\(\int \frac {e^{\frac {1}{9} (9+9 e^{1+x}-9 x+x^4)} (-36-9 x+9 e^{1+x} x+4 x^4)}{9 x^5} \, dx\) [6622]

Optimal result

Integrand size = 46, antiderivative size = 23 \[ \int \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (-36-9 x+9 e^{1+x} x+4 x^4\right )}{9 x^5} \, dx=\frac {e^{1+e^{1+x}-x+\frac {x^4}{9}}}{x^4} \]

[Out]

exp(exp(1)*exp(x)+1/9*x^4-x+1)/x^4

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(23)=46\).

Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12, 2326} \[ \int \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (-36-9 x+9 e^{1+x} x+4 x^4\right )}{9 x^5} \, dx=\frac {e^{\frac {1}{9} \left (x^4-9 x+9 e^{x+1}+9\right )} \left (-4 x^4-9 e^{x+1} x+9 x\right )}{x^5 \left (-4 x^3-9 e^{x+1}+9\right )} \]

[In]

Int[(E^((9 + 9*E^(1 + x) - 9*x + x^4)/9)*(-36 - 9*x + 9*E^(1 + x)*x + 4*x^4))/(9*x^5),x]

[Out]

(E^((9 + 9*E^(1 + x) - 9*x + x^4)/9)*(9*x - 9*E^(1 + x)*x - 4*x^4))/(x^5*(9 - 9*E^(1 + x) - 4*x^3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \int \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (-36-9 x+9 e^{1+x} x+4 x^4\right )}{x^5} \, dx \\ & = \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (9 x-9 e^{1+x} x-4 x^4\right )}{x^5 \left (9-9 e^{1+x}-4 x^3\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (-36-9 x+9 e^{1+x} x+4 x^4\right )}{9 x^5} \, dx=\frac {e^{1+e^{1+x}-x+\frac {x^4}{9}}}{x^4} \]

[In]

Integrate[(E^((9 + 9*E^(1 + x) - 9*x + x^4)/9)*(-36 - 9*x + 9*E^(1 + x)*x + 4*x^4))/(9*x^5),x]

[Out]

E^(1 + E^(1 + x) - x + x^4/9)/x^4

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

method	result	size
risch	\(\frac {{\mathrm e}^{{\mathrm e}^{1+x}+\frac {x^{4}}{9}-x +1}}{x^{4}}\)	\(20\)
norman	\(\frac {{\mathrm e}^{{\mathrm e} \,{\mathrm e}^{x}+\frac {x^{4}}{9}-x +1}}{x^{4}}\)	\(21\)
parallelrisch	\(\frac {{\mathrm e}^{{\mathrm e} \,{\mathrm e}^{x}+\frac {x^{4}}{9}-x +1}}{x^{4}}\)	\(21\)

[In]

int(1/9*(9*x*exp(1)*exp(x)+4*x^4-9*x-36)*exp(exp(1)*exp(x)+1/9*x^4-x+1)/x^5,x,method=_RETURNVERBOSE)

[Out]

exp(exp(1+x)+1/9*x^4-x+1)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (-36-9 x+9 e^{1+x} x+4 x^4\right )}{9 x^5} \, dx=\frac {e^{\left (\frac {1}{9} \, x^{4} - x + e^{\left (x + 1\right )} + 1\right )}}{x^{4}} \]

[In]

integrate(1/9*(9*x*exp(1)*exp(x)+4*x^4-9*x-36)*exp(exp(1)*exp(x)+1/9*x^4-x+1)/x^5,x, algorithm="fricas")

[Out]

e^(1/9*x^4 - x + e^(x + 1) + 1)/x^4

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (-36-9 x+9 e^{1+x} x+4 x^4\right )}{9 x^5} \, dx=\frac {e^{\frac {x^{4}}{9} - x + e e^{x} + 1}}{x^{4}} \]

[In]

integrate(1/9*(9*x*exp(1)*exp(x)+4*x**4-9*x-36)*exp(exp(1)*exp(x)+1/9*x**4-x+1)/x**5,x)

[Out]

exp(x**4/9 - x + E*exp(x) + 1)/x**4

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (-36-9 x+9 e^{1+x} x+4 x^4\right )}{9 x^5} \, dx=\frac {e^{\left (\frac {1}{9} \, x^{4} - x + e^{\left (x + 1\right )} + 1\right )}}{x^{4}} \]

[In]

integrate(1/9*(9*x*exp(1)*exp(x)+4*x^4-9*x-36)*exp(exp(1)*exp(x)+1/9*x^4-x+1)/x^5,x, algorithm="maxima")

[Out]

e^(1/9*x^4 - x + e^(x + 1) + 1)/x^4

Giac [F]

\[ \int \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (-36-9 x+9 e^{1+x} x+4 x^4\right )}{9 x^5} \, dx=\int { \frac {{\left (4 \, x^{4} + 9 \, x e^{\left (x + 1\right )} - 9 \, x - 36\right )} e^{\left (\frac {1}{9} \, x^{4} - x + e^{\left (x + 1\right )} + 1\right )}}{9 \, x^{5}} \,d x } \]

[In]

integrate(1/9*(9*x*exp(1)*exp(x)+4*x^4-9*x-36)*exp(exp(1)*exp(x)+1/9*x^4-x+1)/x^5,x, algorithm="giac")

[Out]

integrate(1/9*(4*x^4 + 9*x*e^(x + 1) - 9*x - 36)*e^(1/9*x^4 - x + e^(x + 1) + 1)/x^5, x)

Mupad [B] (verification not implemented)

Time = 11.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {1}{9} \left (9+9 e^{1+x}-9 x+x^4\right )} \left (-36-9 x+9 e^{1+x} x+4 x^4\right )}{9 x^5} \, dx=\frac {{\mathrm {e}}^{\mathrm {e}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-x}\,\mathrm {e}\,{\mathrm {e}}^{\frac {x^4}{9}}}{x^4} \]

[In]

int(-(exp(exp(1)*exp(x) - x + x^4/9 + 1)*(9*x - 4*x^4 - 9*x*exp(1)*exp(x) + 36))/(9*x^5),x)

[Out]

(exp(exp(1)*exp(x))*exp(-x)*exp(1)*exp(x^4/9))/x^4