### 3.416 $$\int \frac{\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx$$

Optimal. Leaf size=39 $\text{Unintegrable}\left (\frac{\coth (a+b x)}{x},x\right )+\frac{1}{2} \sinh (2 a) \text{Chi}(2 b x)+\frac{1}{2} \cosh (2 a) \text{Shi}(2 b x)$

[Out]

(CoshIntegral[2*b*x]*Sinh[2*a])/2 + (Cosh[2*a]*SinhIntegral[2*b*x])/2 + Unintegrable[Coth[a + b*x]/x, x]

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Rubi [A]  time = 0.104078, 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 = {} $\int \frac{\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[(Cosh[a + b*x]^2*Coth[a + b*x])/x,x]

[Out]

(CoshIntegral[2*b*x]*Sinh[2*a])/2 + (Cosh[2*a]*SinhIntegral[2*b*x])/2 + Defer[Int][Coth[a + b*x]/x, x]

Rubi steps

\begin{align*} \int \frac{\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx &=\int \frac{\coth (a+b x)}{x} \, dx+\int \frac{\cosh (a+b x) \sinh (a+b x)}{x} \, dx\\ &=\int \frac{\coth (a+b x)}{x} \, dx+\int \frac{\sinh (2 a+2 b x)}{2 x} \, dx\\ &=\frac{1}{2} \int \frac{\sinh (2 a+2 b x)}{x} \, dx+\int \frac{\coth (a+b x)}{x} \, dx\\ &=\frac{1}{2} \cosh (2 a) \int \frac{\sinh (2 b x)}{x} \, dx+\frac{1}{2} \sinh (2 a) \int \frac{\cosh (2 b x)}{x} \, dx+\int \frac{\coth (a+b x)}{x} \, dx\\ &=\frac{1}{2} \text{Chi}(2 b x) \sinh (2 a)+\frac{1}{2} \cosh (2 a) \text{Shi}(2 b x)+\int \frac{\coth (a+b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 10.9073, size = 0, normalized size = 0. $\int \frac{\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[(Cosh[a + b*x]^2*Coth[a + b*x])/x,x]

[Out]

Integrate[(Cosh[a + b*x]^2*Coth[a + b*x])/x, x]

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Maple [A]  time = 0.073, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}{\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)^3*csch(b*x+a)/x,x)

[Out]

int(cosh(b*x+a)^3*csch(b*x+a)/x,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \,{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac{1}{4} \,{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \int \frac{1}{x e^{\left (b x + a\right )} + x}\,{d x} + \int \frac{1}{x e^{\left (b x + a\right )} - x}\,{d x} + \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*csch(b*x+a)/x,x, algorithm="maxima")

[Out]

1/4*Ei(2*b*x)*e^(2*a) - 1/4*Ei(-2*b*x)*e^(-2*a) - integrate(1/(x*e^(b*x + a) + x), x) + integrate(1/(x*e^(b*x
+ a) - x), x) + log(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 )^{3} \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)^3*csch(b*x+a)/x,x, algorithm="fricas")

[Out]

integral(cosh(b*x + a)^3*csch(b*x + a)/x, x)

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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.

[In]

integrate(cosh(b*x+a)**3*csch(b*x+a)/x,x)

[Out]

Timed out

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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 )}{x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*csch(b*x+a)/x,x, algorithm="giac")

[Out]

integrate(cosh(b*x + a)^3*csch(b*x + a)/x, x)