3.62.45 \(\int \frac {e^{x^2} (-20 x+30 x^2-10 x^3+10 x^4)+e^{x^2} (-30 x+40 x^2-20 x^3+20 x^4) \log (-x+x^2)+e^{x^2} (-10 x+10 x^2-10 x^3+10 x^4) \log ^2(-x+x^2)}{-4+4 x+e^{x^2} (4 x^2-4 x^3)+e^{2 x^2} (-x^4+x^5)+(e^{x^2} (8 x^2-8 x^3)+e^{2 x^2} (-4 x^4+4 x^5)) \log (-x+x^2)+(e^{x^2} (4 x^2-4 x^3)+e^{2 x^2} (-6 x^4+6 x^5)) \log ^2(-x+x^2)+e^{2 x^2} (-4 x^4+4 x^5) \log ^3(-x+x^2)+e^{2 x^2} (-x^4+x^5) \log ^4(-x+x^2)} \, dx\)

Optimal. Leaf size=27 \[ \frac {5}{2-e^{x^2} \left (x+x \log \left (-x+x^2\right )\right )^2} \]

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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*(-20*x + 30*x^2 - 10*x^3 + 10*x^4) + E^x^2*(-30*x + 40*x^2 - 20*x^3 + 20*x^4)*Log[-x + x^2] + E^x^2
*(-10*x + 10*x^2 - 10*x^3 + 10*x^4)*Log[-x + x^2]^2)/(-4 + 4*x + E^x^2*(4*x^2 - 4*x^3) + E^(2*x^2)*(-x^4 + x^5
) + (E^x^2*(8*x^2 - 8*x^3) + E^(2*x^2)*(-4*x^4 + 4*x^5))*Log[-x + x^2] + (E^x^2*(4*x^2 - 4*x^3) + E^(2*x^2)*(-
6*x^4 + 6*x^5))*Log[-x + x^2]^2 + E^(2*x^2)*(-4*x^4 + 4*x^5)*Log[-x + x^2]^3 + E^(2*x^2)*(-x^4 + x^5)*Log[-x +
x^2]^4),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.13, size = 48, normalized size = 1.78 \begin {gather*} -\frac {5}{-2+e^{x^2} x^2+2 e^{x^2} x^2 \log ((-1+x) x)+e^{x^2} x^2 \log ^2((-1+x) x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x^2*(-20*x + 30*x^2 - 10*x^3 + 10*x^4) + E^x^2*(-30*x + 40*x^2 - 20*x^3 + 20*x^4)*Log[-x + x^2] +
E^x^2*(-10*x + 10*x^2 - 10*x^3 + 10*x^4)*Log[-x + x^2]^2)/(-4 + 4*x + E^x^2*(4*x^2 - 4*x^3) + E^(2*x^2)*(-x^4
+ x^5) + (E^x^2*(8*x^2 - 8*x^3) + E^(2*x^2)*(-4*x^4 + 4*x^5))*Log[-x + x^2] + (E^x^2*(4*x^2 - 4*x^3) + E^(2*x
^2)*(-6*x^4 + 6*x^5))*Log[-x + x^2]^2 + E^(2*x^2)*(-4*x^4 + 4*x^5)*Log[-x + x^2]^3 + E^(2*x^2)*(-x^4 + x^5)*Lo
g[-x + x^2]^4),x]

[Out]

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

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fricas [A]  time = 0.80, size = 49, normalized size = 1.81 \begin {gather*} -\frac {5}{x^{2} e^{\left (x^{2}\right )} \log \left (x^{2} - x\right )^{2} + 2 \, x^{2} e^{\left (x^{2}\right )} \log \left (x^{2} - x\right ) + x^{2} e^{\left (x^{2}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*log(x^2-x)^2+(20*x^4-20*x^3+40*x^2-30*x)*exp(x^2)*log(x^2-x)+(
10*x^4-10*x^3+30*x^2-20*x)*exp(x^2))/((x^5-x^4)*exp(x^2)^2*log(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*log(x^2-x)^3+
((6*x^5-6*x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*log(x^2-x)^2+((4*x^5-4*x^4)*exp(x^2)^2+(-8*x^3+8*x^2)*exp(x
^2))*log(x^2-x)+(x^5-x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2)+4*x-4),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.03, size = 49, normalized size = 1.81 \begin {gather*} -\frac {5}{x^{2} e^{\left (x^{2}\right )} \log \left (x^{2} - x\right )^{2} + 2 \, x^{2} e^{\left (x^{2}\right )} \log \left (x^{2} - x\right ) + x^{2} e^{\left (x^{2}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*log(x^2-x)^2+(20*x^4-20*x^3+40*x^2-30*x)*exp(x^2)*log(x^2-x)+(
10*x^4-10*x^3+30*x^2-20*x)*exp(x^2))/((x^5-x^4)*exp(x^2)^2*log(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*log(x^2-x)^3+
((6*x^5-6*x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*log(x^2-x)^2+((4*x^5-4*x^4)*exp(x^2)^2+(-8*x^3+8*x^2)*exp(x
^2))*log(x^2-x)+(x^5-x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2)+4*x-4),x, algorithm="giac")

[Out]

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

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maple [C]  time = 1.08, size = 703, normalized size = 26.04

method	result	size
risch	\(\frac {20}{8-8 x^{2} {\mathrm e}^{x^{2}} \ln \relax (x )-4 x^{2} {\mathrm e}^{x^{2}}-4 i {\mathrm e}^{x^{2}} \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \left (x -1\right )-4 i {\mathrm e}^{x^{2}} \pi \,x^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \relax (x )-4 i {\mathrm e}^{x^{2}} \pi \,x^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \left (x -1\right )-4 i {\mathrm e}^{x^{2}} \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \relax (x )+4 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 ) {\mathrm e}^{x^{2}}+{\mathrm e}^{x^{2}} \pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{6}-8 \,{\mathrm e}^{x^{2}} x^{2} \ln \relax (x ) \ln \left (x -1\right )-8 x^{2} {\mathrm e}^{x^{2}} \ln \left (x -1\right )-4 \,{\mathrm e}^{x^{2}} x^{2} \ln \relax (x )^{2}-4 \,{\mathrm e}^{x^{2}} x^{2} \ln \left (x -1\right )^{2}+4 \,{\mathrm e}^{x^{2}} \pi ^{2} 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 )^{4}-2 \,{\mathrm e}^{x^{2}} \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}-2 \,{\mathrm e}^{x^{2}} \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}+{\mathrm e}^{x^{2}} \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}-4 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} {\mathrm e}^{x^{2}}+{\mathrm e}^{x^{2}} \pi ^{2} x^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4}-2 \,{\mathrm e}^{x^{2}} \pi ^{2} x^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{5}+{\mathrm e}^{x^{2}} \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4}-2 \,{\mathrm e}^{x^{2}} \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{5}+4 i \pi \,x^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} {\mathrm e}^{x^{2}}-4 i \pi \,x^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} {\mathrm e}^{x^{2}}+4 i {\mathrm e}^{x^{2}} \pi \,x^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} \ln \left (x -1\right )+4 i {\mathrm e}^{x^{2}} \pi \,x^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} \ln \relax (x )+4 i {\mathrm e}^{x^{2}} \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 ) \ln \relax (x )+4 i {\mathrm e}^{x^{2}} \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 ) \ln \left (x -1\right )}\)	\(703\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*ln(x^2-x)^2+(20*x^4-20*x^3+40*x^2-30*x)*exp(x^2)*ln(x^2-x)+(10*x^4-1
0*x^3+30*x^2-20*x)*exp(x^2))/((x^5-x^4)*exp(x^2)^2*ln(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*ln(x^2-x)^3+((6*x^5-6*
x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*ln(x^2-x)^2+((4*x^5-4*x^4)*exp(x^2)^2+(-8*x^3+8*x^2)*exp(x^2))*ln(x^2
-x)+(x^5-x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2)+4*x-4),x,method=_RETURNVERBOSE)

[Out]

20/(8-8*x^2*exp(x^2)*ln(x)-4*x^2*exp(x^2)+exp(x^2)*Pi^2*x^2*csgn(I*x*(x-1))^6-8*exp(x^2)*x^2*ln(x)*ln(x-1)+4*I
*Pi*x^2*csgn(I*x*(x-1))^3*exp(x^2)+exp(x^2)*Pi^2*x^2*csgn(I*(x-1))^2*csgn(I*x*(x-1))^4-2*exp(x^2)*Pi^2*x^2*csg
n(I*(x-1))*csgn(I*x*(x-1))^5+exp(x^2)*Pi^2*x^2*csgn(I*x)^2*csgn(I*x*(x-1))^4-2*exp(x^2)*Pi^2*x^2*csgn(I*x)*csg
n(I*x*(x-1))^5-8*x^2*exp(x^2)*ln(x-1)-4*exp(x^2)*x^2*ln(x)^2-4*exp(x^2)*x^2*ln(x-1)^2-4*I*exp(x^2)*Pi*x^2*csgn
(I*x)*csgn(I*x*(x-1))^2*ln(x-1)-4*I*exp(x^2)*Pi*x^2*csgn(I*(x-1))*csgn(I*x*(x-1))^2*ln(x)-4*I*exp(x^2)*Pi*x^2*
csgn(I*(x-1))*csgn(I*x*(x-1))^2*ln(x-1)-4*I*exp(x^2)*Pi*x^2*csgn(I*x)*csgn(I*x*(x-1))^2*ln(x)+4*exp(x^2)*Pi^2*
x^2*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x-1))^4-2*exp(x^2)*Pi^2*x^2*csgn(I*x)*csgn(I*(x-1))^2*csgn(I*x*(x-1))^3-
2*exp(x^2)*Pi^2*x^2*csgn(I*x)^2*csgn(I*(x-1))*csgn(I*x*(x-1))^3+exp(x^2)*Pi^2*x^2*csgn(I*x)^2*csgn(I*(x-1))^2*
csgn(I*x*(x-1))^2-4*I*Pi*x^2*csgn(I*x)*csgn(I*x*(x-1))^2*exp(x^2)-4*I*Pi*x^2*csgn(I*(x-1))*csgn(I*x*(x-1))^2*e
xp(x^2)+4*I*exp(x^2)*Pi*x^2*csgn(I*x*(x-1))^3*ln(x-1)+4*I*exp(x^2)*Pi*x^2*csgn(I*x*(x-1))^3*ln(x)+4*I*exp(x^2)
*Pi*x^2*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x-1))*ln(x)+4*I*exp(x^2)*Pi*x^2*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x-
1))*ln(x-1)+4*I*Pi*x^2*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x-1))*exp(x^2))

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maxima [B]  time = 0.44, size = 56, normalized size = 2.07 \begin {gather*} -\frac {5}{{\left (x^{2} \log \left (x - 1\right )^{2} + x^{2} \log \relax (x)^{2} + 2 \, x^{2} \log \relax (x) + x^{2} + 2 \, {\left (x^{2} \log \relax (x) + x^{2}\right )} \log \left (x - 1\right )\right )} e^{\left (x^{2}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*log(x^2-x)^2+(20*x^4-20*x^3+40*x^2-30*x)*exp(x^2)*log(x^2-x)+(
10*x^4-10*x^3+30*x^2-20*x)*exp(x^2))/((x^5-x^4)*exp(x^2)^2*log(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*log(x^2-x)^3+
((6*x^5-6*x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*log(x^2-x)^2+((4*x^5-4*x^4)*exp(x^2)^2+(-8*x^3+8*x^2)*exp(x
^2))*log(x^2-x)+(x^5-x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2)+4*x-4),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {{\mathrm {e}}^{x^2}\,\left (-10\,x^4+10\,x^3-10\,x^2+10\,x\right )\,{\ln \left (x^2-x\right )}^2+{\mathrm {e}}^{x^2}\,\left (-20\,x^4+20\,x^3-40\,x^2+30\,x\right )\,\ln \left (x^2-x\right )+{\mathrm {e}}^{x^2}\,\left (-10\,x^4+10\,x^3-30\,x^2+20\,x\right )}{-{\mathrm {e}}^{2\,x^2}\,\left (x^4-x^5\right )\,{\ln \left (x^2-x\right )}^4-{\mathrm {e}}^{2\,x^2}\,\left (4\,x^4-4\,x^5\right )\,{\ln \left (x^2-x\right )}^3+\left ({\mathrm {e}}^{x^2}\,\left (4\,x^2-4\,x^3\right )-{\mathrm {e}}^{2\,x^2}\,\left (6\,x^4-6\,x^5\right )\right )\,{\ln \left (x^2-x\right )}^2+\left ({\mathrm {e}}^{x^2}\,\left (8\,x^2-8\,x^3\right )-{\mathrm {e}}^{2\,x^2}\,\left (4\,x^4-4\,x^5\right )\right )\,\ln \left (x^2-x\right )+4\,x-{\mathrm {e}}^{2\,x^2}\,\left (x^4-x^5\right )+{\mathrm {e}}^{x^2}\,\left (4\,x^2-4\,x^3\right )-4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2)*(20*x - 30*x^2 + 10*x^3 - 10*x^4) + exp(x^2)*log(x^2 - x)^2*(10*x - 10*x^2 + 10*x^3 - 10*x^4) +
exp(x^2)*log(x^2 - x)*(30*x - 40*x^2 + 20*x^3 - 20*x^4))/(4*x - exp(2*x^2)*(x^4 - x^5) + exp(x^2)*(4*x^2 - 4*
x^3) + log(x^2 - x)*(exp(x^2)*(8*x^2 - 8*x^3) - exp(2*x^2)*(4*x^4 - 4*x^5)) + log(x^2 - x)^2*(exp(x^2)*(4*x^2
- 4*x^3) - exp(2*x^2)*(6*x^4 - 6*x^5)) - exp(2*x^2)*log(x^2 - x)^4*(x^4 - x^5) - exp(2*x^2)*log(x^2 - x)^3*(4*
x^4 - 4*x^5) - 4),x)

[Out]

-int((exp(x^2)*(20*x - 30*x^2 + 10*x^3 - 10*x^4) + exp(x^2)*log(x^2 - x)^2*(10*x - 10*x^2 + 10*x^3 - 10*x^4) +
exp(x^2)*log(x^2 - x)*(30*x - 40*x^2 + 20*x^3 - 20*x^4))/(4*x - exp(2*x^2)*(x^4 - x^5) + exp(x^2)*(4*x^2 - 4*
x^3) + log(x^2 - x)*(exp(x^2)*(8*x^2 - 8*x^3) - exp(2*x^2)*(4*x^4 - 4*x^5)) + log(x^2 - x)^2*(exp(x^2)*(4*x^2
- 4*x^3) - exp(2*x^2)*(6*x^4 - 6*x^5)) - exp(2*x^2)*log(x^2 - x)^4*(x^4 - x^5) - exp(2*x^2)*log(x^2 - x)^3*(4*
x^4 - 4*x^5) - 4), x)

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sympy [A]  time = 0.64, size = 36, normalized size = 1.33 \begin {gather*} - \frac {5}{\left (x^{2} \log {\left (x^{2} - x \right )}^{2} + 2 x^{2} \log {\left (x^{2} - x \right )} + x^{2}\right ) e^{x^{2}} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x**4-10*x**3+10*x**2-10*x)*exp(x**2)*ln(x**2-x)**2+(20*x**4-20*x**3+40*x**2-30*x)*exp(x**2)*ln(
x**2-x)+(10*x**4-10*x**3+30*x**2-20*x)*exp(x**2))/((x**5-x**4)*exp(x**2)**2*ln(x**2-x)**4+(4*x**5-4*x**4)*exp(
x**2)**2*ln(x**2-x)**3+((6*x**5-6*x**4)*exp(x**2)**2+(-4*x**3+4*x**2)*exp(x**2))*ln(x**2-x)**2+((4*x**5-4*x**4
)*exp(x**2)**2+(-8*x**3+8*x**2)*exp(x**2))*ln(x**2-x)+(x**5-x**4)*exp(x**2)**2+(-4*x**3+4*x**2)*exp(x**2)+4*x-
4),x)

[Out]

-5/((x**2*log(x**2 - x)**2 + 2*x**2*log(x**2 - x) + x**2)*exp(x**2) - 2)

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