\(\int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx\) [493]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.18 (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 = {} \[ \int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 24.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 114, normalized size of antiderivative = 6.33 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

-2*e^(b*x + a)/(b*x^2*e^(2*b*x + 2*a) - b*x^2) - 8*integrate(1/4*e^(b*x + a)/(x^2*e^(2*b*x + 2*a) + x^2), x) -
8*integrate(1/4/(b*x^3*e^(b*x + a) + b*x^3), x) - 8*integrate(1/4/(b*x^3*e^(b*x + a) - b*x^3), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError >> type

Mupad [N/A]

Not integrable

Time = 2.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int \frac {1}{x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

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

[Out]

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