\(\int \frac {\coth ^2(a+b x)}{x^2} \, dx\) [437]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\coth ^2(a+b x)}{x^2} \, dx=\text {Int}\left (\frac {\coth ^2(a+b x)}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, 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 = {} \[ \int \frac {\coth ^2(a+b x)}{x^2} \, dx=\int \frac {\coth ^2(a+b x)}{x^2} \, dx \]

[In]

Int[Coth[a + b*x]^2/x^2,x]

[Out]

Defer[Int][Coth[a + b*x]^2/x^2, x]

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

Mathematica [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\coth ^2(a+b x)}{x^2} \, dx=\int \frac {\coth ^2(a+b x)}{x^2} \, dx \]

[In]

Integrate[Coth[a + b*x]^2/x^2,x]

[Out]

Integrate[Coth[a + b*x]^2/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.67

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\coth ^2(a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 25.97 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.67 \[ \int \frac {\coth ^2(a+b x)}{x^2} \, dx=\int \frac {\cosh ^{2}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 7.58 \[ \int \frac {\coth ^2(a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

-(b*x*e^(2*b*x + 2*a) - b*x + 2)/(b*x^2*e^(2*b*x + 2*a) - b*x^2) + 2*integrate(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)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\coth ^2(a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\coth ^2(a+b x)}{x^2} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

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

[Out]

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