Optimal. Leaf size=260 \[ -\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}+\frac {3 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {PolyLog}\left (4,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]
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Rubi [A]
time = 0.33, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5901, 5903,
4267, 2611, 6744, 2320, 6724, 5915, 5889, 2317, 2438} \begin {gather*} \frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \end {gather*}
Antiderivative was successfully verified.
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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 )^2} \, dx &=\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {(3 a) \int \frac {x \cosh ^{-1}(a x)^2}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{2 c^2}+\frac {\int \frac {\cosh ^{-1}(a x)^3}{c-a^2 c x^2} \, dx}{2 c}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {3 \int \frac {\cosh ^{-1}(a x)}{-1+a^2 x^2} \, dx}{c^2}-\frac {\text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}-\frac {3 \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}+\frac {3 \text {Subst}\left (\int x \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}-\frac {3 \text {Subst}\left (\int x \text {Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}+\frac {3 \text {Subst}\left (\int x \text {Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \text {Li}_3\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}-\frac {3 \text {Subst}\left (\int \text {Li}_3\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}+\frac {3 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}+\frac {3 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ \end {align*}
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Mathematica [A]
time = 1.43, size = 276, normalized size = 1.06 \begin {gather*} \frac {-\pi ^4+2 \cosh ^{-1}(a x)^4-12 \cosh ^{-1}(a x)^2 \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )-2 \cosh ^{-1}(a x)^3 \text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )+48 \cosh ^{-1}(a x) \log \left (1-e^{-\cosh ^{-1}(a x)}\right )-48 \cosh ^{-1}(a x) \log \left (1+e^{-\cosh ^{-1}(a x)}\right )+8 \cosh ^{-1}(a x)^3 \log \left (1+e^{-\cosh ^{-1}(a x)}\right )-8 \cosh ^{-1}(a x)^3 \log \left (1-e^{\cosh ^{-1}(a x)}\right )-24 \left (-2+\cosh ^{-1}(a x)^2\right ) \text {PolyLog}\left (2,-e^{-\cosh ^{-1}(a x)}\right )-48 \text {PolyLog}\left (2,e^{-\cosh ^{-1}(a x)}\right )-24 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )-48 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{-\cosh ^{-1}(a x)}\right )+48 \cosh ^{-1}(a x) \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )-48 \text {PolyLog}\left (4,-e^{-\cosh ^{-1}(a x)}\right )-48 \text {PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x)^3 \text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )+12 \cosh ^{-1}(a x)^2 \tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )}{16 a c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.91, size = 416, normalized size = 1.60
method | result | size |
derivativedivides | \(\frac {-\frac {\mathrm {arccosh}\left (a x \right )^{2} \left (a x \,\mathrm {arccosh}\left (a x \right )+3 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}+\frac {\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \polylog \left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \polylog \left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}}{a}\) | \(416\) |
default | \(\frac {-\frac {\mathrm {arccosh}\left (a x \right )^{2} \left (a x \,\mathrm {arccosh}\left (a x \right )+3 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}+\frac {\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \polylog \left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \polylog \left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}}{a}\) | \(416\) |
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^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \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 )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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