\(\int \frac {e^{-x} (-4 e^x x+e^{x^3} (-81-18 x-x^2+e^x (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6))+e^{x^3} (81 x+18 x^2-242 x^3-54 x^4-3 x^5) \log (x))}{81 x+18 x^2+x^3} \, dx\) [434]

Optimal result

Integrand size = 108, antiderivative size = 28 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\frac {4}{9+x}+e^{x^3} \left (3-x-e^{-x} \log (x)\right ) \]

[Out]

4/(x+9)+(3-x-ln(x)/exp(x))*exp(x^3)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.80 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1608, 27, 6820, 2258, 2239, 2240, 2250, 6838, 2634} \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=3 e^{x^3}-e^{x^3-x} \log (x)+\frac {x \Gamma \left (\frac {1}{3},-x^3\right )}{3 \sqrt [3]{-x^3}}+\frac {x^4 \Gamma \left (\frac {4}{3},-x^3\right )}{\left (-x^3\right )^{4/3}}+\frac {4}{x+9} \]

[In]

Int[(-4*E^x*x + E^x^3*(-81 - 18*x - x^2 + E^x*(-81*x - 18*x^2 + 728*x^3 - 81*x^4 - 45*x^5 - 3*x^6)) + E^x^3*(8
1*x + 18*x^2 - 242*x^3 - 54*x^4 - 3*x^5)*Log[x])/(E^x*(81*x + 18*x^2 + x^3)),x]

[Out]

3*E^x^3 + 4/(9 + x) + (x*Gamma[1/3, -x^3])/(3*(-x^3)^(1/3)) + (x^4*Gamma[4/3, -x^3])/(-x^3)^(4/3) - E^(-x + x^
3)*Log[x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{x \left (81+18 x+x^2\right )} \, dx \\ & = \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{x (9+x)^2} \, dx \\ & = \int \left (-\frac {e^{-x+x^3}}{x}-\frac {4}{(9+x)^2}+e^{x^3} \left (-1+9 x^2-3 x^3\right )+e^{-x+x^3} \left (1-3 x^2\right ) \log (x)\right ) \, dx \\ & = \frac {4}{9+x}-\int \frac {e^{-x+x^3}}{x} \, dx+\int e^{x^3} \left (-1+9 x^2-3 x^3\right ) \, dx+\int e^{-x+x^3} \left (1-3 x^2\right ) \log (x) \, dx \\ & = \frac {4}{9+x}-e^{-x+x^3} \log (x)+\int \left (-e^{x^3}+9 e^{x^3} x^2-3 e^{x^3} x^3\right ) \, dx \\ & = \frac {4}{9+x}-e^{-x+x^3} \log (x)-3 \int e^{x^3} x^3 \, dx+9 \int e^{x^3} x^2 \, dx-\int e^{x^3} \, dx \\ & = 3 e^{x^3}+\frac {4}{9+x}+\frac {x \Gamma \left (\frac {1}{3},-x^3\right )}{3 \sqrt [3]{-x^3}}+\frac {x^4 \Gamma \left (\frac {4}{3},-x^3\right )}{\left (-x^3\right )^{4/3}}-e^{-x+x^3} \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-e^{x^3} (-3+x)+\frac {4}{9+x}-e^{-x+x^3} \log (x) \]

[In]

Integrate[(-4*E^x*x + E^x^3*(-81 - 18*x - x^2 + E^x*(-81*x - 18*x^2 + 728*x^3 - 81*x^4 - 45*x^5 - 3*x^6)) + E^
x^3*(81*x + 18*x^2 - 242*x^3 - 54*x^4 - 3*x^5)*Log[x])/(E^x*(81*x + 18*x^2 + x^3)),x]

[Out]

-(E^x^3*(-3 + x)) + 4/(9 + x) - E^(-x + x^3)*Log[x]

Maple [A] (verified)

Time = 1.41 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61

method	result	size
risch	\(-\ln \left (x \right ) {\mathrm e}^{x \left (-1+x \right ) \left (1+x \right )}-\frac {x^{2} {\mathrm e}^{x^{3}}+6 \,{\mathrm e}^{x^{3}} x -27 \,{\mathrm e}^{x^{3}}-4}{x +9}\)	\(45\)
parallelrisch	\(-\frac {\left (9 \,{\mathrm e}^{x} {\mathrm e}^{x^{3}} x^{2}+9 x \,{\mathrm e}^{x^{3}} \ln \left (x \right )+54 \,{\mathrm e}^{x} {\mathrm e}^{x^{3}} x +81 \,{\mathrm e}^{x^{3}} \ln \left (x \right )-243 \,{\mathrm e}^{x} {\mathrm e}^{x^{3}}-36 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{9 \left (x +9\right )}\)	\(62\)

[In]

int(((-3*x^5-54*x^4-242*x^3+18*x^2+81*x)*exp(x^3)*ln(x)+((-3*x^6-45*x^5-81*x^4+728*x^3-18*x^2-81*x)*exp(x)-x^2
-18*x-81)*exp(x^3)-4*exp(x)*x)/(x^3+18*x^2+81*x)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)*exp(x*(-1+x)*(1+x))-(x^2*exp(x^3)+6*exp(x^3)*x-27*exp(x^3)-4)/(x+9)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-\frac {{\left ({\left (x + 9\right )} e^{\left (x^{3}\right )} \log \left (x\right ) + {\left (x^{2} + 6 \, x - 27\right )} e^{\left (x^{3} + x\right )} - 4 \, e^{x}\right )} e^{\left (-x\right )}}{x + 9} \]

[In]

integrate(((-3*x^5-54*x^4-242*x^3+18*x^2+81*x)*exp(x^3)*log(x)+((-3*x^6-45*x^5-81*x^4+728*x^3-18*x^2-81*x)*exp
(x)-x^2-18*x-81)*exp(x^3)-4*exp(x)*x)/(x^3+18*x^2+81*x)/exp(x),x, algorithm="fricas")

[Out]

-((x + 9)*e^(x^3)*log(x) + (x^2 + 6*x - 27)*e^(x^3 + x) - 4*e^x)*e^(-x)/(x + 9)

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\left (- x e^{x} + 3 e^{x} - \log {\left (x \right )}\right ) e^{- x} e^{x^{3}} + \frac {4}{x + 9} \]

[In]

integrate(((-3*x**5-54*x**4-242*x**3+18*x**2+81*x)*exp(x**3)*ln(x)+((-3*x**6-45*x**5-81*x**4+728*x**3-18*x**2-
81*x)*exp(x)-x**2-18*x-81)*exp(x**3)-4*exp(x)*x)/(x**3+18*x**2+81*x)/exp(x),x)

[Out]

(-x*exp(x) + 3*exp(x) - log(x))*exp(-x)*exp(x**3) + 4/(x + 9)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-{\left ({\left (x - 3\right )} e^{x} + \log \left (x\right )\right )} e^{\left (x^{3} - x\right )} + \frac {4}{x + 9} \]

[In]

integrate(((-3*x^5-54*x^4-242*x^3+18*x^2+81*x)*exp(x^3)*log(x)+((-3*x^6-45*x^5-81*x^4+728*x^3-18*x^2-81*x)*exp
(x)-x^2-18*x-81)*exp(x^3)-4*exp(x)*x)/(x^3+18*x^2+81*x)/exp(x),x, algorithm="maxima")

[Out]

-((x - 3)*e^x + log(x))*e^(x^3 - x) + 4/(x + 9)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-\frac {x^{2} e^{\left (x^{3}\right )} + x e^{\left (x^{3} - x\right )} \log \left (x\right ) + 6 \, x e^{\left (x^{3}\right )} + 9 \, e^{\left (x^{3} - x\right )} \log \left (x\right ) - 27 \, e^{\left (x^{3}\right )} - 4}{x + 9} \]

[In]

integrate(((-3*x^5-54*x^4-242*x^3+18*x^2+81*x)*exp(x^3)*log(x)+((-3*x^6-45*x^5-81*x^4+728*x^3-18*x^2-81*x)*exp
(x)-x^2-18*x-81)*exp(x^3)-4*exp(x)*x)/(x^3+18*x^2+81*x)/exp(x),x, algorithm="giac")

[Out]

-(x^2*e^(x^3) + x*e^(x^3 - x)*log(x) + 6*x*e^(x^3) + 9*e^(x^3 - x)*log(x) - 27*e^(x^3) - 4)/(x + 9)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\int -\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{x^3}\,\left (18\,x+{\mathrm {e}}^x\,\left (3\,x^6+45\,x^5+81\,x^4-728\,x^3+18\,x^2+81\,x\right )+x^2+81\right )+4\,x\,{\mathrm {e}}^x+{\mathrm {e}}^{x^3}\,\ln \left (x\right )\,\left (3\,x^5+54\,x^4+242\,x^3-18\,x^2-81\,x\right )\right )}{x^3+18\,x^2+81\,x} \,d x \]

[In]

int(-(exp(-x)*(exp(x^3)*(18*x + exp(x)*(81*x + 18*x^2 - 728*x^3 + 81*x^4 + 45*x^5 + 3*x^6) + x^2 + 81) + 4*x*e
xp(x) + exp(x^3)*log(x)*(242*x^3 - 18*x^2 - 81*x + 54*x^4 + 3*x^5)))/(81*x + 18*x^2 + x^3),x)

[Out]

int(-(exp(-x)*(exp(x^3)*(18*x + exp(x)*(81*x + 18*x^2 - 728*x^3 + 81*x^4 + 45*x^5 + 3*x^6) + x^2 + 81) + 4*x*e
xp(x) + exp(x^3)*log(x)*(242*x^3 - 18*x^2 - 81*x + 54*x^4 + 3*x^5)))/(81*x + 18*x^2 + x^3), x)