Optimal. Leaf size=607 \[ -\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {a x-1} \sqrt {a x+1} \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (a x+1) \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \cosh ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \cosh ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 1.08, antiderivative size = 637, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {5713, 5691, 5688, 5715, 3716, 2190, 2531, 2282, 6589, 5716, 260, 261} \[ -\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (a x+1) \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (a x+1)^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (a x+1) \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 261
Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 5688
Rule 5691
Rule 5713
Rule 5715
Rule 5716
Rule 6589
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [C] time = 1.85, size = 363, normalized size = 0.60 \[ -\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \left (\frac {3}{1-a^2 x^2}+\frac {24 \cosh ^{-1}(a x)^2}{a^2 x^2-1}-\frac {9 \cosh ^{-1}(a x)^2}{\left (a^2 x^2-1\right )^2}+96 \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )-48 \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )-60 \log \left (\sqrt {\frac {a x-1}{a x+1}} (a x+1)\right )+\frac {16 a x \left (\frac {a x-1}{a x+1}\right )^{3/2} \cosh ^{-1}(a x)^3}{(a x-1)^3}-\frac {32 a x \sqrt {\frac {a x-1}{a x+1}} \cosh ^{-1}(a x)^3}{a x-1}-\frac {12 a x \sqrt {\frac {a x-1}{a x+1}} \cosh ^{-1}(a x)^3}{(a x-1)^3 (a x+1)^2}-32 \cosh ^{-1}(a x)^3-\frac {6 a x \left (\frac {a x-1}{a x+1}\right )^{3/2} \cosh ^{-1}(a x)}{(a x-1)^3}+\frac {60 a x \sqrt {\frac {a x-1}{a x+1}} \cosh ^{-1}(a x)}{a x-1}+96 \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )+4 i \pi ^3\right )}{60 a c^3 \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{3}}{a^{8} c^{4} x^{8} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{2} c^{4} x^{2} + c^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.59, size = 1319, normalized size = 2.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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