Optimal. Leaf size=387 \[ \frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3}+\frac {3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac {9 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}+\frac {5 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}-\frac {9 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \cosh ^{-1}(a x) \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \text {PolyLog}\left (4,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {9 \text {PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3} \]
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Rubi [A]
time = 0.58, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps
used = 30, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5901, 5903,
4267, 2611, 6744, 2320, 6724, 5915, 5889, 2317, 2438, 75} \begin {gather*} \frac {3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac {5 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac {9 \text {Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {9 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {1}{4 a c^3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\cosh ^{-1}(a x)^2}{4 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4267
Rule 5889
Rule 5901
Rule 5903
Rule 5915
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {(3 a) \int \frac {x \cosh ^{-1}(a x)^2}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{4 c^3}+\frac {3 \int \frac {\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac {\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}+\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {\int \frac {\cosh ^{-1}(a x)}{\left (-1+a^2 x^2\right )^2} \, dx}{2 c^3}+\frac {(9 a) \int \frac {x \cosh ^{-1}(a x)^2}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{8 c^3}+\frac {3 \int \frac {\cosh ^{-1}(a x)^3}{c-a^2 c x^2} \, dx}{8 c^2}\\ &=-\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac {\int \frac {\cosh ^{-1}(a x)}{-1+a^2 x^2} \, dx}{4 c^3}+\frac {9 \int \frac {\cosh ^{-1}(a x)}{-1+a^2 x^2} \, dx}{4 c^3}-\frac {3 \text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}-\frac {a \int \frac {x}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{4 c^3}\\ &=\frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {\text {Subst}\left (\int x \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}-\frac {9 \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}+\frac {9 \text {Subst}\left (\int x \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=\frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3}+\frac {3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {\text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac {\text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int x \text {Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int x \text {Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=\frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3}+\frac {3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \text {Li}_3\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \text {Li}_3\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=\frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3}+\frac {3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}+\frac {5 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}-\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}\\ &=\frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3}+\frac {3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}+\frac {5 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}-\frac {9 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \text {Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {9 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}\\ \end {align*}
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Mathematica [A]
time = 7.31, size = 455, normalized size = 1.18 \begin {gather*} -\frac {3 \pi ^4-6 \cosh ^{-1}(a x)^4-8 \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )+40 \cosh ^{-1}(a x)^2 \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )-4 \cosh ^{-1}(a x) \text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )+6 \cosh ^{-1}(a x)^3 \text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x)^2 \text {csch}^4\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-\cosh ^{-1}(a x)^3 \text {csch}^4\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-160 \cosh ^{-1}(a x) \log \left (1-e^{-\cosh ^{-1}(a x)}\right )+160 \cosh ^{-1}(a x) \log \left (1+e^{-\cosh ^{-1}(a x)}\right )-24 \cosh ^{-1}(a x)^3 \log \left (1+e^{-\cosh ^{-1}(a x)}\right )+24 \cosh ^{-1}(a x)^3 \log \left (1-e^{\cosh ^{-1}(a x)}\right )+8 \left (-20+9 \cosh ^{-1}(a x)^2\right ) \text {PolyLog}\left (2,-e^{-\cosh ^{-1}(a x)}\right )+160 \text {PolyLog}\left (2,e^{-\cosh ^{-1}(a x)}\right )+72 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )+144 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{-\cosh ^{-1}(a x)}\right )-144 \cosh ^{-1}(a x) \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )+144 \text {PolyLog}\left (4,-e^{-\cosh ^{-1}(a x)}\right )+144 \text {PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )-4 \cosh ^{-1}(a x) \text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )+6 \cosh ^{-1}(a x)^3 \text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )+\cosh ^{-1}(a x)^3 \text {sech}^4\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-\frac {16 \cosh ^{-1}(a x)^2 \sinh ^4\left (\frac {1}{2} \cosh ^{-1}(a x)\right )}{\left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^3}+8 \tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )-40 \cosh ^{-1}(a x)^2 \tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )}{64 a c^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 4.28, size = 527, normalized size = 1.36
method | result | size |
derivativedivides | \(\frac {-\frac {9 x^{2} a^{2} \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+3 x^{3} a^{3} \mathrm {arccosh}\left (a x \right )^{3}-11 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-5 \mathrm {arccosh}\left (a x \right )^{3} x a -2 \,\mathrm {arccosh}\left (a x \right ) x^{3} a^{3}+2 \sqrt {a x -1}\, \sqrt {a x +1}+2 a x \,\mathrm {arccosh}\left (a x \right )}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}-\frac {5 \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {3 \mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {9 \polylog \left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \mathrm {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {9 \polylog \left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\) | \(527\) |
default | \(\frac {-\frac {9 x^{2} a^{2} \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+3 x^{3} a^{3} \mathrm {arccosh}\left (a x \right )^{3}-11 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-5 \mathrm {arccosh}\left (a x \right )^{3} x a -2 \,\mathrm {arccosh}\left (a x \right ) x^{3} a^{3}+2 \sqrt {a x -1}\, \sqrt {a x +1}+2 a x \,\mathrm {arccosh}\left (a x \right )}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}-\frac {5 \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {3 \mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {9 \polylog \left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \mathrm {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {9 \polylog \left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\) | \(527\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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.
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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 )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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