Optimal. Leaf size=116 \[ -\frac{e \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}+\frac{e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}-\frac{d \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c} \]
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Rubi [A] time = 0.337515, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5806, 6742, 3303, 3298, 3301, 5448, 12} \[ -\frac{e \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}+\frac{e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}-\frac{d \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c} \]
Antiderivative was successfully verified.
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Rule 5806
Rule 6742
Rule 3303
Rule 3298
Rule 3301
Rule 5448
Rule 12
Rubi steps
\begin{align*} \int \frac{d+e x}{a+b \cosh ^{-1}(c x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(c d+e \cosh (x)) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{c d \sinh (x)}{a+b x}+\frac{e \cosh (x) \sinh (x)}{a+b x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}\\ &=\frac{d \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c}+\frac{e \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 (a+b x)} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}+\frac{\left (d \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c}-\frac{\left (d \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c}\\ &=-\frac{d \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{b c}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac{e \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2}\\ &=-\frac{d \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{b c}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac{\left (e \cosh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2}-\frac{\left (e \sinh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2}\\ &=-\frac{d \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{b c}-\frac{e \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{2 a}{b}\right )}{2 b c^2}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac{e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}\\ \end{align*}
Mathematica [A] time = 0.137259, size = 98, normalized size = 0.84 \[ \frac{-2 c d \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-e \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+2 c d \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )}{2 b c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 120, normalized size = 1. \begin{align*}{\frac{1}{c} \left ({\frac{d}{2\,b}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ) }-{\frac{d}{2\,b}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) }-{\frac{e}{4\,bc}{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\rm arccosh} \left (cx\right )-2\,{\frac{a}{b}} \right ) }+{\frac{e}{4\,bc}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\rm arccosh} \left (cx\right )+2\,{\frac{a}{b}} \right ) } \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*} \int \frac{e x + d}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \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{e x + d}{b \operatorname{arcosh}\left (c x\right ) + a}, 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{d + e x}{a + b \operatorname{acosh}{\left (c x \right )}}\, 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{e x + d}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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