Optimal. Leaf size=35 \[ -\text{CannotIntegrate}\left (\frac{\tanh (a+b x) \text{sech}(a+b x)}{x},x\right )+\sinh (a) \text{Chi}(b x)+\cosh (a) \text{Shi}(b x) \]
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Rubi [A] time = 0.128341, 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 = {} \[ \int \frac{\sinh (a+b x) \tanh ^2(a+b x)}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sinh (a+b x) \tanh ^2(a+b x)}{x} \, dx &=\int \frac{\sinh (a+b x)}{x} \, dx-\int \frac{\text{sech}(a+b x) \tanh (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{sech}(a+b x) \tanh (a+b x)}{x} \, dx\\ &=\text{Chi}(b x) \sinh (a)+\cosh (a) \text{Shi}(b x)-\int \frac{\text{sech}(a+b x) \tanh (a+b x)}{x} \, dx\\ \end{align*}
Mathematica [A] time = 11.5038, size = 0, normalized size = 0. \[ \int \frac{\sinh (a+b x) \tanh ^2(a+b x)}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{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*} -\frac{1}{2} \,{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac{1}{2} \,{\rm Ei}\left (b x\right ) e^{a} + \frac{2 \, e^{\left (b x + a\right )}}{b x e^{\left (2 \, b x + 2 \, a\right )} + b x} + 2 \, \int \frac{e^{\left (b x + a\right )}}{b x^{2} e^{\left (2 \, b x + 2 \, a\right )} + b x^{2}}\,{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{\operatorname{sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}}{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{\sinh ^{3}{\left (a + b x \right )} \operatorname{sech}^{2}{\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{\operatorname{sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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