Optimal. Leaf size=286 \[ \frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt{\cosh ^{-1}(a x)}} \]
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Rubi [A] time = 0.348836, antiderivative size = 295, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5713, 5697, 5780, 5448, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 c (a x+1)^{3/2} (1-a x)^2 \sqrt{c-a^2 c x^2}}{a \sqrt{a x-1} \sqrt{\cosh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5697
Rule 5780
Rule 5448
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\left (c-a^2 c x^2\right )^{3/2}}{\cosh ^{-1}(a x)^{3/2}} \, dx &=-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int \frac{(-1+a x)^{3/2} (1+a x)^{3/2}}{\cosh ^{-1}(a x)^{3/2}} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}-\frac{\left (8 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x \left (-1+a^2 x^2\right )}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}-\frac{\left (8 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}-\frac{\left (8 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sinh (2 x)}{4 \sqrt{x}}+\frac{\sinh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (2 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (2 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (2 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}+\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.467264, size = 239, normalized size = 0.84 \[ -\frac{c \sqrt{c-a^2 c x^2} e^{-4 \cosh ^{-1}(a x)} \left (2 e^{4 \cosh ^{-1}(a x)} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 \cosh ^{-1}(a x)\right )+2 e^{4 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 \cosh ^{-1}(a x)\right )+16 a^2 x^2 e^{4 \cosh ^{-1}(a x)}+4 \sqrt{2 \pi } e^{4 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )-4 \sqrt{2 \pi } e^{4 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )-14 e^{4 \cosh ^{-1}(a x)}-e^{8 \cosh ^{-1}(a x)}-1\right )}{8 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.319, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \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{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}{\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(-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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