\(\int \frac {e^{-2+e^{x+x^2}-6 x+x^2} (1-6 x+2 x^2+e^{x+x^2} (x+2 x^2)+(-6 x+2 x^2+e^{x+x^2} (x+2 x^2)) \log (x))}{x} \, dx\) [10291]

Optimal result

Integrand size = 73, antiderivative size = 21 \[ \int \frac {e^{-2+e^{x+x^2}-6 x+x^2} \left (1-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )+\left (-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )\right ) \log (x)\right )}{x} \, dx=e^{-2+e^{x+x^2}+(-6+x) x} (1+\log (x)) \]

[Out]

exp(x*(-6+x)-2+exp(x^2+x))*(ln(x)+1)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(21)=42\).

Time = 0.61 (sec) , antiderivative size = 95, normalized size of antiderivative = 4.52, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2326} \[ \int \frac {e^{-2+e^{x+x^2}-6 x+x^2} \left (1-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )+\left (-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )\right ) \log (x)\right )}{x} \, dx=\frac {e^{x^2+e^{x^2+x}-6 x-2} \left (-2 x^2-e^{x^2+x} \left (2 x^2+x\right )+\left (-2 x^2-e^{x^2+x} \left (2 x^2+x\right )+6 x\right ) \log (x)+6 x\right )}{x \left (-e^{x^2+x} (2 x+1)-2 x+6\right )} \]

[In]

Int[(E^(-2 + E^(x + x^2) - 6*x + x^2)*(1 - 6*x + 2*x^2 + E^(x + x^2)*(x + 2*x^2) + (-6*x + 2*x^2 + E^(x + x^2)
*(x + 2*x^2))*Log[x]))/x,x]

[Out]

(E^(-2 + E^(x + x^2) - 6*x + x^2)*(6*x - 2*x^2 - E^(x + x^2)*(x + 2*x^2) + (6*x - 2*x^2 - E^(x + x^2)*(x + 2*x
^2))*Log[x]))/(x*(6 - 2*x - E^(x + x^2)*(1 + 2*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 {e^{-2+e^{x+x^2}-6 x+x^2} \left (6 x-2 x^2-e^{x+x^2} \left (x+2 x^2\right )+\left (6 x-2 x^2-e^{x+x^2} \left (x+2 x^2\right )\right ) \log (x)\right )}{x \left (6-2 x-e^{x+x^2} (1+2 x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-2+e^{x+x^2}-6 x+x^2} \left (1-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )+\left (-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )\right ) \log (x)\right )}{x} \, dx=e^{-2+e^{x+x^2}-6 x+x^2} (1+\log (x)) \]

[In]

Integrate[(E^(-2 + E^(x + x^2) - 6*x + x^2)*(1 - 6*x + 2*x^2 + E^(x + x^2)*(x + 2*x^2) + (-6*x + 2*x^2 + E^(x
+ x^2)*(x + 2*x^2))*Log[x]))/x,x]

[Out]

E^(-2 + E^(x + x^2) - 6*x + x^2)*(1 + Log[x])

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

method	result	size
risch	\(\left (\ln \left (x \right )+1\right ) {\mathrm e}^{{\mathrm e}^{\left (1+x \right ) x}+x^{2}-6 x -2}\)	\(21\)
parallelrisch	\(\ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{x^{2}+x}+x^{2}-6 x -2}+{\mathrm e}^{{\mathrm e}^{x^{2}+x}+x^{2}-6 x -2}\)	\(35\)

[In]

int((((2*x^2+x)*exp(x^2+x)+2*x^2-6*x)*ln(x)+(2*x^2+x)*exp(x^2+x)+2*x^2-6*x+1)*exp(exp(x^2+x)+x^2-6*x-2)/x,x,me
thod=_RETURNVERBOSE)

[Out]

(ln(x)+1)*exp(exp((1+x)*x)+x^2-6*x-2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-2+e^{x+x^2}-6 x+x^2} \left (1-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )+\left (-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )\right ) \log (x)\right )}{x} \, dx={\left (\log \left (x\right ) + 1\right )} e^{\left (x^{2} - 6 \, x + e^{\left (x^{2} + x\right )} - 2\right )} \]

[In]

integrate((((2*x^2+x)*exp(x^2+x)+2*x^2-6*x)*log(x)+(2*x^2+x)*exp(x^2+x)+2*x^2-6*x+1)*exp(exp(x^2+x)+x^2-6*x-2)
/x,x, algorithm="fricas")

[Out]

(log(x) + 1)*e^(x^2 - 6*x + e^(x^2 + x) - 2)

Sympy [A] (verification not implemented)

Time = 129.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-2+e^{x+x^2}-6 x+x^2} \left (1-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )+\left (-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )\right ) \log (x)\right )}{x} \, dx=\left (\log {\left (x \right )} + 1\right ) e^{x^{2} - 6 x + e^{x^{2} + x} - 2} \]

[In]

integrate((((2*x**2+x)*exp(x**2+x)+2*x**2-6*x)*ln(x)+(2*x**2+x)*exp(x**2+x)+2*x**2-6*x+1)*exp(exp(x**2+x)+x**2
-6*x-2)/x,x)

[Out]

(log(x) + 1)*exp(x**2 - 6*x + exp(x**2 + x) - 2)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-2+e^{x+x^2}-6 x+x^2} \left (1-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )+\left (-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )\right ) \log (x)\right )}{x} \, dx={\left (\log \left (x\right ) + 1\right )} e^{\left (x^{2} - 6 \, x + e^{\left (x^{2} + x\right )} - 2\right )} \]

[In]

integrate((((2*x^2+x)*exp(x^2+x)+2*x^2-6*x)*log(x)+(2*x^2+x)*exp(x^2+x)+2*x^2-6*x+1)*exp(exp(x^2+x)+x^2-6*x-2)
/x,x, algorithm="maxima")

[Out]

(log(x) + 1)*e^(x^2 - 6*x + e^(x^2 + x) - 2)

Giac [F]

\[ \int \frac {e^{-2+e^{x+x^2}-6 x+x^2} \left (1-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )+\left (-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )\right ) \log (x)\right )}{x} \, dx=\int { \frac {{\left (2 \, x^{2} + {\left (2 \, x^{2} + x\right )} e^{\left (x^{2} + x\right )} + {\left (2 \, x^{2} + {\left (2 \, x^{2} + x\right )} e^{\left (x^{2} + x\right )} - 6 \, x\right )} \log \left (x\right ) - 6 \, x + 1\right )} e^{\left (x^{2} - 6 \, x + e^{\left (x^{2} + x\right )} - 2\right )}}{x} \,d x } \]

[In]

integrate((((2*x^2+x)*exp(x^2+x)+2*x^2-6*x)*log(x)+(2*x^2+x)*exp(x^2+x)+2*x^2-6*x+1)*exp(exp(x^2+x)+x^2-6*x-2)
/x,x, algorithm="giac")

[Out]

integrate((2*x^2 + (2*x^2 + x)*e^(x^2 + x) + (2*x^2 + (2*x^2 + x)*e^(x^2 + x) - 6*x)*log(x) - 6*x + 1)*e^(x^2
- 6*x + e^(x^2 + x) - 2)/x, x)

Mupad [B] (verification not implemented)

Time = 17.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-2+e^{x+x^2}-6 x+x^2} \left (1-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )+\left (-6 x+2 x^2+e^{x+x^2} \left (x+2 x^2\right )\right ) \log (x)\right )}{x} \, dx={\mathrm {e}}^{-6\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x}\,\left (\ln \left (x\right )+1\right ) \]

[In]

int((exp(exp(x + x^2) - 6*x + x^2 - 2)*(log(x)*(exp(x + x^2)*(x + 2*x^2) - 6*x + 2*x^2) - 6*x + exp(x + x^2)*(
x + 2*x^2) + 2*x^2 + 1))/x,x)

[Out]

exp(-6*x)*exp(x^2)*exp(-2)*exp(exp(x^2)*exp(x))*(log(x) + 1)