\(\int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx\) [500]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=4 \text {Int}\left (\frac {\text {csch}^2(2 a+2 b x)}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = 4 \int \frac {\text {csch}^2(2 a+2 b x)}{x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 18.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int \frac {\operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.25 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

-4/(b*x^2*e^(4*b*x + 4*a) - b*x^2) + 16*integrate(1/4/(b*x^3*e^(2*b*x + 2*a) + b*x^3), x) + 16*integrate(1/8/(
b*x^3*e^(b*x + a) + b*x^3), x) - 16*integrate(1/8/(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 = 1.10 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int \frac {1}{x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

int(1/(x^2*cosh(a + b*x)^2*sinh(a + b*x)^2),x)

[Out]

int(1/(x^2*cosh(a + b*x)^2*sinh(a + b*x)^2), x)