\(\int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx\) [458]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx=\text {Int}\left (\frac {\text {csch}(a+b x)}{x^2},x\right )+\text {Int}\left (\frac {\text {csch}^3(a+b x)}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx=\int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

Defer[Int][Csch[a + b*x]/x^2, x] + Defer[Int][Csch[a + b*x]^3/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\text {csch}(a+b x)}{x^2} \, dx+\int \frac {\text {csch}^3(a+b x)}{x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 47.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx=\int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

\[\int \frac {\cosh \left (b x +a \right )^{2} \operatorname {csch}\left (b x +a \right )^{3}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 61.92 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx=\int \frac {\cosh ^{2}{\left (a + b x \right )} \operatorname {csch}^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 159, normalized size of antiderivative = 8.83 \[ \int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

-((b*x*e^(3*a) - 2*e^(3*a))*e^(3*b*x) + (b*x*e^a + 2*e^a)*e^(b*x))/(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) + 2*integrate(1/4*(b^2*x^2 + 6)/(b^2*x^4*e^(b*x + a) + b^2*x^4), x) + 2*integrate(1/4*(b^
2*x^2 + 6)/(b^2*x^4*e^(b*x + a) - b^2*x^4), x)

Giac [N/A]

Not integrable

Time = 2.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\coth ^2(a+b x) \text {csch}(a+b x)}{x^2} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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