Optimal. Leaf size=413 \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.637823, antiderivative size = 428, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5713, 5691, 5688, 5715, 3716, 2190, 2531, 2282, 6589, 5716, 260} \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5691
Rule 5688
Rule 5715
Rule 3716
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 5716
Rule 260
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^2}{\left (-1+a^2 x^2\right )^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (2 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x \cosh ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \coth (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2 \sqrt{c-a^2 c x^2}}-\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{1-a^2 x^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x \cosh ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x \cosh ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x \cosh ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x \cosh ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x \cosh ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \text{Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.942065, size = 258, normalized size = 0.62 \[ \frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left (-24 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )+12 \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )+\frac{6 \cosh ^{-1}(a x)^2}{1-a^2 x^2}+12 \log \left (\sqrt{\frac{a x-1}{a x+1}} (a x+1)\right )-\frac{4 a x \left (\frac{a x-1}{a x+1}\right )^{3/2} \cosh ^{-1}(a x)^3}{(a x-1)^3}+\frac{8 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^3}{a x-1}+8 \cosh ^{-1}(a x)^3-\frac{12 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)}{a x-1}-24 \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )-i \pi ^3\right )}{12 a c^2 \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.315, size = 955, normalized size = 2.3 \begin{align*} \text{result too large to display} \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{\operatorname{arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{3}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, x\right ) \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 \frac{\operatorname{acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}}}\, 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*} \int \frac{\operatorname{arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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