Optimal. Leaf size=53 \[ \frac {\text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {\text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c} \]
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Rubi [A] time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5694, 4182, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {\text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4182
Rule 5694
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{c-a^2 c x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int x \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {\operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}-\frac {\operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {\text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {\text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 1.45 \[ \frac {\text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {\text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {\cosh ^{-1}(a x) \log \left (1-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {\cosh ^{-1}(a x) \log \left (e^{\cosh ^{-1}(a x)}+1\right )}{a c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {arcosh}\left (a x\right )}{a^{2} c x^{2} - c}, x\right ) \]
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 \[ \int -\frac {\operatorname {arcosh}\left (a x\right )}{a^{2} c x^{2} - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 309, normalized size = 5.83 \[ \frac {\arctanh \left (a x \right ) \mathrm {arccosh}\left (a x \right )}{a c}+\frac {2 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {\frac {1}{2}+\frac {a x}{2}}\, \sqrt {-\frac {1}{2}+\frac {a x}{2}}\, \arctanh \left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a c \left (a^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {\frac {1}{2}+\frac {a x}{2}}\, \sqrt {-\frac {1}{2}+\frac {a x}{2}}\, \arctanh \left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a c \left (a^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {\frac {1}{2}+\frac {a x}{2}}\, \sqrt {-\frac {1}{2}+\frac {a x}{2}}\, \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a c \left (a^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {\frac {1}{2}+\frac {a x}{2}}\, \sqrt {-\frac {1}{2}+\frac {a x}{2}}\, \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a c \left (a^{2} x^{2}-1\right )} \]
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 \[ \frac {4 \, {\left (\log \left (a x + 1\right ) - \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right ) - \log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + \log \left (a x - 1\right )^{2}}{8 \, a c} + \frac {\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )}{2 \, a c} + \int \frac {\log \left (a x + 1\right ) - \log \left (a x - 1\right )}{2 \, {\left (a^{3} c x^{3} - a c x + {\left (a^{2} c x^{2} - c\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{c-a^2\,c\,x^2} \,d x \]
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 \[ - \frac {\int \frac {\operatorname {acosh}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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