\(\int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} (3 e^4-3 x)-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} (3 e^8-6 e^4 x+3 x^2)} \, dx\) [6572]

Optimal result

Integrand size = 95, antiderivative size = 22 \[ \int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=3 \left (4+\frac {x}{\left (e^4+e^{4 x/3}-x\right )^2}\right ) \]

[Out]

12+3*x/(exp(4/3*x)-x+exp(4))^2

Rubi [F]

\[ \int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx \]

[In]

Int[(3*E^4 + E^((4*x)/3)*(3 - 8*x) + 3*x)/(E^12 + E^(4*x) + E^((8*x)/3)*(3*E^4 - 3*x) - 3*E^8*x + 3*E^4*x^2 -
x^3 + E^((4*x)/3)*(3*E^8 - 6*E^4*x + 3*x^2)),x]

[Out]

2*(3 + 4*E^4)*Defer[Int][x/(E^4 + E^((4*x)/3) - x)^3, x] - 8*Defer[Int][x/(E^4 + E^((4*x)/3) - x)^2, x] - 8*De
fer[Int][x^2/(E^4 + E^((4*x)/3) - x)^3, x] + 9*Defer[Subst][Defer[Int][(E^4 + E^(4*x) - 3*x)^(-2), x], x, x/3]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx \\ & = \int \left (\frac {2 \left (3+4 e^4-4 x\right ) x}{\left (e^4+e^{4 x/3}-x\right )^3}-\frac {-3+8 x}{\left (-e^4-e^{4 x/3}+x\right )^2}\right ) \, dx \\ & = 2 \int \frac {\left (3+4 e^4-4 x\right ) x}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx-\int \frac {-3+8 x}{\left (-e^4-e^{4 x/3}+x\right )^2} \, dx \\ & = 2 \int \left (\frac {\left (3+4 e^4\right ) x}{\left (e^4+e^{4 x/3}-x\right )^3}-\frac {4 x^2}{\left (e^4+e^{4 x/3}-x\right )^3}\right ) \, dx-\int \left (-\frac {3}{\left (e^4+e^{4 x/3}-x\right )^2}+\frac {8 x}{\left (e^4+e^{4 x/3}-x\right )^2}\right ) \, dx \\ & = 3 \int \frac {1}{\left (e^4+e^{4 x/3}-x\right )^2} \, dx-8 \int \frac {x}{\left (e^4+e^{4 x/3}-x\right )^2} \, dx-8 \int \frac {x^2}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx+\left (2 \left (3+4 e^4\right )\right ) \int \frac {x}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx \\ & = -\left (8 \int \frac {x}{\left (e^4+e^{4 x/3}-x\right )^2} \, dx\right )-8 \int \frac {x^2}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx+9 \text {Subst}\left (\int \frac {1}{\left (e^4+e^{4 x}-3 x\right )^2} \, dx,x,\frac {x}{3}\right )+\left (2 \left (3+4 e^4\right )\right ) \int \frac {x}{\left (e^4+e^{4 x/3}-x\right )^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {3 x}{\left (e^4+e^{4 x/3}-x\right )^2} \]

[In]

Integrate[(3*E^4 + E^((4*x)/3)*(3 - 8*x) + 3*x)/(E^12 + E^(4*x) + E^((8*x)/3)*(3*E^4 - 3*x) - 3*E^8*x + 3*E^4*
x^2 - x^3 + E^((4*x)/3)*(3*E^8 - 6*E^4*x + 3*x^2)),x]

[Out]

(3*x)/(E^4 + E^((4*x)/3) - x)^2

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73

method	result	size
norman	\(\frac {3 x}{\left ({\mathrm e}^{\frac {4 x}{3}}-x +{\mathrm e}^{4}\right )^{2}}\)	\(16\)
risch	\(\frac {3 x}{\left ({\mathrm e}^{\frac {4 x}{3}}-x +{\mathrm e}^{4}\right )^{2}}\)	\(16\)
parallelrisch	\(\frac {3 x}{{\mathrm e}^{8}-2 x \,{\mathrm e}^{4}+2 \,{\mathrm e}^{\frac {4 x}{3}} {\mathrm e}^{4}+x^{2}-2 \,{\mathrm e}^{\frac {4 x}{3}} x +{\mathrm e}^{\frac {8 x}{3}}}\)	\(40\)

[In]

int(((3-8*x)*exp(4/3*x)+3*exp(4)+3*x)/(exp(4/3*x)^3+(3*exp(4)-3*x)*exp(4/3*x)^2+(3*exp(4)^2-6*x*exp(4)+3*x^2)*
exp(4/3*x)+exp(4)^3-3*x*exp(4)^2+3*x^2*exp(4)-x^3),x,method=_RETURNVERBOSE)

[Out]

3*x/(exp(4/3*x)-x+exp(4))^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {3 \, x}{x^{2} - 2 \, x e^{4} - 2 \, {\left (x - e^{4}\right )} e^{\left (\frac {4}{3} \, x\right )} + e^{8} + e^{\left (\frac {8}{3} \, x\right )}} \]

[In]

integrate(((3-8*x)*exp(4/3*x)+3*exp(4)+3*x)/(exp(4/3*x)^3+(3*exp(4)-3*x)*exp(4/3*x)^2+(3*exp(4)^2-6*x*exp(4)+3
*x^2)*exp(4/3*x)+exp(4)^3-3*x*exp(4)^2+3*x^2*exp(4)-x^3),x, algorithm="fricas")

[Out]

3*x/(x^2 - 2*x*e^4 - 2*(x - e^4)*e^(4/3*x) + e^8 + e^(8/3*x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {3 x}{x^{2} - 2 x e^{4} + \left (- 2 x + 2 e^{4}\right ) e^{\frac {4 x}{3}} + e^{\frac {8 x}{3}} + e^{8}} \]

[In]

integrate(((3-8*x)*exp(4/3*x)+3*exp(4)+3*x)/(exp(4/3*x)**3+(3*exp(4)-3*x)*exp(4/3*x)**2+(3*exp(4)**2-6*x*exp(4
)+3*x**2)*exp(4/3*x)+exp(4)**3-3*x*exp(4)**2+3*x**2*exp(4)-x**3),x)

[Out]

3*x/(x**2 - 2*x*exp(4) + (-2*x + 2*exp(4))*exp(4*x/3) + exp(8*x/3) + exp(8))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {3 \, x}{x^{2} - 2 \, x e^{4} - 2 \, {\left (x - e^{4}\right )} e^{\left (\frac {4}{3} \, x\right )} + e^{8} + e^{\left (\frac {8}{3} \, x\right )}} \]

[In]

integrate(((3-8*x)*exp(4/3*x)+3*exp(4)+3*x)/(exp(4/3*x)^3+(3*exp(4)-3*x)*exp(4/3*x)^2+(3*exp(4)^2-6*x*exp(4)+3
*x^2)*exp(4/3*x)+exp(4)^3-3*x*exp(4)^2+3*x^2*exp(4)-x^3),x, algorithm="maxima")

[Out]

3*x/(x^2 - 2*x*e^4 - 2*(x - e^4)*e^(4/3*x) + e^8 + e^(8/3*x))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {3 \, x}{x^{2} - 2 \, x e^{4} - 2 \, x e^{\left (\frac {4}{3} \, x\right )} + e^{8} + e^{\left (\frac {8}{3} \, x\right )} + 2 \, e^{\left (\frac {4}{3} \, x + 4\right )}} \]

[In]

integrate(((3-8*x)*exp(4/3*x)+3*exp(4)+3*x)/(exp(4/3*x)^3+(3*exp(4)-3*x)*exp(4/3*x)^2+(3*exp(4)^2-6*x*exp(4)+3
*x^2)*exp(4/3*x)+exp(4)^3-3*x*exp(4)^2+3*x^2*exp(4)-x^3),x, algorithm="giac")

[Out]

3*x/(x^2 - 2*x*e^4 - 2*x*e^(4/3*x) + e^8 + e^(8/3*x) + 2*e^(4/3*x + 4))

Mupad [B] (verification not implemented)

Time = 13.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {3 e^4+e^{4 x/3} (3-8 x)+3 x}{e^{12}+e^{4 x}+e^{8 x/3} \left (3 e^4-3 x\right )-3 e^8 x+3 e^4 x^2-x^3+e^{4 x/3} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {3\,x}{{\mathrm {e}}^{\frac {8\,x}{3}}+{\mathrm {e}}^8+2\,{\mathrm {e}}^{\frac {4\,x}{3}}\,{\mathrm {e}}^4-2\,x\,{\mathrm {e}}^{\frac {4\,x}{3}}-2\,x\,{\mathrm {e}}^4+x^2} \]

[In]

int((3*x + 3*exp(4) - exp((4*x)/3)*(8*x - 3))/(exp(4*x) + exp(12) + exp((4*x)/3)*(3*exp(8) - 6*x*exp(4) + 3*x^
2) - 3*x*exp(8) - exp((8*x)/3)*(3*x - 3*exp(4)) + 3*x^2*exp(4) - x^3),x)

[Out]

(3*x)/(exp((8*x)/3) + exp(8) + 2*exp((4*x)/3)*exp(4) - 2*x*exp((4*x)/3) - 2*x*exp(4) + x^2)