Optimal. Leaf size=164 \[ \frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {3 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {3 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5901, 5903,
4267, 2317, 2438, 75} \begin {gather*} \frac {3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {3 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {3}{8 a c^3 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 2317
Rule 2438
Rule 4267
Rule 5901
Rule 5903
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {a \int \frac {x}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{4 c^3}+\frac {3 \int \frac {\cosh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}+\frac {x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {(3 a) \int \frac {x}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{8 c^3}+\frac {3 \int \frac {\cosh ^{-1}(a x)}{c-a^2 c x^2} \, dx}{8 c^2}\\ &=\frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 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 )^2}+\frac {3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}-\frac {3 \text {Subst}\left (\int x \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}\\ &=\frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 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 )^2}+\frac {3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {3 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}-\frac {3 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}\\ &=\frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 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 )^2}+\frac {3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {3 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}\\ &=\frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 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 )^2}+\frac {3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {3 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac {3 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}\\ \end {align*}
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Mathematica [A]
time = 1.53, size = 223, normalized size = 1.36 \begin {gather*} \frac {-\frac {2 (-2+a x) \sqrt {1+a x}}{(-1+a x)^{3/2}}+\frac {2 \sqrt {-1+a x} (2+a x)}{(1+a x)^{3/2}}+\frac {6 \cosh ^{-1}(a x)}{(-1+a x)^2}-\frac {6 \cosh ^{-1}(a x)}{(1+a x)^2}+18 \left (-\frac {1}{\sqrt {\frac {-1+a x}{1+a x}}}+\frac {\cosh ^{-1}(a x)}{1-a x}\right )+18 \left (\sqrt {\frac {-1+a x}{1+a x}}-\frac {\cosh ^{-1}(a x)}{1+a x}\right )+9 \cosh ^{-1}(a x) \left (\cosh ^{-1}(a x)-4 \log \left (1-e^{\cosh ^{-1}(a x)}\right )\right )-9 \cosh ^{-1}(a x) \left (\cosh ^{-1}(a x)-4 \log \left (1+e^{\cosh ^{-1}(a x)}\right )\right )+36 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-36 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{96 a c^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 4.39, size = 205, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {-\frac {9 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+9 \,\mathrm {arccosh}\left (a x \right ) a^{3} x^{3}-11 \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \,\mathrm {arccosh}\left (a x \right )}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}}{a}\) | \(205\) |
default | \(\frac {-\frac {9 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+9 \,\mathrm {arccosh}\left (a x \right ) a^{3} x^{3}-11 \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \,\mathrm {arccosh}\left (a x \right )}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}}{a}\) | \(205\) |
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}{\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 = -0.01 \begin {gather*} \int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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