\(\int \frac {\tanh ^3(e+f x)}{c+d x} \, dx\) [14]

Optimal result

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

[Out]

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

[In]

Int[Tanh[e + f*x]^3/(c + d*x),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 26.72 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^3(e+f x)}{c+d x} \, dx=\int \frac {\tanh ^3(e+f x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^3(e+f x)}{c+d x} \, dx=\int { \frac {\tanh \left (f x + e\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\tanh ^3(e+f x)}{c+d x} \, dx=\int \frac {\tanh ^{3}{\left (e + f x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 303, normalized size of antiderivative = 18.94 \[ \int \frac {\tanh ^3(e+f x)}{c+d x} \, dx=\int { \frac {\tanh \left (f x + e\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^3(e+f x)}{c+d x} \, dx=\int { \frac {\tanh \left (f x + e\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.71 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^3(e+f x)}{c+d x} \, dx=\int \frac {{\mathrm {tanh}\left (e+f\,x\right )}^3}{c+d\,x} \,d x \]

[In]

int(tanh(e + f*x)^3/(c + d*x),x)

[Out]

int(tanh(e + f*x)^3/(c + d*x), x)