Optimal. Leaf size=131 \[ \text{PolyLog}\left (2,\frac{e^{\cosh ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right )+\text{PolyLog}\left (2,\frac{e^{\cosh ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )-\frac{1}{2} \cosh ^{-1}(a+b x)^2 \]
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Rubi [A] time = 0.249189, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5866, 5800, 5562, 2190, 2279, 2391} \[ \text{PolyLog}\left (2,\frac{e^{\cosh ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right )+\text{PolyLog}\left (2,\frac{e^{\cosh ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )-\frac{1}{2} \cosh ^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
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Rule 5866
Rule 5800
Rule 5562
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a+b x)}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^{-1}(x)}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \sinh (x)}{-\frac{a}{b}+\frac{\cosh (x)}{b}} \, dx,x,\cosh ^{-1}(a+b x)\right )}{b}\\ &=-\frac{1}{2} \cosh ^{-1}(a+b x)^2+\frac{\operatorname{Subst}\left (\int \frac{e^x x}{-\frac{a}{b}-\frac{\sqrt{-1+a^2}}{b}+\frac{e^x}{b}} \, dx,x,\cosh ^{-1}(a+b x)\right )}{b}+\frac{\operatorname{Subst}\left (\int \frac{e^x x}{-\frac{a}{b}+\frac{\sqrt{-1+a^2}}{b}+\frac{e^x}{b}} \, dx,x,\cosh ^{-1}(a+b x)\right )}{b}\\ &=-\frac{1}{2} \cosh ^{-1}(a+b x)^2+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{a-\sqrt{-1+a^2}}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{a+\sqrt{-1+a^2}}\right )-\operatorname{Subst}\left (\int \log \left (1+\frac{e^x}{\left (-\frac{a}{b}-\frac{\sqrt{-1+a^2}}{b}\right ) b}\right ) \, dx,x,\cosh ^{-1}(a+b x)\right )-\operatorname{Subst}\left (\int \log \left (1+\frac{e^x}{\left (-\frac{a}{b}+\frac{\sqrt{-1+a^2}}{b}\right ) b}\right ) \, dx,x,\cosh ^{-1}(a+b x)\right )\\ &=-\frac{1}{2} \cosh ^{-1}(a+b x)^2+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{a-\sqrt{-1+a^2}}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{a+\sqrt{-1+a^2}}\right )-\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{x}{\left (-\frac{a}{b}-\frac{\sqrt{-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )-\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{x}{\left (-\frac{a}{b}+\frac{\sqrt{-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )\\ &=-\frac{1}{2} \cosh ^{-1}(a+b x)^2+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{a-\sqrt{-1+a^2}}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac{e^{\cosh ^{-1}(a+b x)}}{a+\sqrt{-1+a^2}}\right )+\text{Li}_2\left (\frac{e^{\cosh ^{-1}(a+b x)}}{a-\sqrt{-1+a^2}}\right )+\text{Li}_2\left (\frac{e^{\cosh ^{-1}(a+b x)}}{a+\sqrt{-1+a^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0131778, size = 153, normalized size = 1.17 \[ \text{PolyLog}\left (2,-\frac{e^{\cosh ^{-1}(a+b x)}}{\sqrt{a^2-1}-a}\right )+\text{PolyLog}\left (2,\frac{e^{\cosh ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )+\cosh ^{-1}(a+b x) \log \left (\frac{e^{\cosh ^{-1}(a+b x)}}{b \left (-\frac{\sqrt{a^2-1}}{b}-\frac{a}{b}\right )}+1\right )+\cosh ^{-1}(a+b x) \log \left (\frac{e^{\cosh ^{-1}(a+b x)}}{b \left (\frac{\sqrt{a^2-1}}{b}-\frac{a}{b}\right )}+1\right )-\frac{1}{2} \cosh ^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.081, size = 436, normalized size = 3.3 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (bx+a\right ) \right ) ^{2}}{2}}+{a{\rm arccosh} \left (bx+a\right )\ln \left ({ \left ( \sqrt{{a}^{2}-1}-bx-\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) \left ( a+\sqrt{{a}^{2}-1} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-1}}}}-{a{\rm arccosh} \left (bx+a\right )\ln \left ({ \left ( \sqrt{{a}^{2}-1}+bx+\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) \left ( -a+\sqrt{{a}^{2}-1} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-1}}}}-{\frac{{\rm arccosh} \left (bx+a\right )}{{a}^{2}-1} \left ({a}^{2}-1+a\sqrt{{a}^{2}-1} \right ) \left ( 2\,\ln \left ({\frac{\sqrt{{a}^{2}-1}-bx-\sqrt{bx+a-1}\sqrt{bx+a+1}}{a+\sqrt{{a}^{2}-1}}} \right ){a}^{2}-\ln \left ({ \left ( \sqrt{{a}^{2}-1}-bx-\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) \left ( a+\sqrt{{a}^{2}-1} \right ) ^{-1}} \right ) -\ln \left ({ \left ( \sqrt{{a}^{2}-1}+bx+\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) \left ( -a+\sqrt{{a}^{2}-1} \right ) ^{-1}} \right ) -2\,a\sqrt{{a}^{2}-1}\ln \left ({\frac{\sqrt{{a}^{2}-1}-bx-\sqrt{bx+a-1}\sqrt{bx+a+1}}{a+\sqrt{{a}^{2}-1}}} \right ) \right ) }+{\it dilog} \left ({ \left ( \sqrt{{a}^{2}-1}-bx-\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) \left ( a+\sqrt{{a}^{2}-1} \right ) ^{-1}} \right ) +{\it dilog} \left ({ \left ( \sqrt{{a}^{2}-1}+bx+\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) \left ( -a+\sqrt{{a}^{2}-1} \right ) ^{-1}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{\operatorname{arcosh}\left (b x + a\right )}{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{\operatorname{acosh}{\left (a + b 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{\operatorname{arcosh}\left (b x + a\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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