Optimal. Leaf size=143 \[ \frac{\sqrt{\pi } \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^4}+\frac{\sqrt{\pi } \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^4}-\frac{2 x^3 \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{\cosh ^{-1}(a x)}} \]
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Rubi [A] time = 0.128221, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5666, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^4}+\frac{\sqrt{\pi } \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^4}-\frac{2 x^3 \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{\cosh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 5666
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x^3}{\cosh ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \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 )}{a^4}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^4}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^4}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^4}+\frac{\operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^4}+\frac{\operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^4}+\frac{\operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^4}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^4}+\frac{\operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^4}+\frac{\operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^4}+\frac{\operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^4}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}+\frac{\sqrt{\pi } \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^4}+\frac{\sqrt{\pi } \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a^4}+\frac{\sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.159266, size = 124, normalized size = 0.87 \[ -\frac{-\sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 \cosh ^{-1}(a x)\right )-\sqrt{2} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 \cosh ^{-1}(a x)\right )+\sqrt{2} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 \cosh ^{-1}(a x)\right )+\sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 \cosh ^{-1}(a x)\right )+2 \sinh \left (2 \cosh ^{-1}(a x)\right )+\sinh \left (4 \cosh ^{-1}(a x)\right )}{4 a^4 \sqrt{\cosh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{3}{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{3}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\operatorname{acosh}^{\frac{3}{2}}{\left (a x \right )}}\, dx \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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