\(\int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx\) [10]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\tanh ^2(e+f x)}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable(tanh(f*x+e)^2/(d*x+c)^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx=\int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx \]

[In]

Int[Tanh[e + f*x]^2/(c + d*x)^2,x]

[Out]

Defer[Int][Tanh[e + f*x]^2/(c + d*x)^2, x]

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

Mathematica [N/A]

Not integrable

Time = 24.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx=\int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx \]

[In]

Integrate[Tanh[e + f*x]^2/(c + d*x)^2,x]

[Out]

Integrate[Tanh[e + f*x]^2/(c + d*x)^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \frac {\tanh \left (f x +e \right )^{2}}{\left (d x +c \right )^{2}}d x\]

[In]

int(tanh(f*x+e)^2/(d*x+c)^2,x)

[Out]

int(tanh(f*x+e)^2/(d*x+c)^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx=\int { \frac {\tanh \left (f x + e\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(tanh(f*x+e)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(tanh(f*x + e)^2/(d^2*x^2 + 2*c*d*x + c^2), x)

Sympy [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx=\int \frac {\tanh ^{2}{\left (e + f x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(tanh(f*x+e)**2/(d*x+c)**2,x)

[Out]

Integral(tanh(e + f*x)**2/(c + d*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 195, normalized size of antiderivative = 12.19 \[ \int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx=\int { \frac {\tanh \left (f x + e\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(tanh(f*x+e)^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

4*d*integrate(1/(d^3*f*x^3 + 3*c*d^2*f*x^2 + 3*c^2*d*f*x + c^3*f + (d^3*f*x^3*e^(2*e) + 3*c*d^2*f*x^2*e^(2*e)
+ 3*c^2*d*f*x*e^(2*e) + c^3*f*e^(2*e))*e^(2*f*x)), x) - (d*f*x + c*f + (d*f*x*e^(2*e) + c*f*e^(2*e))*e^(2*f*x)
- 2*d)/(d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f + (d^3*f*x^2*e^(2*e) + 2*c*d^2*f*x*e^(2*e) + c^2*d*f*e^(2*e))*e^(2*
f*x))

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx=\int { \frac {\tanh \left (f x + e\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(tanh(f*x+e)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(tanh(f*x + e)^2/(d*x + c)^2, x)

Mupad [N/A]

Not integrable

Time = 1.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^2(e+f x)}{(c+d x)^2} \, dx=\int \frac {{\mathrm {tanh}\left (e+f\,x\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(tanh(e + f*x)^2/(c + d*x)^2,x)

[Out]

int(tanh(e + f*x)^2/(c + d*x)^2, x)