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

Optimal. Leaf size=21 \[ e^{-2+e^{x+x^2}+(-6+x) x} (1+\log (x)) \]

[Out]

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

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(21)=42\).
time = 1.07, antiderivative size = 95, normalized size of antiderivative = 4.52, number of steps used = 1, number of rules used = 1, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2326} \begin {gather*} \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 )} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.30, size = 22, normalized size = 1.05 \begin {gather*} e^{-2+e^{x+x^2}-6 x+x^2} (1+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [A]
time = 0.04, size = 21, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[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((x+1)*x)+x^2-6*x-2)

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Maxima [A]
time = 0.32, size = 20, normalized size = 0.95 \begin {gather*} {\left (\log \left (x\right ) + 1\right )} e^{\left (x^{2} - 6 \, x + e^{\left (x^{2} + x\right )} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.35, size = 20, normalized size = 0.95 \begin {gather*} {\left (\log \left (x\right ) + 1\right )} e^{\left (x^{2} - 6 \, x + e^{\left (x^{2} + x\right )} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 126.09, size = 20, normalized size = 0.95 \begin {gather*} \left (\log {\left (x \right )} + 1\right ) e^{x^{2} - 6 x + e^{x^{2} + x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 7.03, size = 23, normalized size = 1.10 \begin {gather*} {\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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