### 3.429 $$\int \frac{\coth (a+b x) \text{csch}(a+b x)}{x} \, dx$$

Optimal. Leaf size=18 $\text{CannotIntegrate}\left (\frac{\coth (a+b x) \text{csch}(a+b x)}{x},x\right )$

[Out]

CannotIntegrate[(Coth[a + b*x]*Csch[a + b*x])/x, x]

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Rubi [A]  time = 0.141851, 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 = {} $\int \frac{\coth (a+b x) \text{csch}(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[(Coth[a + b*x]*Csch[a + b*x])/x,x]

[Out]

Defer[Int][(Coth[a + b*x]*Csch[a + b*x])/x, x]

Rubi steps

\begin{align*} \int \frac{\coth (a+b x) \text{csch}(a+b x)}{x} \, dx &=\int \frac{\coth (a+b x) \text{csch}(a+b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 31.313, size = 0, normalized size = 0. $\int \frac{\coth (a+b x) \text{csch}(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[(Coth[a + b*x]*Csch[a + b*x])/x,x]

[Out]

Integrate[(Coth[a + b*x]*Csch[a + b*x])/x, x]

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Maple [A]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cosh \left ( bx+a \right ) \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)*csch(b*x+a)^2/x,x)

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*csch(b*x+a)^2/x,x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (a + b x \right )} \operatorname{csch}^{2}{\left (a + b x \right )}}{x}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (b x + a\right ) \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)*csch(b*x+a)^2/x,x, algorithm="giac")

[Out]

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