\(\int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx\) [5]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\tanh (e+f x)}{(c+d x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 14, 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 = {} \[ \int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx=\int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx \]

[In]

Int[Tanh[e + f*x]/(c + d*x)^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 24.63 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx=\int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx=\int { \frac {\tanh \left (f x + e\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx=\int \frac {\tanh {\left (e + f x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 71, normalized size of antiderivative = 5.07 \[ \int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx=\int { \frac {\tanh \left (f x + e\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx=\int { \frac {\tanh \left (f x + e\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.68 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\tanh (e+f x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {tanh}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(tanh(e + f*x)/(c + d*x)^2,x)

[Out]

int(tanh(e + f*x)/(c + d*x)^2, x)