3.360 \(\int \frac{\sinh (a+b x) \tanh (a+b x)}{x} \, dx\)

Optimal. Leaf size=29 \[ -\text{Unintegrable}\left (\frac{\text{sech}(a+b x)}{x},x\right )+\cosh (a) \text{Chi}(b x)+\sinh (a) \text{Shi}(b x) \]

[Out]

Cosh[a]*CoshIntegral[b*x] + Sinh[a]*SinhIntegral[b*x] - Unintegrable[Sech[a + b*x]/x, x]

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Rubi [A]  time = 0.0859443, 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 = {} \[ \int \frac{\sinh (a+b x) \tanh (a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Sinh[a + b*x]*Tanh[a + b*x])/x,x]

[Out]

Cosh[a]*CoshIntegral[b*x] + Sinh[a]*SinhIntegral[b*x] - Defer[Int][Sech[a + b*x]/x, x]

Rubi steps

\begin{align*} \int \frac{\sinh (a+b x) \tanh (a+b x)}{x} \, dx &=\int \frac{\cosh (a+b x)}{x} \, dx-\int \frac{\text{sech}(a+b x)}{x} \, dx\\ &=\cosh (a) \int \frac{\cosh (b x)}{x} \, dx+\sinh (a) \int \frac{\sinh (b x)}{x} \, dx-\int \frac{\text{sech}(a+b x)}{x} \, dx\\ &=\cosh (a) \text{Chi}(b x)+\sinh (a) \text{Shi}(b x)-\int \frac{\text{sech}(a+b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 8.73769, size = 0, normalized size = 0. \[ \int \frac{\sinh (a+b x) \tanh (a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Sinh[a + b*x]*Tanh[a + b*x])/x,x]

[Out]

Integrate[(Sinh[a + b*x]*Tanh[a + b*x])/x, x]

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Maple [A]  time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm sech} \left (bx+a\right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a)*sinh(b*x+a)^2/x,x)

[Out]

int(sech(b*x+a)*sinh(b*x+a)^2/x,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)*sinh(b*x+a)^2/x,x, algorithm="maxima")

[Out]

integrate(sech(b*x + a)*sinh(b*x + a)^2/x, x)

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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 ) \sinh \left (b x + a\right )^{2}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)*sinh(b*x+a)^2/x,x, algorithm="fricas")

[Out]

integral(sech(b*x + a)*sinh(b*x + a)^2/x, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)*sinh(b*x+a)**2/x,x)

[Out]

Integral(sinh(a + b*x)**2*sech(a + b*x)/x, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)*sinh(b*x+a)^2/x,x, algorithm="giac")

[Out]

integrate(sech(b*x + a)*sinh(b*x + a)^2/x, x)