### 3.451 $$\int \frac{\coth (a+b x) \text{csch}^2(a+b x)}{x^2} \, dx$$

Optimal. Leaf size=20 $\text{CannotIntegrate}\left (\frac{\coth (a+b x) \text{csch}^2(a+b x)}{x^2},x\right )$

[Out]

CannotIntegrate[(Coth[a + b*x]*Csch[a + b*x]^2)/x^2, x]

________________________________________________________________________________________

Rubi [A]  time = 0.236077, 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 = {} $\int \frac{\coth (a+b x) \text{csch}^2(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 19.7865, size = 0, normalized size = 0. $\int \frac{\coth (a+b x) \text{csch}^2(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cosh \left ( bx+a \right ) \left ({\rm csch} \left (bx+a\right ) \right ) ^{3}}{{x}^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left ({\left (b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 1\right )}}{b^{2} x^{3} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} x^{3} e^{\left (2 \, b x + 2 \, a\right )} + b^{2} x^{3}} - 12 \, \int \frac{1}{4 \,{\left (b^{2} x^{4} e^{\left (b x + a\right )} + b^{2} x^{4}\right )}}\,{d x} + 12 \, \int \frac{1}{4 \,{\left (b^{2} x^{4} e^{\left (b x + a\right )} - b^{2} x^{4}\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)^3/x^2,x, algorithm="maxima")

[Out]

-2*((b*x*e^(2*a) - e^(2*a))*e^(2*b*x) + 1)/(b^2*x^3*e^(4*b*x + 4*a) - 2*b^2*x^3*e^(2*b*x + 2*a) + b^2*x^3) - 1
2*integrate(1/4/(b^2*x^4*e^(b*x + a) + b^2*x^4), x) + 12*integrate(1/4/(b^2*x^4*e^(b*x + a) - b^2*x^4), x)

________________________________________________________________________________________

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 )^{3}}{x^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*csch(b*x+a)^3/x^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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 )^{3}}{x^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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