3.74.15 \(\int \frac {e^{x+x^2} (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6))+e^{x+x^2} (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6) \log (x)+(e^{x+x^2} (x^4+2 x^5+e^4 (-x^4-2 x^5))+e^{x+x^2} (-x^4-2 x^5) \log (x)) \log (-1+e^4+\log (x))}{-9+6 x^3-x^6+e^4 (9-6 x^3+x^6)+(9-6 x^3+x^6) \log (x)+(-6 x^2+2 x^5+e^4 (6 x^2-2 x^5)+(6 x^2-2 x^5) \log (x)) \log (-1+e^4+\log (x))+(-x^4+e^4 x^4+x^4 \log (x)) \log ^2(-1+e^4+\log (x))} \, dx\)

Optimal. Leaf size=31 \[ \frac {e^{x+x^2} x}{-\frac {3}{x}+x \left (x-\log \left (-1+e^4+\log (x)\right )\right )} \]

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Rubi [F]  time = 13.28, 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 {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(x + x^2)*(6*x + 3*x^2 + 7*x^3 + x^4 - x^5 - 2*x^6 + E^4*(-6*x - 3*x^2 - 6*x^3 - x^4 + x^5 + 2*x^6)) +
E^(x + x^2)*(-6*x - 3*x^2 - 6*x^3 - x^4 + x^5 + 2*x^6)*Log[x] + (E^(x + x^2)*(x^4 + 2*x^5 + E^4*(-x^4 - 2*x^5)
) + E^(x + x^2)*(-x^4 - 2*x^5)*Log[x])*Log[-1 + E^4 + Log[x]])/(-9 + 6*x^3 - x^6 + E^4*(9 - 6*x^3 + x^6) + (9
- 6*x^3 + x^6)*Log[x] + (-6*x^2 + 2*x^5 + E^4*(6*x^2 - 2*x^5) + (6*x^2 - 2*x^5)*Log[x])*Log[-1 + E^4 + Log[x]]
+ (-x^4 + E^4*x^4 + x^4*Log[x])*Log[-1 + E^4 + Log[x]]^2),x]

[Out]

6*(1 - E^4)*Defer[Int][(E^(x + x^2)*x)/((-1 + E^4 + Log[x])*(-3 + x^3 - x^2*Log[-1 + E^4 + Log[x]])^2), x] + D
efer[Int][(E^(x + x^2)*x^3)/((-1 + E^4 + Log[x])*(-3 + x^3 - x^2*Log[-1 + E^4 + Log[x]])^2), x] + (1 - E^4)*De
fer[Int][(E^(x + x^2)*x^4)/((-1 + E^4 + Log[x])*(-3 + x^3 - x^2*Log[-1 + E^4 + Log[x]])^2), x] - 6*Defer[Int][
(E^(x + x^2)*x*Log[x])/((-1 + E^4 + Log[x])*(-3 + x^3 - x^2*Log[-1 + E^4 + Log[x]])^2), x] - Defer[Int][(E^(x
+ x^2)*x^4*Log[x])/((-1 + E^4 + Log[x])*(-3 + x^3 - x^2*Log[-1 + E^4 + Log[x]])^2), x] + Defer[Int][(E^(x + x^
2)*x^2)/(-3 + x^3 - x^2*Log[-1 + E^4 + Log[x]]), x] + 2*Defer[Int][(E^(x + x^2)*x^3)/(-3 + x^3 - x^2*Log[-1 +
E^4 + Log[x]]), x]

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 33, normalized size = 1.06 \begin {gather*} -\frac {e^{x+x^2} x^2}{3-x^3+x^2 \log \left (-1+e^4+\log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x + x^2)*(6*x + 3*x^2 + 7*x^3 + x^4 - x^5 - 2*x^6 + E^4*(-6*x - 3*x^2 - 6*x^3 - x^4 + x^5 + 2*x^
6)) + E^(x + x^2)*(-6*x - 3*x^2 - 6*x^3 - x^4 + x^5 + 2*x^6)*Log[x] + (E^(x + x^2)*(x^4 + 2*x^5 + E^4*(-x^4 -
2*x^5)) + E^(x + x^2)*(-x^4 - 2*x^5)*Log[x])*Log[-1 + E^4 + Log[x]])/(-9 + 6*x^3 - x^6 + E^4*(9 - 6*x^3 + x^6)
+ (9 - 6*x^3 + x^6)*Log[x] + (-6*x^2 + 2*x^5 + E^4*(6*x^2 - 2*x^5) + (6*x^2 - 2*x^5)*Log[x])*Log[-1 + E^4 + L
og[x]] + (-x^4 + E^4*x^4 + x^4*Log[x])*Log[-1 + E^4 + Log[x]]^2),x]

[Out]

-((E^(x + x^2)*x^2)/(3 - x^3 + x^2*Log[-1 + E^4 + Log[x]]))

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fricas [A]  time = 0.97, size = 29, normalized size = 0.94 \begin {gather*} \frac {x^{2} e^{\left (x^{2} + x\right )}}{x^{3} - x^{2} \log \left (e^{4} + \log \relax (x) - 1\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^5-x^4)*exp(x^2+x)*log(x)+((-2*x^5-x^4)*exp(4)+2*x^5+x^4)*exp(x^2+x))*log(log(x)+exp(4)-1)+(2
*x^6+x^5-x^4-6*x^3-3*x^2-6*x)*exp(x^2+x)*log(x)+((2*x^6+x^5-x^4-6*x^3-3*x^2-6*x)*exp(4)-2*x^6-x^5+x^4+7*x^3+3*
x^2+6*x)*exp(x^2+x))/((x^4*log(x)+x^4*exp(4)-x^4)*log(log(x)+exp(4)-1)^2+((-2*x^5+6*x^2)*log(x)+(-2*x^5+6*x^2)
*exp(4)+2*x^5-6*x^2)*log(log(x)+exp(4)-1)+(x^6-6*x^3+9)*log(x)+(x^6-6*x^3+9)*exp(4)-x^6+6*x^3-9),x, algorithm=
"fricas")

[Out]

x^2*e^(x^2 + x)/(x^3 - x^2*log(e^4 + log(x) - 1) - 3)

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giac [A]  time = 5.12, size = 29, normalized size = 0.94 \begin {gather*} \frac {x^{2} e^{\left (x^{2} + x\right )}}{x^{3} - x^{2} \log \left (e^{4} + \log \relax (x) - 1\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^5-x^4)*exp(x^2+x)*log(x)+((-2*x^5-x^4)*exp(4)+2*x^5+x^4)*exp(x^2+x))*log(log(x)+exp(4)-1)+(2
*x^6+x^5-x^4-6*x^3-3*x^2-6*x)*exp(x^2+x)*log(x)+((2*x^6+x^5-x^4-6*x^3-3*x^2-6*x)*exp(4)-2*x^6-x^5+x^4+7*x^3+3*
x^2+6*x)*exp(x^2+x))/((x^4*log(x)+x^4*exp(4)-x^4)*log(log(x)+exp(4)-1)^2+((-2*x^5+6*x^2)*log(x)+(-2*x^5+6*x^2)
*exp(4)+2*x^5-6*x^2)*log(log(x)+exp(4)-1)+(x^6-6*x^3+9)*log(x)+(x^6-6*x^3+9)*exp(4)-x^6+6*x^3-9),x, algorithm=
"giac")

[Out]

x^2*e^(x^2 + x)/(x^3 - x^2*log(e^4 + log(x) - 1) - 3)

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maple [A]  time = 0.06, size = 30, normalized size = 0.97

method	result	size
risch	\(\frac {x^{2} {\mathrm e}^{\left (x +1\right ) x}}{x^{3}-x^{2} \ln \left (\ln \relax (x )+{\mathrm e}^{4}-1\right )-3}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^5-x^4)*exp(x^2+x)*ln(x)+((-2*x^5-x^4)*exp(4)+2*x^5+x^4)*exp(x^2+x))*ln(ln(x)+exp(4)-1)+(2*x^6+x^5-
x^4-6*x^3-3*x^2-6*x)*exp(x^2+x)*ln(x)+((2*x^6+x^5-x^4-6*x^3-3*x^2-6*x)*exp(4)-2*x^6-x^5+x^4+7*x^3+3*x^2+6*x)*e
xp(x^2+x))/((x^4*ln(x)+x^4*exp(4)-x^4)*ln(ln(x)+exp(4)-1)^2+((-2*x^5+6*x^2)*ln(x)+(-2*x^5+6*x^2)*exp(4)+2*x^5-
6*x^2)*ln(ln(x)+exp(4)-1)+(x^6-6*x^3+9)*ln(x)+(x^6-6*x^3+9)*exp(4)-x^6+6*x^3-9),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.47, size = 29, normalized size = 0.94 \begin {gather*} \frac {x^{2} e^{\left (x^{2} + x\right )}}{x^{3} - x^{2} \log \left (e^{4} + \log \relax (x) - 1\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^5-x^4)*exp(x^2+x)*log(x)+((-2*x^5-x^4)*exp(4)+2*x^5+x^4)*exp(x^2+x))*log(log(x)+exp(4)-1)+(2
*x^6+x^5-x^4-6*x^3-3*x^2-6*x)*exp(x^2+x)*log(x)+((2*x^6+x^5-x^4-6*x^3-3*x^2-6*x)*exp(4)-2*x^6-x^5+x^4+7*x^3+3*
x^2+6*x)*exp(x^2+x))/((x^4*log(x)+x^4*exp(4)-x^4)*log(log(x)+exp(4)-1)^2+((-2*x^5+6*x^2)*log(x)+(-2*x^5+6*x^2)
*exp(4)+2*x^5-6*x^2)*log(log(x)+exp(4)-1)+(x^6-6*x^3+9)*log(x)+(x^6-6*x^3+9)*exp(4)-x^6+6*x^3-9),x, algorithm=
"maxima")

[Out]

x^2*e^(x^2 + x)/(x^3 - x^2*log(e^4 + log(x) - 1) - 3)

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mupad [B]  time = 4.96, size = 268, normalized size = 8.65 \begin {gather*} -\frac {x^3\,\left (6\,{\mathrm {e}}^{x^2+x}-12\,{\mathrm {e}}^{x^2+x+4}+6\,{\mathrm {e}}^{x^2+x+8}-12\,{\mathrm {e}}^{x^2+x}\,\ln \relax (x)+12\,{\mathrm {e}}^{x^2+x+4}\,\ln \relax (x)+6\,{\mathrm {e}}^{x^2+x}\,{\ln \relax (x)}^2\right )-x^5\,\left ({\mathrm {e}}^{x^2+x+4}-{\mathrm {e}}^{x^2+x}+{\mathrm {e}}^{x^2+x}\,\ln \relax (x)\right )+x^6\,\left ({\mathrm {e}}^{x^2+x}-2\,{\mathrm {e}}^{x^2+x+4}+{\mathrm {e}}^{x^2+x+8}-2\,{\mathrm {e}}^{x^2+x}\,\ln \relax (x)+2\,{\mathrm {e}}^{x^2+x+4}\,\ln \relax (x)+{\mathrm {e}}^{x^2+x}\,{\ln \relax (x)}^2\right )}{\left (x^2\,\ln \left ({\mathrm {e}}^4+\ln \relax (x)-1\right )-x^3+3\right )\,\left (6\,x+6\,x\,{\ln \relax (x)}^2-x^3\,\ln \relax (x)-2\,x^4\,\ln \relax (x)-12\,x\,{\mathrm {e}}^4+6\,x\,{\mathrm {e}}^8+x^4\,{\ln \relax (x)}^2-x^3\,{\mathrm {e}}^4-2\,x^4\,{\mathrm {e}}^4+x^4\,{\mathrm {e}}^8-12\,x\,\ln \relax (x)+x^3+x^4+12\,x\,{\mathrm {e}}^4\,\ln \relax (x)+2\,x^4\,{\mathrm {e}}^4\,\ln \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + x^2)*(6*x - exp(4)*(6*x + 3*x^2 + 6*x^3 + x^4 - x^5 - 2*x^6) + 3*x^2 + 7*x^3 + x^4 - x^5 - 2*x^6)
+ log(exp(4) + log(x) - 1)*(exp(x + x^2)*(x^4 - exp(4)*(x^4 + 2*x^5) + 2*x^5) - exp(x + x^2)*log(x)*(x^4 + 2*
x^5)) - exp(x + x^2)*log(x)*(6*x + 3*x^2 + 6*x^3 + x^4 - x^5 - 2*x^6))/(log(exp(4) + log(x) - 1)^2*(x^4*log(x)
+ x^4*exp(4) - x^4) + exp(4)*(x^6 - 6*x^3 + 9) + log(x)*(x^6 - 6*x^3 + 9) + 6*x^3 - x^6 + log(exp(4) + log(x)
- 1)*(log(x)*(6*x^2 - 2*x^5) + exp(4)*(6*x^2 - 2*x^5) - 6*x^2 + 2*x^5) - 9),x)

[Out]

-(x^3*(6*exp(x + x^2) - 12*exp(x + x^2 + 4) + 6*exp(x + x^2 + 8) - 12*exp(x + x^2)*log(x) + 12*exp(x + x^2 + 4
)*log(x) + 6*exp(x + x^2)*log(x)^2) - x^5*(exp(x + x^2 + 4) - exp(x + x^2) + exp(x + x^2)*log(x)) + x^6*(exp(x
+ x^2) - 2*exp(x + x^2 + 4) + exp(x + x^2 + 8) - 2*exp(x + x^2)*log(x) + 2*exp(x + x^2 + 4)*log(x) + exp(x +
x^2)*log(x)^2))/((x^2*log(exp(4) + log(x) - 1) - x^3 + 3)*(6*x + 6*x*log(x)^2 - x^3*log(x) - 2*x^4*log(x) - 12
*x*exp(4) + 6*x*exp(8) + x^4*log(x)^2 - x^3*exp(4) - 2*x^4*exp(4) + x^4*exp(8) - 12*x*log(x) + x^3 + x^4 + 12*
x*exp(4)*log(x) + 2*x^4*exp(4)*log(x)))

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sympy [A]  time = 0.75, size = 27, normalized size = 0.87 \begin {gather*} \frac {x^{2} e^{x^{2} + x}}{x^{3} - x^{2} \log {\left (\log {\relax (x )} - 1 + e^{4} \right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**5-x**4)*exp(x**2+x)*ln(x)+((-2*x**5-x**4)*exp(4)+2*x**5+x**4)*exp(x**2+x))*ln(ln(x)+exp(4)-
1)+(2*x**6+x**5-x**4-6*x**3-3*x**2-6*x)*exp(x**2+x)*ln(x)+((2*x**6+x**5-x**4-6*x**3-3*x**2-6*x)*exp(4)-2*x**6-
x**5+x**4+7*x**3+3*x**2+6*x)*exp(x**2+x))/((x**4*ln(x)+x**4*exp(4)-x**4)*ln(ln(x)+exp(4)-1)**2+((-2*x**5+6*x**
2)*ln(x)+(-2*x**5+6*x**2)*exp(4)+2*x**5-6*x**2)*ln(ln(x)+exp(4)-1)+(x**6-6*x**3+9)*ln(x)+(x**6-6*x**3+9)*exp(4
)-x**6+6*x**3-9),x)

[Out]

x**2*exp(x**2 + x)/(x**3 - x**2*log(log(x) - 1 + exp(4)) - 3)

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