Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{\text{csch}(a+b x)}{x},x\right )+\sinh (a) \text{Chi}(b x)+\cosh (a) \text{Shi}(b x) \]
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Rubi [A] time = 0.0801649, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cosh (a+b x) \coth (a+b x)}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cosh (a+b x) \coth (a+b x)}{x} \, dx &=\int \frac{\text{csch}(a+b x)}{x} \, dx+\int \frac{\sinh (a+b x)}{x} \, dx\\ &=\cosh (a) \int \frac{\sinh (b x)}{x} \, dx+\sinh (a) \int \frac{\cosh (b x)}{x} \, dx+\int \frac{\text{csch}(a+b x)}{x} \, dx\\ &=\text{Chi}(b x) \sinh (a)+\cosh (a) \text{Shi}(b x)+\int \frac{\text{csch}(a+b x)}{x} \, dx\\ \end{align*}
Mathematica [A] time = 25.9794, size = 0, normalized size = 0. \[ \int \frac{\cosh (a+b x) \coth (a+b x)}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}{\rm csch} \left (bx+a\right )}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (b x + a\right )^{2} \operatorname{csch}\left (b x + a\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cosh \left (b x + a\right )^{2} \operatorname{csch}\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{2}{\left (a + b x \right )} \operatorname{csch}{\left (a + b x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (b x + a\right )^{2} \operatorname{csch}\left (b x + a\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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