### 3.417 $$\int \frac{\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx$$

Optimal. Leaf size=51 $\text{Unintegrable}\left (\frac{\coth (a+b x)}{x^2},x\right )+b \cosh (2 a) \text{Chi}(2 b x)+b \sinh (2 a) \text{Shi}(2 b x)-\frac{\sinh (2 a+2 b x)}{2 x}$

[Out]

b*Cosh[2*a]*CoshIntegral[2*b*x] - Sinh[2*a + 2*b*x]/(2*x) + b*Sinh[2*a]*SinhIntegral[2*b*x] + Unintegrable[Cot
h[a + b*x]/x^2, x]

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Rubi [A]  time = 0.1278, 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 = {} $\int \frac{\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 16.4086, size = 0, normalized size = 0. $\int \frac{\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x\right ) + \frac{1}{2} \, b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x\right ) - \frac{1}{x} - \int \frac{1}{x^{2} e^{\left (b x + a\right )} + x^{2}}\,{d x} + \int \frac{1}{x^{2} e^{\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)^3*csch(b*x+a)/x^2,x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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