Optimal. Leaf size=163 \[ \frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]
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Rubi [A] time = 0.29, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5689, 5718, 207, 5694, 4182, 2531, 2282, 6589} \[ \frac {\cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 207
Rule 2282
Rule 2531
Rule 4182
Rule 5689
Rule 5694
Rule 5718
Rule 6589
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {a \int \frac {x \cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c^2}+\frac {\int \frac {\cosh ^{-1}(a x)^2}{c-a^2 c x^2} \, dx}{2 c}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{c^2}-\frac {\operatorname {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\operatorname {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}-\frac {\operatorname {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\operatorname {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}+\frac {\operatorname {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.94, size = 191, normalized size = 1.17 \[ \frac {-8 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{-\cosh ^{-1}(a x)}\right )+8 \cosh ^{-1}(a x) \text {Li}_2\left (e^{-\cosh ^{-1}(a x)}\right )-8 \text {Li}_3\left (-e^{-\cosh ^{-1}(a x)}\right )+8 \text {Li}_3\left (e^{-\cosh ^{-1}(a x)}\right )-4 \cosh ^{-1}(a x)^2 \log \left (1-e^{-\cosh ^{-1}(a x)}\right )+4 \cosh ^{-1}(a x)^2 \log \left (e^{-\cosh ^{-1}(a x)}+1\right )+4 \cosh ^{-1}(a x) \tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )-4 \cosh ^{-1}(a x) \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )+\cosh ^{-1}(a x)^2 \left (-\text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )-\cosh ^{-1}(a x)^2 \text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )+8 \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )}{8 a c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )^{2}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 288, normalized size = 1.77 \[ -\frac {\mathrm {arccosh}\left (a x \right )^{2} x}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{a \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 a \,c^{2}}-\frac {\mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}+\frac {\polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}+\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 a \,c^{2}}+\frac {\mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}-\frac {\polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}-\frac {2 \arctanh \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}} \]
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 \[ -\frac {{\left (2 \, a x - {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{4 \, {\left (a^{3} c^{2} x^{2} - a c^{2}\right )}} - \int -\frac {{\left (2 \, a^{3} x^{3} + {\left (2 \, a^{2} x^{2} - {\left (a^{3} x^{3} - a x\right )} \log \left (a x + 1\right ) + {\left (a^{3} x^{3} - a x\right )} \log \left (a x - 1\right )\right )} \sqrt {a x + 1} \sqrt {a x - 1} - 2 \, a x - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{2 \, {\left (a^{5} c^{2} x^{5} - 2 \, a^{3} c^{2} x^{3} + a c^{2} x + {\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]
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 \[ \frac {\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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