3.4.16 \(\int \frac {1-x+e^x (2 x-2 x^2)+e^{2 x} (-2 x+2 x^3)+(-2 x+2 x^2+e^x (4 x-2 x^2-2 x^3)) \log (x)+(-2 x+2 x^2) \log ^2(x)+(x-e^{2 x} x^2+2 e^x x^2 \log (x)-x^2 \log ^2(x)) \log (-x+e^{2 x} x^2-2 e^x x^2 \log (x)+x^2 \log ^2(x))}{-x+2 x^2-x^3+e^{2 x} (x^2-2 x^3+x^4)+e^x (-2 x^2+4 x^3-2 x^4) \log (x)+(x^2-2 x^3+x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {\log \left (-x+x^2 \left (e^x-\log (x)\right )^2\right )}{-1+x} \]

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Rubi [F]  time = 17.68, 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 {1-x+e^x \left (2 x-2 x^2\right )+e^{2 x} \left (-2 x+2 x^3\right )+\left (-2 x+2 x^2+e^x \left (4 x-2 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+2 x^2\right ) \log ^2(x)+\left (x-e^{2 x} x^2+2 e^x x^2 \log (x)-x^2 \log ^2(x)\right ) \log \left (-x+e^{2 x} x^2-2 e^x x^2 \log (x)+x^2 \log ^2(x)\right )}{-x+2 x^2-x^3+e^{2 x} \left (x^2-2 x^3+x^4\right )+e^x \left (-2 x^2+4 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

4*Log[1 - x] - 2*Log[x] + 3*Defer[Int][1/((-1 + x)*(-1 + E^(2*x)*x - 2*E^x*x*Log[x] + x*Log[x]^2)), x] - 2*Def
er[Int][E^x/((-1 + x)*(-1 + E^(2*x)*x - 2*E^x*x*Log[x] + x*Log[x]^2)), x] - Defer[Int][1/(x*(-1 + E^(2*x)*x -
2*E^x*x*Log[x] + x*Log[x]^2)), x] + 2*Defer[Int][(E^x*Log[x])/(-1 + E^(2*x)*x - 2*E^x*x*Log[x] + x*Log[x]^2),
x] + 2*Defer[Int][Log[x]/((-1 + x)*(-1 + E^(2*x)*x - 2*E^x*x*Log[x] + x*Log[x]^2)), x] + 2*Defer[Int][(E^x*Log
[x])/((-1 + x)*(-1 + E^(2*x)*x - 2*E^x*x*Log[x] + x*Log[x]^2)), x] - 2*Defer[Int][Log[x]^2/(-1 + E^(2*x)*x - 2
*E^x*x*Log[x] + x*Log[x]^2), x] - 2*Defer[Int][Log[x]^2/((-1 + x)*(-1 + E^(2*x)*x - 2*E^x*x*Log[x] + x*Log[x]^
2)), x] - Defer[Int][Log[-x + E^(2*x)*x^2 - 2*E^x*x^2*Log[x] + x^2*Log[x]^2]/(1 - x)^2, x]

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 32, normalized size = 1.28 \begin {gather*} \frac {\log \left (x \left (-1+e^{2 x} x-2 e^x x \log (x)+x \log ^2(x)\right )\right )}{-1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.65, size = 36, normalized size = 1.44 \begin {gather*} \frac {\log \left (-2 \, x^{2} e^{x} \log \relax (x) + x^{2} \log \relax (x)^{2} + x^{2} e^{\left (2 \, x\right )} - x\right )}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*log(x)^2+2*x^2*exp(x)*log(x)-exp(x)^2*x^2+x)*log(x^2*log(x)^2-2*x^2*exp(x)*log(x)+exp(x)^2*x^
2-x)+(2*x^2-2*x)*log(x)^2+((-2*x^3-2*x^2+4*x)*exp(x)+2*x^2-2*x)*log(x)+(2*x^3-2*x)*exp(x)^2+(-2*x^2+2*x)*exp(x
)-x+1)/((x^4-2*x^3+x^2)*log(x)^2+(-2*x^4+4*x^3-2*x^2)*exp(x)*log(x)+(x^4-2*x^3+x^2)*exp(x)^2-x^3+2*x^2-x),x, a
lgorithm="fricas")

[Out]

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

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giac [A]  time = 0.58, size = 31, normalized size = 1.24 \begin {gather*} \frac {\log \left (-2 \, x e^{x} \log \relax (x) + x \log \relax (x)^{2} + x e^{\left (2 \, x\right )} - 1\right ) + \log \relax (x)}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*log(x)^2+2*x^2*exp(x)*log(x)-exp(x)^2*x^2+x)*log(x^2*log(x)^2-2*x^2*exp(x)*log(x)+exp(x)^2*x^
2-x)+(2*x^2-2*x)*log(x)^2+((-2*x^3-2*x^2+4*x)*exp(x)+2*x^2-2*x)*log(x)+(2*x^3-2*x)*exp(x)^2+(-2*x^2+2*x)*exp(x
)-x+1)/((x^4-2*x^3+x^2)*log(x)^2+(-2*x^4+4*x^3-2*x^2)*exp(x)*log(x)+(x^4-2*x^3+x^2)*exp(x)^2-x^3+2*x^2-x),x, a
lgorithm="giac")

[Out]

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

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maple [C]  time = 0.32, size = 220, normalized size = 8.80

method	result	size
risch	\(\frac {\ln \left (-1+\left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} \ln \relax (x )+\ln \relax (x )^{2}\right ) x \right )}{x -1}+\frac {-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (1-\left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} \ln \relax (x )+\ln \relax (x )^{2}\right ) x \right )\right ) \mathrm {csgn}\left (i x \left (1-\left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} \ln \relax (x )+\ln \relax (x )^{2}\right ) x \right )\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (1-\left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} \ln \relax (x )+\ln \relax (x )^{2}\right ) x \right )\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (1-\left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} \ln \relax (x )+\ln \relax (x )^{2}\right ) x \right )\right ) \mathrm {csgn}\left (i x \left (1-\left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} \ln \relax (x )+\ln \relax (x )^{2}\right ) x \right )\right )^{2}+i \pi \mathrm {csgn}\left (i x \left (1-\left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} \ln \relax (x )+\ln \relax (x )^{2}\right ) x \right )\right )^{3}+2 \ln \relax (x )}{2 x -2}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2*ln(x)^2+2*x^2*exp(x)*ln(x)-exp(x)^2*x^2+x)*ln(x^2*ln(x)^2-2*x^2*exp(x)*ln(x)+exp(x)^2*x^2-x)+(2*x^2
-2*x)*ln(x)^2+((-2*x^3-2*x^2+4*x)*exp(x)+2*x^2-2*x)*ln(x)+(2*x^3-2*x)*exp(x)^2+(-2*x^2+2*x)*exp(x)-x+1)/((x^4-
2*x^3+x^2)*ln(x)^2+(-2*x^4+4*x^3-2*x^2)*exp(x)*ln(x)+(x^4-2*x^3+x^2)*exp(x)^2-x^3+2*x^2-x),x,method=_RETURNVER
BOSE)

[Out]

1/(x-1)*ln(-1+(exp(2*x)-2*exp(x)*ln(x)+ln(x)^2)*x)+1/2*(-I*Pi*csgn(I*x)*csgn(I*(1-(exp(2*x)-2*exp(x)*ln(x)+ln(
x)^2)*x))*csgn(I*x*(1-(exp(2*x)-2*exp(x)*ln(x)+ln(x)^2)*x))+I*Pi*csgn(I*x)*csgn(I*x*(1-(exp(2*x)-2*exp(x)*ln(x
)+ln(x)^2)*x))^2-I*Pi*csgn(I*(1-(exp(2*x)-2*exp(x)*ln(x)+ln(x)^2)*x))*csgn(I*x*(1-(exp(2*x)-2*exp(x)*ln(x)+ln(
x)^2)*x))^2+I*Pi*csgn(I*x*(1-(exp(2*x)-2*exp(x)*ln(x)+ln(x)^2)*x))^3+2*ln(x))/(x-1)

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maxima [A]  time = 0.87, size = 31, normalized size = 1.24 \begin {gather*} \frac {\log \left (-2 \, x e^{x} \log \relax (x) + x \log \relax (x)^{2} + x e^{\left (2 \, x\right )} - 1\right ) + \log \relax (x)}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*log(x)^2+2*x^2*exp(x)*log(x)-exp(x)^2*x^2+x)*log(x^2*log(x)^2-2*x^2*exp(x)*log(x)+exp(x)^2*x^
2-x)+(2*x^2-2*x)*log(x)^2+((-2*x^3-2*x^2+4*x)*exp(x)+2*x^2-2*x)*log(x)+(2*x^3-2*x)*exp(x)^2+(-2*x^2+2*x)*exp(x
)-x+1)/((x^4-2*x^3+x^2)*log(x)^2+(-2*x^4+4*x^3-2*x^2)*exp(x)*log(x)+(x^4-2*x^3+x^2)*exp(x)^2-x^3+2*x^2-x),x, a
lgorithm="maxima")

[Out]

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

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mupad [B]  time = 0.70, size = 36, normalized size = 1.44 \begin {gather*} \frac {\ln \left (x^2\,{\mathrm {e}}^{2\,x}-x+x^2\,{\ln \relax (x)}^2-2\,x^2\,{\mathrm {e}}^x\,\ln \relax (x)\right )}{x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2*exp(2*x) - x + x^2*log(x)^2 - 2*x^2*exp(x)*log(x))/(x - 1)

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sympy [A]  time = 1.62, size = 34, normalized size = 1.36 \begin {gather*} \frac {\log {\left (x^{2} e^{2 x} - 2 x^{2} e^{x} \log {\relax (x )} + x^{2} \log {\relax (x )}^{2} - x \right )}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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