Optimal. Leaf size=211 \[ \frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \]
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Rubi [A] time = 0.392167, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5713, 5683, 5676, 5670, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5683
Rule 5676
Rule 5670
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \, dx &=\frac{\sqrt{a^2-x^2} \int \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \, dx}{\sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}-\frac{\sqrt{a^2-x^2} \int \frac{\sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx}{2 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{\sqrt{a^2-x^2} \int \frac{x}{\sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{4 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{\left (a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}\left (\frac{x}{a}\right )\right )}{4 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{\left (a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\cosh ^{-1}\left (\frac{x}{a}\right )\right )}{4 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{\left (a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}\left (\frac{x}{a}\right )\right )}{8 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (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 )}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{\left (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 )}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{\left (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 )}{8 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{\left (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 )}{8 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ &=\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}+\frac{a \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}-\frac{a \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}}\\ \end{align*}
Mathematica [A] time = 0.137909, size = 121, normalized size = 0.57 \[ -\frac{a^2 \sqrt{a^2-x^2} \left (3 \sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},2 \cosh ^{-1}\left (\frac{x}{a}\right )\right )+3 \sqrt{2} \sqrt{-\cosh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-2 \cosh ^{-1}\left (\frac{x}{a}\right )\right )+16 \cosh ^{-1}\left (\frac{x}{a}\right )^2\right )}{48 \sqrt{\frac{x-a}{a+x}} (a+x) \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.47, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{a}^{2}-{x}^{2}}\sqrt{{\rm arccosh} \left ({\frac{x}{a}}\right )}\, 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}} \sqrt{\operatorname{arcosh}\left (\frac{x}{a}\right )}\,{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 \sqrt{- \left (- a + x\right ) \left (a + x\right )} \sqrt{\operatorname{acosh}{\left (\frac{x}{a} \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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