\(\int \frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx\) [520]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.25 (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 = {} \[ \int \frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int \frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 43.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int \frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int \frac {\operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 226, normalized size of antiderivative = 11.30 \[ \int \frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{3} \operatorname {sech}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

(2*b*x*e^(3*b*x + 3*a) - (3*b*x*e^(5*a) - 2*e^(5*a))*e^(5*b*x) - (3*b*x*e^a + 2*e^a)*e^(b*x))/(b^2*x^3*e^(6*b*
x + 6*a) - b^2*x^3*e^(4*b*x + 4*a) - b^2*x^3*e^(2*b*x + 2*a) + b^2*x^3) - 32*integrate(3/64*(b^2*x^2 - 2)/(b^2
*x^4*e^(b*x + a) + b^2*x^4), x) - 32*integrate(3/64*(b^2*x^2 - 2)/(b^2*x^4*e^(b*x + a) - b^2*x^4), x) - 32*int
egrate(1/8*e^(b*x + a)/(b*x^3*e^(2*b*x + 2*a) + b*x^3), x)

Giac [N/A]

Not integrable

Time = 20.10 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {csch}\left (b x + a\right )^{3} \operatorname {sech}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 2.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx=\int \frac {1}{x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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