\(\int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx\) [375]

Optimal result

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

[Out]

Unintegrable(sech(b*x+a)/x^2,x)-Unintegrable(sech(b*x+a)^3/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

Defer[Int][Sech[a + b*x]/x^2, x] - Defer[Int][Sech[a + b*x]^3/x^2, x]

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

Mathematica [N/A]

Not integrable

Time = 11.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.79 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int \frac {\sinh ^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 133, normalized size of antiderivative = 7.39 \[ \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.63 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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