\(\int \frac {\tanh ^2(a+b x)}{x^2} \, dx\) [368]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\tanh ^2(a+b x)}{x^2} \, dx=\text {Int}\left (\frac {\tanh ^2(a+b x)}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, 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 = {} \[ \int \frac {\tanh ^2(a+b x)}{x^2} \, dx=\int \frac {\tanh ^2(a+b x)}{x^2} \, dx \]

[In]

Int[Tanh[a + b*x]^2/x^2,x]

[Out]

Defer[Int][Tanh[a + b*x]^2/x^2, x]

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

Mathematica [N/A]

Not integrable

Time = 10.98 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\tanh ^2(a+b x)}{x^2} \, dx=\int \frac {\tanh ^2(a+b x)}{x^2} \, dx \]

[In]

Integrate[Tanh[a + b*x]^2/x^2,x]

[Out]

Integrate[Tanh[a + b*x]^2/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.67

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\tanh ^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.67 \[ \int \frac {\tanh ^2(a+b x)}{x^2} \, dx=\int \frac {\sinh ^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 5.67 \[ \int \frac {\tanh ^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\tanh ^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\tanh ^2(a+b x)}{x^2} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \]

[In]

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

[Out]

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