Optimal. Leaf size=60 \[ \frac{\text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a+b x)}\right )}{2 d}-\frac{\cosh ^{-1}(a+b x)^2}{2 d}+\frac{\cosh ^{-1}(a+b x) \log \left (e^{2 \cosh ^{-1}(a+b x)}+1\right )}{d} \]
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Rubi [A] time = 0.0898679, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5866, 12, 5660, 3718, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a+b x)}\right )}{2 d}-\frac{\cosh ^{-1}(a+b x)^2}{2 d}+\frac{\cosh ^{-1}(a+b x) \log \left (e^{2 \cosh ^{-1}(a+b x)}+1\right )}{d} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 12
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b \cosh ^{-1}(x)}{d x} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^{-1}(x)}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}(a+b x)\right )}{d}\\ &=-\frac{\cosh ^{-1}(a+b x)^2}{2 d}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a+b x)\right )}{d}\\ &=-\frac{\cosh ^{-1}(a+b x)^2}{2 d}+\frac{\cosh ^{-1}(a+b x) \log \left (1+e^{2 \cosh ^{-1}(a+b x)}\right )}{d}-\frac{\operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a+b x)\right )}{d}\\ &=-\frac{\cosh ^{-1}(a+b x)^2}{2 d}+\frac{\cosh ^{-1}(a+b x) \log \left (1+e^{2 \cosh ^{-1}(a+b x)}\right )}{d}-\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a+b x)}\right )}{2 d}\\ &=-\frac{\cosh ^{-1}(a+b x)^2}{2 d}+\frac{\cosh ^{-1}(a+b x) \log \left (1+e^{2 \cosh ^{-1}(a+b x)}\right )}{d}+\frac{\text{Li}_2\left (-e^{2 \cosh ^{-1}(a+b x)}\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0427777, size = 53, normalized size = 0.88 \[ \frac{\cosh ^{-1}(a+b x) \left (\cosh ^{-1}(a+b x)+2 \log \left (e^{-2 \cosh ^{-1}(a+b x)}+1\right )\right )-\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a+b x)}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.039, size = 85, normalized size = 1.4 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (bx+a\right ) \right ) ^{2}}{2\,d}}+{\frac{{\rm arccosh} \left (bx+a\right )}{d}\ln \left ( 1+ \left ( bx+a+\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) ^{2} \right ) }+{\frac{1}{2\,d}{\it polylog} \left ( 2,- \left ( bx+a+\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) ^{2} \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{b \operatorname{arcosh}\left (b x + a\right )}{b d x + a d}, 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*} \frac{b \int \frac{\operatorname{acosh}{\left (a + b x \right )}}{a + b x}\, dx}{d} \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 )}{d x + \frac{a d}{b}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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