\(\int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx\) [486]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.20 (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 = {} \[ \int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx=\int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx \]

[In]

Int[(Csch[a + b*x]*Sech[a + b*x]^3)/x^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 26.47 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx=\int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx \]

[In]

Integrate[(Csch[a + b*x]*Sech[a + b*x]^3)/x^2,x]

[Out]

Integrate[(Csch[a + b*x]*Sech[a + b*x]^3)/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx=\int \frac {\operatorname {csch}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 155, normalized size of antiderivative = 8.61 \[ \int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

2*((b*x*e^(2*a) - e^(2*a))*e^(2*b*x) - 1)/(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) + 16
*integrate(1/8*(b^2*x^2 - 3)/(b^2*x^4*e^(2*b*x + 2*a) + b^2*x^4), x) - 16*integrate(1/16/(x^2*e^(b*x + a) + x^
2), x) + 16*integrate(1/16/(x^2*e^(b*x + a) - x^2), x)

Giac [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\text {csch}(a+b x) \text {sech}^3(a+b x)}{x^2} \, dx=\int \frac {1}{x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

[In]

int(1/(x^2*cosh(a + b*x)^3*sinh(a + b*x)),x)

[Out]

int(1/(x^2*cosh(a + b*x)^3*sinh(a + b*x)), x)