Optimal. Leaf size=46 \[ \text{CannotIntegrate}\left (\frac{\coth (a+b x) \text{csch}(a+b x)}{x^2},x\right )+b \sinh (a) \text{Chi}(b x)+b \cosh (a) \text{Shi}(b x)-\frac{\cosh (a+b x)}{x} \]
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Rubi [A] time = 0.136694, 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 = {} \[ \int \frac{\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx &=\int \frac{\cosh (a+b x)}{x^2} \, dx+\int \frac{\coth (a+b x) \text{csch}(a+b x)}{x^2} \, dx\\ &=-\frac{\cosh (a+b x)}{x}+b \int \frac{\sinh (a+b x)}{x} \, dx+\int \frac{\coth (a+b x) \text{csch}(a+b x)}{x^2} \, dx\\ &=-\frac{\cosh (a+b x)}{x}+(b \cosh (a)) \int \frac{\sinh (b x)}{x} \, dx+(b \sinh (a)) \int \frac{\cosh (b x)}{x} \, dx+\int \frac{\coth (a+b x) \text{csch}(a+b x)}{x^2} \, dx\\ &=-\frac{\cosh (a+b x)}{x}+b \text{Chi}(b x) \sinh (a)+b \cosh (a) \text{Shi}(b x)+\int \frac{\coth (a+b x) \text{csch}(a+b x)}{x^2} \, dx\\ \end{align*}
Mathematica [A] time = 20.4555, size = 0, normalized size = 0. \[ \int \frac{\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ({\rm csch} \left (bx+a\right ) \right ) ^{2}}{{x}^{2}}}\, 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*} -\frac{1}{2} \, b e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + \frac{1}{2} \, b e^{a} \Gamma \left (-1, -b x\right ) - \frac{2 \, e^{\left (b x + a\right )}}{b x^{2} e^{\left (2 \, b x + 2 \, a\right )} - b x^{2}} - 2 \, \int \frac{1}{b x^{3} e^{\left (b x + a\right )} + b x^{3}}\,{d x} - 2 \, \int \frac{1}{b x^{3} e^{\left (b x + a\right )} - b x^{3}}\,{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 )^{3} \operatorname{csch}\left (b x + a\right )^{2}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )^{3} \operatorname{csch}\left (b x + a\right )^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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