\(\int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx\) [472]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx=2 \text {Int}\left (\frac {\text {csch}(2 a+2 b x)}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 13.79 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {csch}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx=\int \frac {1}{x^2\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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