### 3.444 $$\int \frac{\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx$$

Optimal. Leaf size=46 $\text{CannotIntegrate}\left (\frac{\coth (a+b x) \text{csch}(a+b x)}{x^2},x\right )+b \sinh (a) \text{Chi}(b x)+b \cosh (a) \text{Shi}(b x)-\frac{\cosh (a+b x)}{x}$

[Out]

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

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Rubi [A]  time = 0.136694, 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 = {} $\int \frac{\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 20.4555, size = 0, normalized size = 0. $\int \frac{\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

[Out]

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

[Out]

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