\(\int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx\) [505]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx=\text {Int}\left (\frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.21 (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 = {} \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx=\int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 49.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx=\int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx=\int \frac {\operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 214, normalized size of antiderivative = 10.70 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

-(2*b*x*e^(3*b*x + 3*a) + (3*b*x*e^(5*a) - e^(5*a))*e^(5*b*x) + (3*b*x*e^a + e^a)*e^(b*x))/(b^2*x^2*e^(6*b*x +
6*a) + b^2*x^2*e^(4*b*x + 4*a) - b^2*x^2*e^(2*b*x + 2*a) - b^2*x^2) - 32*integrate(1/32*(3*b^2*x^2*e^a - 2*e^
a)*e^(b*x)/(b^2*x^3*e^(2*b*x + 2*a) + b^2*x^3), x) - 32*integrate(1/32/(b*x^2*e^(b*x + a) + b*x^2), x) - 32*in
tegrate(1/32/(b*x^2*e^(b*x + a) - b*x^2), x)

Giac [N/A]

Not integrable

Time = 2.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^3(a+b x)}{x} \, dx=\int \frac {1}{x\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

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

[Out]

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