Optimal. Leaf size=607 \[ -\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \text {PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.62, antiderivative size = 607, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used =
{5901, 5899, 5913, 3797, 2221, 2611, 2320, 6724, 5912, 5914, 5900, 266, 5902, 74, 267}
\begin {gather*} -\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {a x-1} \sqrt {a x+1} \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 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} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (a x+1) \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \cosh ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \cosh ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 74
Rule 266
Rule 267
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5899
Rule 5900
Rule 5901
Rule 5902
Rule 5912
Rule 5913
Rule 5914
Rule 6724
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{\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)^3}{(-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)^3}{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)^3}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (3 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 )^3} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{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)^3}{15 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{(-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)^3}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (4 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}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{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)^3}{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 {\cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{5 c^3 \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)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^2} \, dx}{10 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)^2}{1-a^2 x^2} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{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)^3}{15 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\left (8 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x^2 \coth (x) \, dx,x,\cosh ^{-1}(a x)\right )}{5 a c^3 \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}{5 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (4 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{1-a^2 x^2} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{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)^3}{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)^3}{15 a c^3 \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^3 \sqrt {c-a^2 c x^2}}+\frac {\left (16 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{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)^3}{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)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \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^3 \sqrt {c-a^2 c x^2}}+\frac {\left (16 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{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)^3}{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)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \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^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (8 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{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)^3}{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)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \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^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{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)^3}{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)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \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^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 1.33, size = 363, normalized size = 0.60 \begin {gather*} -\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (4 i \pi ^3+\frac {3}{1-a^2 x^2}+\frac {60 a x \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)}{-1+a x}-\frac {6 a x \left (\frac {-1+a x}{1+a x}\right )^{3/2} \cosh ^{-1}(a x)}{(-1+a x)^3}-\frac {9 \cosh ^{-1}(a x)^2}{\left (-1+a^2 x^2\right )^2}+\frac {24 \cosh ^{-1}(a x)^2}{-1+a^2 x^2}-32 \cosh ^{-1}(a x)^3-\frac {32 a x \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)^3}{-1+a x}+\frac {16 a x \left (\frac {-1+a x}{1+a x}\right )^{3/2} \cosh ^{-1}(a x)^3}{(-1+a x)^3}-\frac {12 a x \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)^3}{(-1+a x)^3 (1+a x)^2}+96 \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )-60 \log \left (\sqrt {\frac {-1+a x}{1+a x}} (1+a x)\right )+96 \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )-48 \text {PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )\right )}{60 a c^3 \sqrt {c-a^2 c x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1318\) vs.
\(2(564)=1128\).
time = 2.85, size = 1319, normalized size = 2.17
method | result | size |
default | \(\text {Expression too large to display}\) | \(1319\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________