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

Optimal. Leaf size=26 \[ \frac {\left (-3 e^x+x\right ) \left (\log (x)+\frac {x}{x+\log (x)}\right )}{x+x^2} \]

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Rubi [F]  time = 27.64, 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 {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\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[(-x^2 + E^x*(3*x + 3*x^2 - 3*x^4) + (3*x^2 + 2*x^3 - x^4 + E^x*(-6*x - 3*x^2 - 3*x^4))*Log[x] + (x + x^2 -
2*x^3 + E^x*(-3 + 3*x + 6*x^2 - 6*x^3))*Log[x]^2 + (-x^2 + E^x*(3 + 3*x - 3*x^2))*Log[x]^3)/(x^4 + 2*x^5 + x^
6 + (2*x^3 + 4*x^4 + 2*x^5)*Log[x] + (x^2 + 2*x^3 + x^4)*Log[x]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x \left (x+3 e^x \left (-1-x+x^3\right )\right )-x (1+x) \left ((-3+x) x+3 e^x \left (2-x+x^2\right )\right ) \log (x)-(-1+x) \left (x (1+2 x)+e^x \left (-3+6 x^2\right )\right ) \log ^2(x)-\left (x^2+3 e^x \left (-1-x+x^2\right )\right ) \log ^3(x)}{x^2 (1+x)^2 (x+\log (x))^2} \, dx\\ &=\int \left (-\frac {1}{(1+x)^2 (x+\log (x))^2}-\frac {(-3+x) \log (x)}{(1+x) (x+\log (x))^2}-\frac {(-1+x) (1+2 x) \log ^2(x)}{x (1+x)^2 (x+\log (x))^2}-\frac {\log ^3(x)}{(1+x)^2 (x+\log (x))^2}-\frac {3 e^x \left (-x-x^2+x^4+2 x \log (x)+x^2 \log (x)+x^4 \log (x)+\log ^2(x)-x \log ^2(x)-2 x^2 \log ^2(x)+2 x^3 \log ^2(x)-\log ^3(x)-x \log ^3(x)+x^2 \log ^3(x)\right )}{x^2 (1+x)^2 (x+\log (x))^2}\right ) \, dx\\ &=-\left (3 \int \frac {e^x \left (-x-x^2+x^4+2 x \log (x)+x^2 \log (x)+x^4 \log (x)+\log ^2(x)-x \log ^2(x)-2 x^2 \log ^2(x)+2 x^3 \log ^2(x)-\log ^3(x)-x \log ^3(x)+x^2 \log ^3(x)\right )}{x^2 (1+x)^2 (x+\log (x))^2} \, dx\right )-\int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {(-3+x) \log (x)}{(1+x) (x+\log (x))^2} \, dx-\int \frac {(-1+x) (1+2 x) \log ^2(x)}{x (1+x)^2 (x+\log (x))^2} \, dx-\int \frac {\log ^3(x)}{(1+x)^2 (x+\log (x))^2} \, dx\\ &=-\left (3 \int \frac {e^x \left (x \left (-1-x+x^3\right )+x \left (2+x+x^3\right ) \log (x)+\left (1-x-2 x^2+2 x^3\right ) \log ^2(x)+\left (-1-x+x^2\right ) \log ^3(x)\right )}{x^2 (1+x)^2 (x+\log (x))^2} \, dx\right )-\int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \left (-\frac {2 x}{(1+x)^2}+\frac {\log (x)}{(1+x)^2}-\frac {x^3}{(1+x)^2 (x+\log (x))^2}+\frac {3 x^2}{(1+x)^2 (x+\log (x))}\right ) \, dx-\int \left (-\frac {(-3+x) x}{(1+x) (x+\log (x))^2}+\frac {-3+x}{(1+x) (x+\log (x))}\right ) \, dx-\int \left (\frac {(-1+x) (1+2 x)}{x (1+x)^2}+\frac {x \left (-1-x+2 x^2\right )}{(1+x)^2 (x+\log (x))^2}-\frac {2 \left (-1-x+2 x^2\right )}{(1+x)^2 (x+\log (x))}\right ) \, dx\\ &=2 \int \frac {x}{(1+x)^2} \, dx+2 \int \frac {-1-x+2 x^2}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {x^2}{(1+x)^2 (x+\log (x))} \, dx-3 \int \left (\frac {e^x}{x^2 (1+x)}+\frac {e^x \left (-1-x+x^2\right ) \log (x)}{x^2 (1+x)^2}-\frac {e^x}{x (x+\log (x))^2}+\frac {e^x x}{(1+x)^2 (x+\log (x))}\right ) \, dx-\int \frac {(-1+x) (1+2 x)}{x (1+x)^2} \, dx-\int \frac {\log (x)}{(1+x)^2} \, dx-\int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+\int \frac {x^3}{(1+x)^2 (x+\log (x))^2} \, dx+\int \frac {(-3+x) x}{(1+x) (x+\log (x))^2} \, dx-\int \frac {x \left (-1-x+2 x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {-3+x}{(1+x) (x+\log (x))} \, dx\\ &=-\frac {x \log (x)}{1+x}+2 \int \left (-\frac {1}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx+2 \int \left (\frac {2}{x+\log (x)}+\frac {2}{(1+x)^2 (x+\log (x))}-\frac {5}{(1+x) (x+\log (x))}\right ) \, dx-3 \int \frac {e^x}{x^2 (1+x)} \, dx-3 \int \frac {e^x \left (-1-x+x^2\right ) \log (x)}{x^2 (1+x)^2} \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx-3 \int \frac {e^x x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \left (\frac {1}{x+\log (x)}+\frac {1}{(1+x)^2 (x+\log (x))}-\frac {2}{(1+x) (x+\log (x))}\right ) \, dx+\int \frac {1}{1+x} \, dx-\int \left (-\frac {1}{x}-\frac {2}{(1+x)^2}+\frac {3}{1+x}\right ) \, dx-\int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+\int \left (-\frac {2}{(x+\log (x))^2}+\frac {x}{(x+\log (x))^2}-\frac {1}{(1+x)^2 (x+\log (x))^2}+\frac {3}{(1+x) (x+\log (x))^2}\right ) \, dx+\int \left (-\frac {4}{(x+\log (x))^2}+\frac {x}{(x+\log (x))^2}+\frac {4}{(1+x) (x+\log (x))^2}\right ) \, dx-\int \left (-\frac {5}{(x+\log (x))^2}+\frac {2 x}{(x+\log (x))^2}-\frac {2}{(1+x)^2 (x+\log (x))^2}+\frac {7}{(1+x) (x+\log (x))^2}\right ) \, dx-\int \left (\frac {1}{x+\log (x)}-\frac {4}{(1+x) (x+\log (x))}\right ) \, dx\\ &=\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+3 \int \frac {e^x}{x^2 (1+x)} \, dx-3 \int \left (\frac {e^x}{x^2}-\frac {e^x}{x}+\frac {e^x}{1+x}\right ) \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx-3 \int \left (-\frac {e^x}{(1+x)^2 (x+\log (x))}+\frac {e^x}{(1+x) (x+\log (x))}\right ) \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ &=\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-3 \int \frac {e^x}{x^2} \, dx+3 \int \frac {e^x}{x} \, dx-3 \int \frac {e^x}{1+x} \, dx+3 \int \left (\frac {e^x}{x^2}-\frac {e^x}{x}+\frac {e^x}{1+x}\right ) \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+3 \int \frac {e^x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {e^x}{(1+x) (x+\log (x))} \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ &=\frac {3 e^x}{x}+3 \text {Ei}(x)-\frac {3 \text {Ei}(1+x)}{e}+\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+3 \int \frac {e^x}{x^2} \, dx-2 \left (3 \int \frac {e^x}{x} \, dx\right )+3 \int \frac {e^x}{1+x} \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+3 \int \frac {e^x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {e^x}{(1+x) (x+\log (x))} \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ &=-3 \text {Ei}(x)+\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+3 \int \frac {e^x}{x} \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+3 \int \frac {e^x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {e^x}{(1+x) (x+\log (x))} \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ &=\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+3 \int \frac {e^x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {e^x}{(1+x) (x+\log (x))} \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 32, normalized size = 1.23 \begin {gather*} \frac {\left (-3 e^x+x\right ) \left (x+x \log (x)+\log ^2(x)\right )}{x (1+x) (x+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 50, normalized size = 1.92 \begin {gather*} \frac {{\left (x - 3 \, e^{x}\right )} \log \relax (x)^{2} + x^{2} - 3 \, x e^{x} + {\left (x^{2} - 3 \, x e^{x}\right )} \log \relax (x)}{x^{3} + x^{2} + {\left (x^{2} + x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.20, size = 56, normalized size = 2.15 \begin {gather*} \frac {x^{2} \log \relax (x) - 3 \, x e^{x} \log \relax (x) + x \log \relax (x)^{2} - 3 \, e^{x} \log \relax (x)^{2} + x^{2} - 3 \, x e^{x}}{x^{3} + x^{2} \log \relax (x) + x^{2} + x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 37, normalized size = 1.42

method	result	size
risch	\(\frac {\left (x -3 \,{\mathrm e}^{x}\right ) \ln \relax (x )}{\left (x +1\right ) x}+\frac {x -3 \,{\mathrm e}^{x}}{\left (x +1\right ) \left (x +\ln \relax (x )\right )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x-3*exp(x))/(x+1)/x*ln(x)+1/(x+1)*(x-3*exp(x))/(x+ln(x))

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maxima [A]  time = 0.49, size = 48, normalized size = 1.85 \begin {gather*} \frac {x^{2} \log \relax (x) + x \log \relax (x)^{2} + x^{2} - 3 \, {\left (x \log \relax (x) + \log \relax (x)^{2} + x\right )} e^{x}}{x^{3} + x^{2} + {\left (x^{2} + x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.48, size = 60, normalized size = 2.31 \begin {gather*} \frac {x}{x^{2} + x + \left (x + 1\right ) \log {\relax (x )}} + \frac {\left (- 3 x \log {\relax (x )} - 3 x - 3 \log {\relax (x )}^{2}\right ) e^{x}}{x^{3} + x^{2} \log {\relax (x )} + x^{2} + x \log {\relax (x )}} + \frac {\log {\relax (x )}}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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