### 3.410 $$\int \frac{\cosh (a+b x) \coth (a+b x)}{x^2} \, dx$$

Optimal. Leaf size=40 $\text{Unintegrable}\left (\frac{\text{csch}(a+b x)}{x^2},x\right )+b \cosh (a) \text{Chi}(b x)+b \sinh (a) \text{Shi}(b x)-\frac{\sinh (a+b x)}{x}$

[Out]

b*Cosh[a]*CoshIntegral[b*x] - Sinh[a + b*x]/x + b*Sinh[a]*SinhIntegral[b*x] + Unintegrable[Csch[a + b*x]/x^2,
x]

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Rubi [A]  time = 0.0987742, 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 = {} $\int \frac{\cosh (a+b x) \coth (a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

b*Cosh[a]*CoshIntegral[b*x] - Sinh[a + b*x]/x + b*Sinh[a]*SinhIntegral[b*x] + Defer[Int][Csch[a + b*x]/x^2, x]

Rubi steps

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

Mathematica [A]  time = 36.9795, size = 0, normalized size = 0. $\int \frac{\cosh (a+b x) \coth (a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}{\rm csch} \left (bx+a\right )}{{x}^{2}}}\, 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^2,x)

[Out]

int(cosh(b*x+a)^2*csch(b*x+a)/x^2,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^{2}}\,{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^2,x, algorithm="maxima")

[Out]

integrate(cosh(b*x + a)^2*csch(b*x + a)/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 )^{2} \operatorname{csch}\left (b x + a\right )}{x^{2}}, 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^2,x, algorithm="fricas")

[Out]

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

[Out]

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