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

Optimal. Leaf size=22 \[ 3 \left (4+\frac {x}{\left (e^4+e^{4 x/3}-x\right )^2}\right ) \]

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Rubi [F]  time = 0.76, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 19, normalized size = 0.86 \begin {gather*} \frac {3 x}{\left (e^4+e^{4 x/3}-x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 1.12, size = 32, normalized size = 1.45 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 1.69, size = 35, normalized size = 1.59 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.91, size = 16, normalized size = 0.73

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.43, size = 32, normalized size = 1.45 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 4.91, size = 35, normalized size = 1.59 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.15, size = 37, normalized size = 1.68 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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