Optimal. Leaf size=330 \[ \frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{a x-1} \sqrt{a x+1}}+\frac{5 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.707116, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5713, 5683, 5676, 5664, 5759, 5670, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{a x-1} \sqrt{a x+1}}+\frac{5 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5683
Rule 5676
Rule 5664
Rule 5759
Rule 5670
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\sqrt{c-a^2 c x^2} \int \frac{\cosh ^{-1}(a x)^{5/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (5 a \sqrt{c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^{3/2} \, dx}{4 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 a^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2 \sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{16 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{32 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 a \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{64 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{5 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{5 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{5 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 \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 )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{5 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 \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 )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 \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 )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{5 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{5 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.476998, size = 148, normalized size = 0.45 \[ -\frac{\sqrt{-c (a x-1) (a x+1)} \left (-105 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+105 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+8 \sqrt{\cosh ^{-1}(a x)} \left (64 \cosh ^{-1}(a x)^3+140 \cosh \left (2 \cosh ^{-1}(a x)\right ) \cosh ^{-1}(a x)-7 \left (16 \cosh ^{-1}(a x)^2+15\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )\right )}{3584 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.51, size = 0, normalized size = 0. \begin{align*} \int \sqrt{-{a}^{2}c{x}^{2}+c} \left ({\rm arccosh} \left (ax\right ) \right ) ^{{\frac{5}{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} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{\frac{5}{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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