Optimal. Leaf size=429 \[ -\frac{8 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{30 c^3 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}-\frac{16 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.671582, antiderivative size = 459, normalized size of antiderivative = 1.07, number of steps used = 14, 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, 2279, 2391, 5716, 39, 40} \[ -\frac{8 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{30 c^3 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^3 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c^3 (1-a x)^2 (a x+1)^2 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}-\frac{16 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5713
Rule 5691
Rule 5688
Rule 5715
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rule 5716
Rule 39
Rule 40
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{(-1+a x)^{7/2} (1+a x)^{7/2}} \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^2}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{5 c^3 \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)}{\left (-1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^3 (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{1}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{10 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (8 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (8 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)}{\left (-1+a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x}{30 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^3 (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{1}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (16 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)}{1-a^2 x^2} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{30 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (16 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\cosh ^{-1}(a x)\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{30 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (32 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{30 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{16 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (16 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{30 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{16 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (8 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{30 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{16 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.41474, size = 220, normalized size = 0.51 \[ -\frac{-16 \sqrt{\frac{a x-1}{a x+1}} (a x+1) \text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(a x)}\right )+a x \left (\frac{1}{1-a^2 x^2}+10\right )+2 \left (a x \left (\frac{4}{a^2 x^2-1}-\frac{3}{\left (a^2 x^2-1\right )^2}+8 \sqrt{\frac{a x-1}{a x+1}}-8\right )+8 \sqrt{\frac{a x-1}{a x+1}}\right ) \cosh ^{-1}(a x)^2+\frac{\left (\frac{a x-1}{a x+1}\right )^{3/2} \cosh ^{-1}(a x) \left (8 a^2 x^2+32 \left (a^2 x^2-1\right )^2 \log \left (1-e^{-2 \cosh ^{-1}(a x)}\right )-11\right )}{(a x-1)^3}}{30 a c^3 \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.255, size = 794, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )^{2}}{a^{8} c^{4} x^{8} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{2} c^{4} x^{2} + c^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]