Optimal. Leaf size=316 \[ \frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3 a \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}} \]
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Rubi [A] time = 0.727819, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5713, 5683, 5676, 5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ \frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3 a \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5683
Rule 5676
Rule 5664
Rule 5781
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2} \, dx &=\frac{\sqrt{a^2-x^2} \int \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2} \, dx}{\sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{\sqrt{a^2-x^2} \int \frac{\cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx}{2 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{\left (3 \sqrt{a^2-x^2}\right ) \int x \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \, dx}{4 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (3 \sqrt{a^2-x^2}\right ) \int \frac{x^2}{\sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{16 a^2 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}\left (\frac{x}{a}\right )\right )}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}\left (\frac{x}{a}\right )\right )}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{3 a \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}\left (\frac{x}{a}\right )\right )}{32 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{3 a \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{3 a \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{32 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{32 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{3 a \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{3 a \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{3 a \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ \end{align*}
Mathematica [A] time = 0.363241, size = 144, normalized size = 0.46 \[ \frac{a^2 \sqrt{a^2-x^2} \left (15 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )+15 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )-8 \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \left (16 \cosh ^{-1}\left (\frac{x}{a}\right )^2+15 \cosh \left (2 \cosh ^{-1}\left (\frac{x}{a}\right )\right )-20 \cosh ^{-1}\left (\frac{x}{a}\right ) \sinh \left (2 \cosh ^{-1}\left (\frac{x}{a}\right )\right )\right )\right )}{640 \sqrt{\frac{x-a}{a+x}} (a+x)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.448, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm arccosh} \left ({\frac{x}{a}}\right ) \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}-{x}^{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 \sqrt{a^{2} - x^{2}} \operatorname{arcosh}\left (\frac{x}{a}\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(-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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