Optimal. Leaf size=55 \[ -\frac{1}{2} b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right ) \]
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Rubi [A] time = 0.0766328, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5660, 3718, 2190, 2279, 2391} \[ \frac{1}{2} b \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right ) \]
Warning: Unable to verify antiderivative.
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Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx &=\operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-b \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac{1}{2} b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac{1}{2} b \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0401445, size = 48, normalized size = 0.87 \[ \frac{1}{2} b \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )-\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right )+a \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.03, size = 75, normalized size = 1.4 \begin{align*} a\ln \left ( cx \right ) -{\frac{b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2}}+b{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) +{\frac{b}{2}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{x}\,{d x} + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcosh}\left (c x\right ) + a}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acosh}{\left (c x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (c x\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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