### 3.443 $$\int \frac{\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx$$

Optimal. Leaf size=33 $\text{CannotIntegrate}\left (\frac{\coth (a+b x) \text{csch}(a+b x)}{x},x\right )+\cosh (a) \text{Chi}(b x)+\sinh (a) \text{Shi}(b x)$

[Out]

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

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Rubi [A]  time = 0.104907, 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 = {} $\int \frac{\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 23.097, size = 0, normalized size = 0. $\int \frac{\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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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}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^3*csch(b*x+a)^2/x,x)

[Out]

int(cosh(b*x+a)^3*csch(b*x+a)^2/x,x)

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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} - \int \frac{1}{b x^{2} e^{\left (b x + a\right )} + b x^{2}}\,{d x} - \int \frac{1}{b x^{2} e^{\left (b x + a\right )} - b x^{2}}\,{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)^2/x,x, algorithm="maxima")

[Out]

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

[Out]

integral(cosh(b*x + a)^3*csch(b*x + a)^2/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)**2/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 )^{2}}{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)^2/x,x, algorithm="giac")

[Out]

integrate(cosh(b*x + a)^3*csch(b*x + a)^2/x, x)