)}}-\frac{128 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{16 \sqrt{\pi } \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{4 \sqrt{2 \pi } \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{16 \sqrt{\pi } \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac{4 \sqrt{2 \pi } \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^4}\\ \end{align*}
Mathematica [A] time = 0.555799, size = 291, normalized size = 1.19 \[ \frac{e^{-4 \cosh ^{-1}(a x)} \left (128 e^{4 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-4 \cosh ^{-1}(a x)\right )-8 e^{2 \cosh ^{-1}(a x)} \left (-4 \sqrt{2} e^{2 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-2 \cosh ^{-1}(a x)\right )+4 \sqrt{2} e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},2 \cosh ^{-1}(a x)\right )+4 e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-4 \cosh ^{-1}(a x)^2+3 a x \sqrt{\frac{a x-1}{a x+1}} (a x+1) e^{2 \cosh ^{-1}(a x)}+e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)\right )-128 e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},4 \cosh ^{-1}(a x)\right )-3 e^{8 \cosh ^{-1}(a x)}-64 e^{8 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2+64 \cosh ^{-1}(a x)^2-8 e^{8 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-8 \cosh ^{-1}(a x)+3\right )}{120 a^4 \cosh ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \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^{3}}{\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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