3.92.61 \(\int \frac {-6 e^{2 x^2-2 x^3+2 (x-x^2) \log (x)} x^2+e^{x^2-x^3+(x-x^2) \log (x)} ((1+5 x+3 x^2-x^3-6 x^4) \log (\frac {21}{5})+(x-4 x^3) \log (\frac {21}{5}) \log (x))}{18 e^{2 x^2-2 x^3+2 (x-x^2) \log (x)} x^2-12 e^{x^2-x^3+(x-x^2) \log (x)} x \log (\frac {21}{5})+2 \log ^2(\frac {21}{5})} \, dx\)

Optimal. Leaf size=33 \[ \frac {\frac {1}{2}+x}{-3+\frac {e^{-\left (\left (x-x^2\right ) (x+\log (x))\right )} \log \left (\frac {21}{5}\right )}{x}} \]

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Rubi [F]  time = 36.14, 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 {-6 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2+e^{x^2-x^3+\left (x-x^2\right ) \log (x)} \left (\left (1+5 x+3 x^2-x^3-6 x^4\right ) \log \left (\frac {21}{5}\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{18 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2-12 e^{x^2-x^3+\left (x-x^2\right ) \log (x)} x \log \left (\frac {21}{5}\right )+2 \log ^2\left (\frac {21}{5}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6*E^(2*x^2 - 2*x^3 + 2*(x - x^2)*Log[x])*x^2 + E^(x^2 - x^3 + (x - x^2)*Log[x])*((1 + 5*x + 3*x^2 - x^3
- 6*x^4)*Log[21/5] + (x - 4*x^3)*Log[21/5]*Log[x]))/(18*E^(2*x^2 - 2*x^3 + 2*(x - x^2)*Log[x])*x^2 - 12*E^(x^2
- x^3 + (x - x^2)*Log[x])*x*Log[21/5] + 2*Log[21/5]^2),x]

[Out]

(3*Defer[Int][(E^(2*x^2)*x^(1 + 2*x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5])^2, x])/2 + (9*Defer[Int][(E^
(2*x^2)*x^(2 + 2*x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5])^2, x])/2 + (3*Log[x]*Defer[Int][(E^(2*x^2)*x^
(2 + 2*x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5])^2, x])/2 + (9*Defer[Int][(E^(2*x^2)*x^(3 + 2*x))/(3*E^x
^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5])^2, x])/2 - (3*Defer[Int][(E^(2*x^2)*x^(4 + 2*x))/(3*E^x^2*x^(1 + x) - E^
x^3*x^x^2*Log[21/5])^2, x])/2 - 6*Log[x]*Defer[Int][(E^(2*x^2)*x^(4 + 2*x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*L
og[21/5])^2, x] - 9*Defer[Int][(E^(2*x^2)*x^(5 + 2*x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5])^2, x] - Def
er[Int][(E^x^2*x^x)/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5]), x]/2 - (5*Defer[Int][(E^x^2*x^(1 + x))/(3*E^x
^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5]), x])/2 - (Log[x]*Defer[Int][(E^x^2*x^(1 + x))/(3*E^x^2*x^(1 + x) - E^x^3
*x^x^2*Log[21/5]), x])/2 - (3*Defer[Int][(E^x^2*x^(2 + x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5]), x])/2
+ Defer[Int][(E^x^2*x^(3 + x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5]), x]/2 + 2*Log[x]*Defer[Int][(E^x^2*
x^(3 + x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5]), x] + 3*Defer[Int][(E^x^2*x^(4 + x))/(3*E^x^2*x^(1 + x)
- E^x^3*x^x^2*Log[21/5]), x] + Defer[Int][Defer[Int][(E^x^2*x^(1 + x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[2
1/5]), x]/x, x]/2 - 2*Defer[Int][Defer[Int][(E^x^2*x^(3 + x))/(3*E^x^2*x^(1 + x) - E^x^3*x^x^2*Log[21/5]), x]/
x, x] - (3*Defer[Int][Defer[Int][(E^(2*x^2)*x^(2 + 2*x))/(-3*E^x^2*x^(1 + x) + E^x^3*x^x^2*Log[21/5])^2, x]/x,
x])/2 + 6*Defer[Int][Defer[Int][(E^(2*x^2)*x^(4 + 2*x))/(-3*E^x^2*x^(1 + x) + E^x^3*x^x^2*Log[21/5])^2, x]/x,
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{x^2} x^x \left (-6 e^{x^2} x^{2+x}-e^{x^3} x^{x^2} \left (-1-5 x-3 x^2+x^3+6 x^4\right ) \log \left (\frac {21}{5}\right )-e^{x^3} x^{1+x^2} \left (-1+4 x^2\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{2 \left (3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{x^2} x^x \left (-6 e^{x^2} x^{2+x}-e^{x^3} x^{x^2} \left (-1-5 x-3 x^2+x^3+6 x^4\right ) \log \left (\frac {21}{5}\right )-e^{x^3} x^{1+x^2} \left (-1+4 x^2\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{\left (3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {3 e^{2 x^2} x^{1+2 x} (1+2 x) \left (-1-x-x^2+3 x^3-x \log (x)+2 x^2 \log (x)\right )}{\left (3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}+\frac {e^{x^2} x^x \left (-1-5 x-3 x^2+x^3+6 x^4-x \log (x)+4 x^3 \log (x)\right )}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{x^2} x^x \left (-1-5 x-3 x^2+x^3+6 x^4-x \log (x)+4 x^3 \log (x)\right )}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )} \, dx-\frac {3}{2} \int \frac {e^{2 x^2} x^{1+2 x} (1+2 x) \left (-1-x-x^2+3 x^3-x \log (x)+2 x^2 \log (x)\right )}{\left (3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {e^{x^2} x^x}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}-\frac {5 e^{x^2} x^{1+x}}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}-\frac {3 e^{x^2} x^{2+x}}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}+\frac {e^{x^2} x^{3+x}}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}+\frac {6 e^{x^2} x^{4+x}}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}-\frac {e^{x^2} x^{1+x} \log (x)}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}+\frac {4 e^{x^2} x^{3+x} \log (x)}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}\right ) \, dx-\frac {3}{2} \int \left (\frac {e^{2 x^2} x^{1+2 x} \left (-1-x-x^2+3 x^3-x \log (x)+2 x^2 \log (x)\right )}{\left (3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}+\frac {2 e^{2 x^2} x^{2+2 x} \left (-1-x-x^2+3 x^3-x \log (x)+2 x^2 \log (x)\right )}{\left (3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{x^2} x^x}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )} \, dx\right )+\frac {1}{2} \int \frac {e^{x^2} x^{3+x}}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )} \, dx-\frac {1}{2} \int \frac {e^{x^2} x^{1+x} \log (x)}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )} \, dx-\frac {3}{2} \int \frac {e^{x^2} x^{2+x}}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )} \, dx-\frac {3}{2} \int \frac {e^{2 x^2} x^{1+2 x} \left (-1-x-x^2+3 x^3-x \log (x)+2 x^2 \log (x)\right )}{\left (3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2} \, dx+2 \int \frac {e^{x^2} x^{3+x} \log (x)}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )} \, dx-\frac {5}{2} \int \frac {e^{x^2} x^{1+x}}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )} \, dx+3 \int \frac {e^{x^2} x^{4+x}}{3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )} \, dx-3 \int \frac {e^{2 x^2} x^{2+2 x} \left (-1-x-x^2+3 x^3-x \log (x)+2 x^2 \log (x)\right )}{\left (3 e^{x^2} x^{1+x}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 49, normalized size = 1.48 \begin {gather*} \frac {e^{x^2} x^{1+x} (1+2 x)}{2 \left (-3 e^{x^2} x^{1+x}+e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*E^(2*x^2 - 2*x^3 + 2*(x - x^2)*Log[x])*x^2 + E^(x^2 - x^3 + (x - x^2)*Log[x])*((1 + 5*x + 3*x^2
- x^3 - 6*x^4)*Log[21/5] + (x - 4*x^3)*Log[21/5]*Log[x]))/(18*E^(2*x^2 - 2*x^3 + 2*(x - x^2)*Log[x])*x^2 - 12*
E^(x^2 - x^3 + (x - x^2)*Log[x])*x*Log[21/5] + 2*Log[21/5]^2),x]

[Out]

(E^x^2*x^(1 + x)*(1 + 2*x))/(2*(-3*E^x^2*x^(1 + x) + E^x^3*x^x^2*Log[21/5]))

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fricas [B]  time = 0.57, size = 62, normalized size = 1.88 \begin {gather*} -\frac {6 \, x^{2} e^{\left (-x^{3} + x^{2} - {\left (x^{2} - x\right )} \log \relax (x)\right )} + \log \left (\frac {21}{5}\right )}{6 \, {\left (3 \, x e^{\left (-x^{3} + x^{2} - {\left (x^{2} - x\right )} \log \relax (x)\right )} - \log \left (\frac {21}{5}\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2+((-4*x^3+x)*log(21/5)*log(x)+(-6*x^4-x^3+3*x^2+5*x+1)*log(21/
5))*exp((-x^2+x)*log(x)-x^3+x^2))/(18*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2-12*x*log(21/5)*exp((-x^2+x)*log(x)-x^
3+x^2)+2*log(21/5)^2),x, algorithm="fricas")

[Out]

-1/6*(6*x^2*e^(-x^3 + x^2 - (x^2 - x)*log(x)) + log(21/5))/(3*x*e^(-x^3 + x^2 - (x^2 - x)*log(x)) - log(21/5))

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giac [B]  time = 1.06, size = 68, normalized size = 2.06 \begin {gather*} -\frac {6 \, x^{2} e^{\left (-x^{3} - x^{2} \log \relax (x) + x^{2} + x \log \relax (x)\right )} + \log \left (21\right ) - \log \relax (5)}{6 \, {\left (3 \, x e^{\left (-x^{3} - x^{2} \log \relax (x) + x^{2} + x \log \relax (x)\right )} - \log \left (21\right ) + \log \relax (5)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2+((-4*x^3+x)*log(21/5)*log(x)+(-6*x^4-x^3+3*x^2+5*x+1)*log(21/
5))*exp((-x^2+x)*log(x)-x^3+x^2))/(18*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2-12*x*log(21/5)*exp((-x^2+x)*log(x)-x^
3+x^2)+2*log(21/5)^2),x, algorithm="giac")

[Out]

-1/6*(6*x^2*e^(-x^3 - x^2*log(x) + x^2 + x*log(x)) + log(21) - log(5))/(3*x*e^(-x^3 - x^2*log(x) + x^2 + x*log
(x)) - log(21) + log(5))

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maple [B]  time = 0.07, size = 62, normalized size = 1.88

method	result	size
risch	\(-\frac {x}{3}+\frac {2 x \ln \relax (7)-2 x \ln \relax (5)+2 x \ln \relax (3)+\ln \relax (7)-\ln \relax (5)+\ln \relax (3)}{-18 x \,x^{-x \left (x -1\right )} {\mathrm e}^{-x^{2} \left (x -1\right )}+6 \ln \relax (7)-6 \ln \relax (5)+6 \ln \relax (3)}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^2*exp((-x^2+x)*ln(x)-x^3+x^2)^2+((-4*x^3+x)*ln(21/5)*ln(x)+(-6*x^4-x^3+3*x^2+5*x+1)*ln(21/5))*exp((-
x^2+x)*ln(x)-x^3+x^2))/(18*x^2*exp((-x^2+x)*ln(x)-x^3+x^2)^2-12*x*ln(21/5)*exp((-x^2+x)*ln(x)-x^3+x^2)+2*ln(21
/5)^2),x,method=_RETURNVERBOSE)

[Out]

-1/3*x+1/6*(2*x*ln(7)-2*x*ln(5)+2*x*ln(3)+ln(7)-ln(5)+ln(3))/(-3*x*x^(-x*(x-1))*exp(-x^2*(x-1))+ln(7)-ln(5)+ln
(3))

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maxima [A]  time = 0.54, size = 54, normalized size = 1.64 \begin {gather*} \frac {{\left (2 \, x^{2} + x\right )} e^{\left (x^{2} + x \log \relax (x)\right )}}{2 \, {\left ({\left (\log \relax (7) - \log \relax (5) + \log \relax (3)\right )} e^{\left (x^{3} + x^{2} \log \relax (x)\right )} - 3 \, x e^{\left (x^{2} + x \log \relax (x)\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2+((-4*x^3+x)*log(21/5)*log(x)+(-6*x^4-x^3+3*x^2+5*x+1)*log(21/
5))*exp((-x^2+x)*log(x)-x^3+x^2))/(18*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2-12*x*log(21/5)*exp((-x^2+x)*log(x)-x^
3+x^2)+2*log(21/5)^2),x, algorithm="maxima")

[Out]

1/2*(2*x^2 + x)*e^(x^2 + x*log(x))/((log(7) - log(5) + log(3))*e^(x^3 + x^2*log(x)) - 3*x*e^(x^2 + x*log(x)))

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mupad [B]  time = 8.25, size = 56, normalized size = 1.70 \begin {gather*} \frac {x^{x^2}\,\ln \left (\frac {21}{5}\right )+6\,x^x\,x^2\,{\mathrm {e}}^{x^2-x^3}}{6\,\left (x^{x^2}\,\ln \left (\frac {21}{5}\right )-3\,x\,x^x\,{\mathrm {e}}^{x^2-x^3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x)*(x - x^2) + x^2 - x^3)*(log(21/5)*(5*x + 3*x^2 - x^3 - 6*x^4 + 1) + log(21/5)*log(x)*(x - 4*x^
3)) - 6*x^2*exp(2*log(x)*(x - x^2) + 2*x^2 - 2*x^3))/(18*x^2*exp(2*log(x)*(x - x^2) + 2*x^2 - 2*x^3) + 2*log(2
1/5)^2 - 12*x*exp(log(x)*(x - x^2) + x^2 - x^3)*log(21/5)),x)

[Out]

(x^(x^2)*log(21/5) + 6*x^x*x^2*exp(x^2 - x^3))/(6*(x^(x^2)*log(21/5) - 3*x*x^x*exp(x^2 - x^3)))

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sympy [B]  time = 0.59, size = 53, normalized size = 1.61 \begin {gather*} - \frac {x}{3} + \frac {- 2 x \log {\left (21 \right )} + 2 x \log {\relax (5 )} - \log {\left (21 \right )} + \log {\relax (5 )}}{18 x e^{- x^{3} + x^{2} + \left (- x^{2} + x\right ) \log {\relax (x )}} - 6 \log {\left (21 \right )} + 6 \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**2*exp((-x**2+x)*ln(x)-x**3+x**2)**2+((-4*x**3+x)*ln(21/5)*ln(x)+(-6*x**4-x**3+3*x**2+5*x+1)*l
n(21/5))*exp((-x**2+x)*ln(x)-x**3+x**2))/(18*x**2*exp((-x**2+x)*ln(x)-x**3+x**2)**2-12*x*ln(21/5)*exp((-x**2+x
)*ln(x)-x**3+x**2)+2*ln(21/5)**2),x)

[Out]

-x/3 + (-2*x*log(21) + 2*x*log(5) - log(21) + log(5))/(18*x*exp(-x**3 + x**2 + (-x**2 + x)*log(x)) - 6*log(21)
+ 6*log(5))

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