\(\int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx\) [15]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\tanh ^3(e+f x)}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable(tanh(f*x+e)^3/(d*x+c)^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx=\int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx \]

[In]

Int[Tanh[e + f*x]^3/(c + d*x)^2,x]

[Out]

Defer[Int][Tanh[e + f*x]^3/(c + d*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 22.53 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx=\int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx \]

[In]

Integrate[Tanh[e + f*x]^3/(c + d*x)^2,x]

[Out]

Integrate[Tanh[e + f*x]^3/(c + d*x)^2, x]

Maple [N/A] (verified)

Not integrable

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

\[\int \frac {\tanh \left (f x +e \right )^{3}}{\left (d x +c \right )^{2}}d x\]

[In]

int(tanh(f*x+e)^3/(d*x+c)^2,x)

[Out]

int(tanh(f*x+e)^3/(d*x+c)^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx=\int { \frac {\tanh \left (f x + e\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(tanh(f*x+e)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(tanh(f*x + e)^3/(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 \[ \int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx=\int \frac {\tanh ^{3}{\left (e + f x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(tanh(f*x+e)**3/(d*x+c)**2,x)

[Out]

Integral(tanh(e + f*x)**3/(c + d*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 501, normalized size of antiderivative = 31.31 \[ \int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx=\int { \frac {\tanh \left (f x + e\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(tanh(f*x+e)^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

-(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 + 2*d^2 + (d^2*f^2*x^2*e^(4*e) + 2*c*d*f^2*x*e^(4*e) + c^2*f^2*e^(4*e))*
e^(4*f*x) + 2*(d^2*f^2*x^2*e^(2*e) + c^2*f^2*e^(2*e) - c*d*f*e^(2*e) + d^2*e^(2*e) + (2*c*d*f^2*e^(2*e) - d^2*
f*e^(2*e))*x)*e^(2*f*x))/(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2 + (d^4*f^2*x^3*e^(4*e) +
3*c*d^3*f^2*x^2*e^(4*e) + 3*c^2*d^2*f^2*x*e^(4*e) + c^3*d*f^2*e^(4*e))*e^(4*f*x) + 2*(d^4*f^2*x^3*e^(2*e) + 3
*c*d^3*f^2*x^2*e^(2*e) + 3*c^2*d^2*f^2*x*e^(2*e) + c^3*d*f^2*e^(2*e))*e^(2*f*x)) - integrate(2*(d^2*f^2*x^2 +
2*c*d*f^2*x + c^2*f^2 + 3*d^2)/(d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d*f^2*x + c^4*f^2 +
(d^4*f^2*x^4*e^(2*e) + 4*c*d^3*f^2*x^3*e^(2*e) + 6*c^2*d^2*f^2*x^2*e^(2*e) + 4*c^3*d*f^2*x*e^(2*e) + c^4*f^2*e
^(2*e))*e^(2*f*x)), x)

Giac [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx=\int { \frac {\tanh \left (f x + e\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(tanh(f*x+e)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(tanh(f*x + e)^3/(d*x + c)^2, x)

Mupad [N/A]

Not integrable

Time = 1.76 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^3(e+f x)}{(c+d x)^2} \, dx=\int \frac {{\mathrm {tanh}\left (e+f\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(tanh(e + f*x)^3/(c + d*x)^2,x)

[Out]

int(tanh(e + f*x)^3/(c + d*x)^2, x)