\(\int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx\) [353]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.16 (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 = {} \[ \int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx=\int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 22.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx=\int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.86 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx=\int \frac {\sinh {\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 5.61 \[ \int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )}{x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\text {sech}^2(a+b x) \tanh (a+b x)}{x} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x\right )}{x\,{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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