3.52.36 \(\int \frac {e^x (2 x-4 x^2)+e^{x+x^2} (x-2 x^2)+(e^x (4 x-2 x^2-2 x^3)+e^{x+x^2} (2 x-x^2-3 x^3+2 x^4)) \log (-x+x^2)+e^x (-9 x^2+4 x^4+4 x^5+x^6) \log ^2(-x+x^2)}{-4+e^{2 x^2} (-1+x)+4 x+e^{x^2} (-4+4 x)+(36 x-12 x^2-20 x^3-4 x^4+e^{x^2} (18 x-6 x^2-10 x^3-2 x^4)) \log (-x+x^2)+(-81 x^2-27 x^3+54 x^4+42 x^5+11 x^6+x^7) \log ^2(-x+x^2)} \, dx\)

Optimal. Leaf size=35 \[ \frac {e^x x}{(3+x)^2-\frac {2+e^{x^2}}{x \log \left (-x+x^2\right )}} \]

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Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(2*x - 4*x^2) + E^(x + x^2)*(x - 2*x^2) + (E^x*(4*x - 2*x^2 - 2*x^3) + E^(x + x^2)*(2*x - x^2 - 3*x^3
+ 2*x^4))*Log[-x + x^2] + E^x*(-9*x^2 + 4*x^4 + 4*x^5 + x^6)*Log[-x + x^2]^2)/(-4 + E^(2*x^2)*(-1 + x) + 4*x
+ E^x^2*(-4 + 4*x) + (36*x - 12*x^2 - 20*x^3 - 4*x^4 + E^x^2*(18*x - 6*x^2 - 10*x^3 - 2*x^4))*Log[-x + x^2] +
(-81*x^2 - 27*x^3 + 54*x^4 + 42*x^5 + 11*x^6 + x^7)*Log[-x + x^2]^2),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.19, size = 37, normalized size = 1.06 \begin {gather*} \frac {e^x x^2 \log ((-1+x) x)}{-2-e^{x^2}+x (3+x)^2 \log ((-1+x) x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(2*x - 4*x^2) + E^(x + x^2)*(x - 2*x^2) + (E^x*(4*x - 2*x^2 - 2*x^3) + E^(x + x^2)*(2*x - x^2 -
3*x^3 + 2*x^4))*Log[-x + x^2] + E^x*(-9*x^2 + 4*x^4 + 4*x^5 + x^6)*Log[-x + x^2]^2)/(-4 + E^(2*x^2)*(-1 + x)
+ 4*x + E^x^2*(-4 + 4*x) + (36*x - 12*x^2 - 20*x^3 - 4*x^4 + E^x^2*(18*x - 6*x^2 - 10*x^3 - 2*x^4))*Log[-x + x
^2] + (-81*x^2 - 27*x^3 + 54*x^4 + 42*x^5 + 11*x^6 + x^7)*Log[-x + x^2]^2),x]

[Out]

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

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fricas [A]  time = 0.55, size = 60, normalized size = 1.71 \begin {gather*} \frac {x^{2} e^{\left (x^{2} + x\right )} \log \left (x^{2} - x\right )}{{\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} e^{\left (x^{2}\right )} \log \left (x^{2} - x\right ) - e^{\left (2 \, x^{2}\right )} - 2 \, e^{\left (x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6+4*x^5+4*x^4-9*x^2)*exp(x)*log(x^2-x)^2+((2*x^4-3*x^3-x^2+2*x)*exp(x)*exp(x^2)+(-2*x^3-2*x^2+4*
x)*exp(x))*log(x^2-x)+(-2*x^2+x)*exp(x)*exp(x^2)+(-4*x^2+2*x)*exp(x))/((x^7+11*x^6+42*x^5+54*x^4-27*x^3-81*x^2
)*log(x^2-x)^2+((-2*x^4-10*x^3-6*x^2+18*x)*exp(x^2)-4*x^4-20*x^3-12*x^2+36*x)*log(x^2-x)+(x-1)*exp(x^2)^2+(4*x
-4)*exp(x^2)+4*x-4),x, algorithm="fricas")

[Out]

x^2*e^(x^2 + x)*log(x^2 - x)/((x^3 + 6*x^2 + 9*x)*e^(x^2)*log(x^2 - x) - e^(2*x^2) - 2*e^(x^2))

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giac [B]  time = 0.90, size = 72, normalized size = 2.06 \begin {gather*} \frac {x^{2} e^{x} \log \left (x - 1\right ) + x^{2} e^{x} \log \relax (x)}{x^{3} \log \left (x - 1\right ) + x^{3} \log \relax (x) + 6 \, x^{2} \log \left (x - 1\right ) + 6 \, x^{2} \log \relax (x) + 9 \, x \log \left (x - 1\right ) + 9 \, x \log \relax (x) - e^{\left (x^{2}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6+4*x^5+4*x^4-9*x^2)*exp(x)*log(x^2-x)^2+((2*x^4-3*x^3-x^2+2*x)*exp(x)*exp(x^2)+(-2*x^3-2*x^2+4*
x)*exp(x))*log(x^2-x)+(-2*x^2+x)*exp(x)*exp(x^2)+(-4*x^2+2*x)*exp(x))/((x^7+11*x^6+42*x^5+54*x^4-27*x^3-81*x^2
)*log(x^2-x)^2+((-2*x^4-10*x^3-6*x^2+18*x)*exp(x^2)-4*x^4-20*x^3-12*x^2+36*x)*log(x^2-x)+(x-1)*exp(x^2)^2+(4*x
-4)*exp(x^2)+4*x-4),x, algorithm="giac")

[Out]

(x^2*e^x*log(x - 1) + x^2*e^x*log(x))/(x^3*log(x - 1) + x^3*log(x) + 6*x^2*log(x - 1) + 6*x^2*log(x) + 9*x*log
(x - 1) + 9*x*log(x) - e^(x^2) - 2)

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maple [C]  time = 0.90, size = 348, normalized size = 9.94

method	result	size
risch	\(\frac {x \,{\mathrm e}^{x}}{\left (3+x \right )^{2}}-\frac {2 x \,{\mathrm e}^{x} \left ({\mathrm e}^{x^{2}}+2\right )}{\left (x^{2}+6 x +9\right ) \left (-i \pi \,x^{3} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}+6 i \pi \,x^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}+i \pi \,x^{3} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}+i \pi \,x^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )+9 i \pi x \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}-i \pi \,x^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}+9 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )-9 i \pi x \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}-6 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}+6 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )-9 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}-6 i \pi \,x^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}-2 x^{3} \ln \relax (x )-2 x^{3} \ln \left (x -1\right )-12 x^{2} \ln \relax (x )-12 x^{2} \ln \left (x -1\right )-18 x \ln \relax (x )-18 \ln \left (x -1\right ) x +2 \,{\mathrm e}^{x^{2}}+4\right )}\)	\(348\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6+4*x^5+4*x^4-9*x^2)*exp(x)*ln(x^2-x)^2+((2*x^4-3*x^3-x^2+2*x)*exp(x)*exp(x^2)+(-2*x^3-2*x^2+4*x)*exp(
x))*ln(x^2-x)+(-2*x^2+x)*exp(x)*exp(x^2)+(-4*x^2+2*x)*exp(x))/((x^7+11*x^6+42*x^5+54*x^4-27*x^3-81*x^2)*ln(x^2
-x)^2+((-2*x^4-10*x^3-6*x^2+18*x)*exp(x^2)-4*x^4-20*x^3-12*x^2+36*x)*ln(x^2-x)+(x-1)*exp(x^2)^2+(4*x-4)*exp(x^
2)+4*x-4),x,method=_RETURNVERBOSE)

[Out]

x/(3+x)^2*exp(x)-2*x*exp(x)*(exp(x^2)+2)/(x^2+6*x+9)/(-I*Pi*x^3*csgn(I*(x-1))*csgn(I*x*(x-1))^2+6*I*Pi*x^2*csg
n(I*x*(x-1))^3+I*Pi*x^3*csgn(I*x*(x-1))^3+I*Pi*x^3*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x-1))+9*I*Pi*x*csgn(I*x*(
x-1))^3-I*Pi*x^3*csgn(I*x)*csgn(I*x*(x-1))^2+9*I*Pi*x*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x-1))-9*I*Pi*x*csgn(I*
(x-1))*csgn(I*x*(x-1))^2-6*I*Pi*x^2*csgn(I*x)*csgn(I*x*(x-1))^2+6*I*Pi*x^2*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x
-1))-9*I*Pi*x*csgn(I*x)*csgn(I*x*(x-1))^2-6*I*Pi*x^2*csgn(I*(x-1))*csgn(I*x*(x-1))^2-2*x^3*ln(x)-2*x^3*ln(x-1)
-12*x^2*ln(x)-12*x^2*ln(x-1)-18*x*ln(x)-18*ln(x-1)*x+2*exp(x^2)+4)

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maxima [A]  time = 0.49, size = 62, normalized size = 1.77 \begin {gather*} \frac {x^{2} e^{x} \log \left (x - 1\right ) + x^{2} e^{x} \log \relax (x)}{{\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} \log \left (x - 1\right ) + {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} \log \relax (x) - e^{\left (x^{2}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6+4*x^5+4*x^4-9*x^2)*exp(x)*log(x^2-x)^2+((2*x^4-3*x^3-x^2+2*x)*exp(x)*exp(x^2)+(-2*x^3-2*x^2+4*
x)*exp(x))*log(x^2-x)+(-2*x^2+x)*exp(x)*exp(x^2)+(-4*x^2+2*x)*exp(x))/((x^7+11*x^6+42*x^5+54*x^4-27*x^3-81*x^2
)*log(x^2-x)^2+((-2*x^4-10*x^3-6*x^2+18*x)*exp(x^2)-4*x^4-20*x^3-12*x^2+36*x)*log(x^2-x)+(x-1)*exp(x^2)^2+(4*x
-4)*exp(x^2)+4*x-4),x, algorithm="maxima")

[Out]

(x^2*e^x*log(x - 1) + x^2*e^x*log(x))/((x^3 + 6*x^2 + 9*x)*log(x - 1) + (x^3 + 6*x^2 + 9*x)*log(x) - e^(x^2) -
2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (x^6+4\,x^5+4\,x^4-9\,x^2\right )\,{\ln \left (x^2-x\right )}^2+\left ({\mathrm {e}}^{x^2}\,{\mathrm {e}}^x\,\left (2\,x^4-3\,x^3-x^2+2\,x\right )-{\mathrm {e}}^x\,\left (2\,x^3+2\,x^2-4\,x\right )\right )\,\ln \left (x^2-x\right )+{\mathrm {e}}^x\,\left (2\,x-4\,x^2\right )+{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x\,\left (x-2\,x^2\right )}{\left (x^7+11\,x^6+42\,x^5+54\,x^4-27\,x^3-81\,x^2\right )\,{\ln \left (x^2-x\right )}^2+\left (36\,x-{\mathrm {e}}^{x^2}\,\left (2\,x^4+10\,x^3+6\,x^2-18\,x\right )-12\,x^2-20\,x^3-4\,x^4\right )\,\ln \left (x^2-x\right )+4\,x+{\mathrm {e}}^{2\,x^2}\,\left (x-1\right )+{\mathrm {e}}^{x^2}\,\left (4\,x-4\right )-4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(2*x - 4*x^2) - log(x^2 - x)*(exp(x)*(2*x^2 - 4*x + 2*x^3) - exp(x^2)*exp(x)*(2*x - x^2 - 3*x^3 +
2*x^4)) + exp(x^2)*exp(x)*(x - 2*x^2) + exp(x)*log(x^2 - x)^2*(4*x^4 - 9*x^2 + 4*x^5 + x^6))/(4*x - log(x^2 -
x)*(exp(x^2)*(6*x^2 - 18*x + 10*x^3 + 2*x^4) - 36*x + 12*x^2 + 20*x^3 + 4*x^4) + log(x^2 - x)^2*(54*x^4 - 27*x
^3 - 81*x^2 + 42*x^5 + 11*x^6 + x^7) + exp(2*x^2)*(x - 1) + exp(x^2)*(4*x - 4) - 4),x)

[Out]

int((exp(x)*(2*x - 4*x^2) - log(x^2 - x)*(exp(x)*(2*x^2 - 4*x + 2*x^3) - exp(x^2)*exp(x)*(2*x - x^2 - 3*x^3 +
2*x^4)) + exp(x^2)*exp(x)*(x - 2*x^2) + exp(x)*log(x^2 - x)^2*(4*x^4 - 9*x^2 + 4*x^5 + x^6))/(4*x - log(x^2 -
x)*(exp(x^2)*(6*x^2 - 18*x + 10*x^3 + 2*x^4) - 36*x + 12*x^2 + 20*x^3 + 4*x^4) + log(x^2 - x)^2*(54*x^4 - 27*x
^3 - 81*x^2 + 42*x^5 + 11*x^6 + x^7) + exp(2*x^2)*(x - 1) + exp(x^2)*(4*x - 4) - 4), x)

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sympy [B]  time = 0.59, size = 53, normalized size = 1.51 \begin {gather*} - \frac {x^{2} e^{x} \log {\left (x^{2} - x \right )}}{- x^{3} \log {\left (x^{2} - x \right )} - 6 x^{2} \log {\left (x^{2} - x \right )} - 9 x \log {\left (x^{2} - x \right )} + e^{x^{2}} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**6+4*x**5+4*x**4-9*x**2)*exp(x)*ln(x**2-x)**2+((2*x**4-3*x**3-x**2+2*x)*exp(x)*exp(x**2)+(-2*x**
3-2*x**2+4*x)*exp(x))*ln(x**2-x)+(-2*x**2+x)*exp(x)*exp(x**2)+(-4*x**2+2*x)*exp(x))/((x**7+11*x**6+42*x**5+54*
x**4-27*x**3-81*x**2)*ln(x**2-x)**2+((-2*x**4-10*x**3-6*x**2+18*x)*exp(x**2)-4*x**4-20*x**3-12*x**2+36*x)*ln(x
**2-x)+(x-1)*exp(x**2)**2+(4*x-4)*exp(x**2)+4*x-4),x)

[Out]

-x**2*exp(x)*log(x**2 - x)/(-x**3*log(x**2 - x) - 6*x**2*log(x**2 - x) - 9*x*log(x**2 - x) + exp(x**2) + 2)

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