Optimal. Leaf size=433 \[ \frac{3 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{3 \pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{3 \pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt{\cosh ^{-1}(a x)}} \]
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Rubi [A] time = 0.460933, antiderivative size = 444, normalized size of antiderivative = 1.03, number of steps used = 20, 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} \[ \frac{3 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{3 \pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{3 \pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 c^2 (a x+1)^{5/2} (1-a x)^3 \sqrt{c-a^2 c x^2}}{a \sqrt{a x-1} \sqrt{\cosh ^{-1}(a x)}} \]
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 )^{5/2}}{\cosh ^{-1}(a x)^{3/2}} \, dx &=\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{(-1+a x)^{5/2} (1+a x)^{5/2}}{\cosh ^{-1}(a x)^{3/2}} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}+\frac{\left (12 a c^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{x \left (-1+a^2 x^2\right )^2}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}+\frac{\left (12 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^5(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}+\frac{\left (12 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{5 \sinh (2 x)}{32 \sqrt{x}}-\frac{\sinh (4 x)}{8 \sqrt{x}}+\frac{\sinh (6 x)}{32 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}+\frac{\left (3 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (6 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c^2 \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 )}{2 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 c^2 \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 )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}-\frac{\left (3 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-6 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{6 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 c^2 \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 )}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c^2 \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 )}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 c^2 \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 )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 c^2 \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 )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}-\frac{\left (3 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-6 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{6 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{2 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{2 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt{c-a^2 c x^2}}{a \sqrt{-1+a x} \sqrt{\cosh ^{-1}(a x)}}+\frac{3 c^2 \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 c^2 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{c^2 \sqrt{\frac{3 \pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 c^2 \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{15 c^2 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c^2 \sqrt{\frac{3 \pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 1.19608, size = 411, normalized size = 0.95 \[ \frac{c^2 \sqrt{c-a^2 c x^2} e^{-6 \cosh ^{-1}(a x)} \left (\sqrt{6} e^{6 \cosh ^{-1}(a x)} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-6 \cosh ^{-1}(a x)\right )-12 e^{6 \cosh ^{-1}(a x)} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 \cosh ^{-1}(a x)\right )-\sqrt{2} e^{6 \cosh ^{-1}(a x)} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 \cosh ^{-1}(a x)\right )-\sqrt{2} e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 \cosh ^{-1}(a x)\right )-12 e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 \cosh ^{-1}(a x)\right )+\sqrt{6} e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},6 \cosh ^{-1}(a x)\right )-64 a^2 x^2 e^{6 \cosh ^{-1}(a x)}-16 \sqrt{2 \pi } e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+16 \sqrt{2 \pi } e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+6 e^{2 \cosh ^{-1}(a x)}+e^{4 \cosh ^{-1}(a x)}+52 e^{6 \cosh ^{-1}(a x)}+e^{8 \cosh ^{-1}(a x)}+6 e^{10 \cosh ^{-1}(a x)}-e^{12 \cosh ^{-1}(a x)}-1\right )}{32 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.322, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{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{5}{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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