### 3.409 $$\int \frac{\cosh (a+b x) \coth (a+b x)}{x} \, dx$$

Optimal. Leaf size=27 $\text{Unintegrable}\left (\frac{\text{csch}(a+b x)}{x},x\right )+\sinh (a) \text{Chi}(b x)+\cosh (a) \text{Shi}(b x)$

[Out]

CoshIntegral[b*x]*Sinh[a] + Cosh[a]*SinhIntegral[b*x] + Unintegrable[Csch[a + b*x]/x, 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.

[In]

Int[(Cosh[a + b*x]*Coth[a + b*x])/x,x]

[Out]

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

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.

[In]

Integrate[(Cosh[a + b*x]*Coth[a + b*x])/x,x]

[Out]

Integrate[(Cosh[a + b*x]*Coth[a + b*x])/x, x]

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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.

[In]

int(cosh(b*x+a)^2*csch(b*x+a)/x,x)

[Out]

int(cosh(b*x+a)^2*csch(b*x+a)/x,x)

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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.

[In]

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

[Out]

integrate(cosh(b*x + a)^2*csch(b*x + a)/x, x)

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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.

[In]

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

[Out]

integral(cosh(b*x + a)^2*csch(b*x + a)/x, x)

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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.

[In]

integrate(cosh(b*x+a)**2*csch(b*x+a)/x,x)

[Out]

Integral(cosh(a + b*x)**2*csch(a + b*x)/x, x)

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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.

[In]

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

[Out]

integrate(cosh(b*x + a)^2*csch(b*x + a)/x, x)