3.11.88 \(\int \frac {(-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x) (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6))}{(-4+e^{\frac {x}{3+x^2+x^3}}+x) (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6)+e^{\frac {x}{3+x^2+x^3}} (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7))} \, dx\)

Optimal. Leaf size=25 \[ 3-\frac {x}{-4+e^{\frac {x}{3+x^2 (1+x)}}+x} \]

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Rubi [F]  time = 5.51, 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 {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-12 + 3*E^(x/(3 + x^2 + x^3)) + 2*x)*(36 + 24*x^2 + 24*x^3 + 4*x^4 + 8*x^5 + 4*x^6 + E^(x/(3 + x^2 + x^3
))*(-9 + 3*x - 6*x^2 - 7*x^3 - 3*x^4 - 2*x^5 - x^6)))/((-4 + E^(x/(3 + x^2 + x^3)) + x)*(432 - 180*x + 306*x^2
+ 168*x^3 - 60*x^4 + 88*x^5 + 10*x^6 - 16*x^7 + 2*x^8 + E^((2*x)/(3 + x^2 + x^3))*(27 + 18*x^2 + 18*x^3 + 3*x
^4 + 6*x^5 + 3*x^6) + E^(x/(3 + x^2 + x^3))*(-216 + 45*x - 144*x^2 - 114*x^3 + 6*x^4 - 43*x^5 - 14*x^6 + 5*x^7
))),x]

[Out]

Defer[Int][x/(-4 + E^(x/(3 + x^2 + x^3)) + x)^2, x] - Defer[Int][(-4 + E^(x/(3 + x^2 + x^3)) + x)^(-1), x] - 1
5*Defer[Int][1/((-4 + E^(x/(3 + x^2 + x^3)) + x)^2*(3 + x^2 + x^3)^2), x] + 39*Defer[Int][x/((-4 + E^(x/(3 + x
^2 + x^3)) + x)^2*(3 + x^2 + x^3)^2), x] - 14*Defer[Int][x^2/((-4 + E^(x/(3 + x^2 + x^3)) + x)^2*(3 + x^2 + x^
3)^2), x] - 3*Defer[Int][1/((-4 + E^(x/(3 + x^2 + x^3)) + x)*(3 + x^2 + x^3)^2), x] + 9*Defer[Int][x/((-4 + E^
(x/(3 + x^2 + x^3)) + x)*(3 + x^2 + x^3)^2), x] - Defer[Int][x^2/((-4 + E^(x/(3 + x^2 + x^3)) + x)*(3 + x^2 +
x^3)^2), x] + 5*Defer[Int][1/((-4 + E^(x/(3 + x^2 + x^3)) + x)^2*(3 + x^2 + x^3)), x] - 9*Defer[Int][x/((-4 +
E^(x/(3 + x^2 + x^3)) + x)^2*(3 + x^2 + x^3)), x] + 2*Defer[Int][x^2/((-4 + E^(x/(3 + x^2 + x^3)) + x)^2*(3 +
x^2 + x^3)), x] + Defer[Int][1/((-4 + E^(x/(3 + x^2 + x^3)) + x)*(3 + x^2 + x^3)), x] - 2*Defer[Int][x/((-4 +
E^(x/(3 + x^2 + x^3)) + x)*(3 + x^2 + x^3)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (3+x^2+x^3\right )^2-e^{\frac {x}{3+x^2+x^3}} \left (9-3 x+6 x^2+7 x^3+3 x^4+2 x^5+x^6\right )}{\left (4-e^{\frac {x}{3+x^2+x^3}}-x\right )^2 \left (3+x^2+x^3\right )^2} \, dx\\ &=\int \left (\frac {x \left (21-3 x+2 x^2-x^3+3 x^4+2 x^5+x^6\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}-\frac {9-3 x+6 x^2+7 x^3+3 x^4+2 x^5+x^6}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}\right ) \, dx\\ &=\int \frac {x \left (21-3 x+2 x^2-x^3+3 x^4+2 x^5+x^6\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx-\int \frac {9-3 x+6 x^2+7 x^3+3 x^4+2 x^5+x^6}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx\\ &=-\int \left (\frac {1}{-4+e^{\frac {x}{3+x^2+x^3}}+x}+\frac {3-9 x+x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}+\frac {-1+2 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )}\right ) \, dx+\int \left (\frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2}-\frac {15-39 x+14 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}+\frac {5-9 x+2 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )}\right ) \, dx\\ &=\int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2} \, dx-\int \frac {1}{-4+e^{\frac {x}{3+x^2+x^3}}+x} \, dx-\int \frac {3-9 x+x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx-\int \frac {15-39 x+14 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx-\int \frac {-1+2 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )} \, dx+\int \frac {5-9 x+2 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )} \, dx\\ &=\int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2} \, dx-\int \frac {1}{-4+e^{\frac {x}{3+x^2+x^3}}+x} \, dx-\int \left (\frac {15}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}-\frac {39 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}+\frac {14 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}\right ) \, dx-\int \left (\frac {3}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}-\frac {9 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}+\frac {x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}\right ) \, dx+\int \left (\frac {5}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )}-\frac {9 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )}+\frac {2 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )}\right ) \, dx-\int \left (-\frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )}+\frac {2 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )}\right ) \, dx\\ &=2 \int \frac {x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )} \, dx-2 \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )} \, dx-3 \int \frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx+5 \int \frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )} \, dx+9 \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx-9 \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )} \, dx-14 \int \frac {x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx-15 \int \frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx+39 \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx+\int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2} \, dx-\int \frac {1}{-4+e^{\frac {x}{3+x^2+x^3}}+x} \, dx-\int \frac {x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx+\int \frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 22, normalized size = 0.88 \begin {gather*} -\frac {x}{-4+e^{\frac {x}{3+x^2+x^3}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-12 + 3*E^(x/(3 + x^2 + x^3)) + 2*x)*(36 + 24*x^2 + 24*x^3 + 4*x^4 + 8*x^5 + 4*x^6 + E^(x/(3 + x^2
+ x^3))*(-9 + 3*x - 6*x^2 - 7*x^3 - 3*x^4 - 2*x^5 - x^6)))/((-4 + E^(x/(3 + x^2 + x^3)) + x)*(432 - 180*x + 3
06*x^2 + 168*x^3 - 60*x^4 + 88*x^5 + 10*x^6 - 16*x^7 + 2*x^8 + E^((2*x)/(3 + x^2 + x^3))*(27 + 18*x^2 + 18*x^3
+ 3*x^4 + 6*x^5 + 3*x^6) + E^(x/(3 + x^2 + x^3))*(-216 + 45*x - 144*x^2 - 114*x^3 + 6*x^4 - 43*x^5 - 14*x^6 +
5*x^7))),x]

[Out]

-(x/(-4 + E^(x/(3 + x^2 + x^3)) + x))

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fricas [A]  time = 0.63, size = 21, normalized size = 0.84 \begin {gather*} -\frac {x}{x + e^{\left (\frac {x}{x^{3} + x^{2} + 3}\right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6-2*x^5-3*x^4-7*x^3-6*x^2+3*x-9)*exp(x/(x^3+x^2+3))+4*x^6+8*x^5+4*x^4+24*x^3+24*x^2+36)*exp(x)/
((3*x^6+6*x^5+3*x^4+18*x^3+18*x^2+27)*exp(x/(x^3+x^2+3))^2+(5*x^7-14*x^6-43*x^5+6*x^4-114*x^3-144*x^2+45*x-216
)*exp(x/(x^3+x^2+3))+2*x^8-16*x^7+10*x^6+88*x^5-60*x^4+168*x^3+306*x^2-180*x+432)/exp(-log((3*exp(x/(x^3+x^2+3
))+2*x-12)/(exp(x/(x^3+x^2+3))+x-4))+x),x, algorithm="fricas")

[Out]

-x/(x + e^(x/(x^3 + x^2 + 3)) - 4)

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giac [A]  time = 1.13, size = 21, normalized size = 0.84 \begin {gather*} -\frac {x}{x + e^{\left (\frac {x}{x^{3} + x^{2} + 3}\right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6-2*x^5-3*x^4-7*x^3-6*x^2+3*x-9)*exp(x/(x^3+x^2+3))+4*x^6+8*x^5+4*x^4+24*x^3+24*x^2+36)*exp(x)/
((3*x^6+6*x^5+3*x^4+18*x^3+18*x^2+27)*exp(x/(x^3+x^2+3))^2+(5*x^7-14*x^6-43*x^5+6*x^4-114*x^3-144*x^2+45*x-216
)*exp(x/(x^3+x^2+3))+2*x^8-16*x^7+10*x^6+88*x^5-60*x^4+168*x^3+306*x^2-180*x+432)/exp(-log((3*exp(x/(x^3+x^2+3
))+2*x-12)/(exp(x/(x^3+x^2+3))+x-4))+x),x, algorithm="giac")

[Out]

-x/(x + e^(x/(x^3 + x^2 + 3)) - 4)

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maple [C]  time = 0.23, size = 197, normalized size = 7.88

method	result	size
risch	\(-\frac {x \,{\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (\frac {3 \,{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}}{2}+x -6\right )}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (\frac {3 \,{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}}{2}+x -6\right )}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\right )+\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (\frac {3 \,{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}}{2}+x -6\right )}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\right )+\mathrm {csgn}\left (i \left (\frac {3 \,{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}}{2}+x -6\right )\right )\right )}{2}}}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^6-2*x^5-3*x^4-7*x^3-6*x^2+3*x-9)*exp(x/(x^3+x^2+3))+4*x^6+8*x^5+4*x^4+24*x^3+24*x^2+36)*exp(x)/((3*x^
6+6*x^5+3*x^4+18*x^3+18*x^2+27)*exp(x/(x^3+x^2+3))^2+(5*x^7-14*x^6-43*x^5+6*x^4-114*x^3-144*x^2+45*x-216)*exp(
x/(x^3+x^2+3))+2*x^8-16*x^7+10*x^6+88*x^5-60*x^4+168*x^3+306*x^2-180*x+432)/exp(-ln((3*exp(x/(x^3+x^2+3))+2*x-
12)/(exp(x/(x^3+x^2+3))+x-4))+x),x,method=_RETURNVERBOSE)

[Out]

-1/(exp(x/(x^3+x^2+3))+x-4)*x*exp(-1/2*I*Pi*csgn(I/(exp(x/(x^3+x^2+3))+x-4)*(3/2*exp(x/(x^3+x^2+3))+x-6))*(-cs
gn(I/(exp(x/(x^3+x^2+3))+x-4)*(3/2*exp(x/(x^3+x^2+3))+x-6))+csgn(I/(exp(x/(x^3+x^2+3))+x-4)))*(-csgn(I/(exp(x/
(x^3+x^2+3))+x-4)*(3/2*exp(x/(x^3+x^2+3))+x-6))+csgn(I*(3/2*exp(x/(x^3+x^2+3))+x-6))))

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maxima [A]  time = 0.45, size = 21, normalized size = 0.84 \begin {gather*} -\frac {x}{x + e^{\left (\frac {x}{x^{3} + x^{2} + 3}\right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6-2*x^5-3*x^4-7*x^3-6*x^2+3*x-9)*exp(x/(x^3+x^2+3))+4*x^6+8*x^5+4*x^4+24*x^3+24*x^2+36)*exp(x)/
((3*x^6+6*x^5+3*x^4+18*x^3+18*x^2+27)*exp(x/(x^3+x^2+3))^2+(5*x^7-14*x^6-43*x^5+6*x^4-114*x^3-144*x^2+45*x-216
)*exp(x/(x^3+x^2+3))+2*x^8-16*x^7+10*x^6+88*x^5-60*x^4+168*x^3+306*x^2-180*x+432)/exp(-log((3*exp(x/(x^3+x^2+3
))+2*x-12)/(exp(x/(x^3+x^2+3))+x-4))+x),x, algorithm="maxima")

[Out]

-x/(x + e^(x/(x^3 + x^2 + 3)) - 4)

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mupad [B]  time = 1.34, size = 21, normalized size = 0.84 \begin {gather*} -\frac {x}{x+{\mathrm {e}}^{\frac {x}{x^3+x^2+3}}-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log((2*x + 3*exp(x/(x^2 + x^3 + 3)) - 12)/(x + exp(x/(x^2 + x^3 + 3)) - 4)) - x)*exp(x)*(24*x^2 + 24*
x^3 + 4*x^4 + 8*x^5 + 4*x^6 - exp(x/(x^2 + x^3 + 3))*(6*x^2 - 3*x + 7*x^3 + 3*x^4 + 2*x^5 + x^6 + 9) + 36))/(e
xp((2*x)/(x^2 + x^3 + 3))*(18*x^2 + 18*x^3 + 3*x^4 + 6*x^5 + 3*x^6 + 27) - 180*x - exp(x/(x^2 + x^3 + 3))*(144
*x^2 - 45*x + 114*x^3 - 6*x^4 + 43*x^5 + 14*x^6 - 5*x^7 + 216) + 306*x^2 + 168*x^3 - 60*x^4 + 88*x^5 + 10*x^6
- 16*x^7 + 2*x^8 + 432),x)

[Out]

-x/(x + exp(x/(x^2 + x^3 + 3)) - 4)

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sympy [A]  time = 0.48, size = 17, normalized size = 0.68 \begin {gather*} - \frac {x}{x + e^{\frac {x}{x^{3} + x^{2} + 3}} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**6-2*x**5-3*x**4-7*x**3-6*x**2+3*x-9)*exp(x/(x**3+x**2+3))+4*x**6+8*x**5+4*x**4+24*x**3+24*x**2
+36)*exp(x)/((3*x**6+6*x**5+3*x**4+18*x**3+18*x**2+27)*exp(x/(x**3+x**2+3))**2+(5*x**7-14*x**6-43*x**5+6*x**4-
114*x**3-144*x**2+45*x-216)*exp(x/(x**3+x**2+3))+2*x**8-16*x**7+10*x**6+88*x**5-60*x**4+168*x**3+306*x**2-180*
x+432)/exp(-ln((3*exp(x/(x**3+x**2+3))+2*x-12)/(exp(x/(x**3+x**2+3))+x-4))+x),x)

[Out]

-x/(x + exp(x/(x**3 + x**2 + 3)) - 4)

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