\(\int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx\) [66]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx=\text {Int}\left (\frac {(a+b \tanh (e+f x))^3}{c+d x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx=\int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 32.80 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx=\int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.60 \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx=\int { \frac {{\left (b \tanh \left (f x + e\right ) + a\right )}^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

integral((b^3*tanh(f*x + e)^3 + 3*a*b^2*tanh(f*x + e)^2 + 3*a^2*b*tanh(f*x + e) + a^3)/(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 1.40 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx=\int \frac {\left (a + b \tanh {\left (e + f x \right )}\right )^{3}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 455, normalized size of antiderivative = 22.75 \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx=\int { \frac {{\left (b \tanh \left (f x + e\right ) + a\right )}^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

a^3*log(d*x + c)/d + (3*a^2*b + 3*a*b^2 + b^3)*log(d*x + c)/d + (6*a*b^2*d*f*x + 6*a*b^2*c*f - b^3*d + (6*a*b^
2*c*f*e^(2*e) + (2*c*f*e^(2*e) - d*e^(2*e))*b^3 + 2*(3*a*b^2*d*f*e^(2*e) + b^3*d*f*e^(2*e))*x)*e^(2*f*x))/(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)) - integrate(2*(3*a^2*b*c^2*f^2 - 3*a
*b^2*c*d*f + (c^2*f^2 + d^2)*b^3 + (3*a^2*b*d^2*f^2 + b^3*d^2*f^2)*x^2 + (6*a^2*b*c*d*f^2 + 2*b^3*c*d*f^2 - 3*
a*b^2*d^2*f)*x)/(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 = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx=\int { \frac {{\left (b \tanh \left (f x + e\right ) + a\right )}^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx=\int \frac {{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^3}{c+d\,x} \,d x \]

[In]

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

[Out]

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