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

Optimal. Leaf size=30 \[ \frac {(-x+\log (x)) \left (-x+\log \left (e^6+x+x \left (e^3+x \log (x)\right )\right )\right )}{x} \]

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Rubi [F]  time = 2.98, 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 {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*E^6)/(3*x^3) + (1 + E^3)/(2*x^2) - 3/x + x - 3*Log[x] - (2*Log[x])/x + 2*Defer[Int][(-E^6 - (1 + E^3)*x - x
^2*Log[x])^(-1), x] + 2*E^3*Defer[Int][(-E^6 - (1 + E^3)*x - x^2*Log[x])^(-1), x] - (3 + 3*E^3 - E^6)*Defer[In
t][(-E^6 - (1 + E^3)*x - x^2*Log[x])^(-1), x] + E^6*Defer[Int][1/(x*(-E^6 - (1 + E^3)*x - x^2*Log[x])), x] + E
^6*Defer[Int][(E^6 + (1 + E^3)*x + x^2*Log[x])^(-1), x] + 2*E^12*Defer[Int][1/(x^4*(E^6 + (1 + E^3)*x + x^2*Lo
g[x])), x] + 3*E^6*(1 + E^3)*Defer[Int][1/(x^3*(E^6 + (1 + E^3)*x + x^2*Log[x])), x] + 2*(1 + E^3)^2*Defer[Int
][1/(x^2*(E^6 + (1 + E^3)*x + x^2*Log[x])), x] - (1 + 2*E^3 + 2*E^6)*Defer[Int][1/(x^2*(E^6 + (1 + E^3)*x + x^
2*Log[x])), x] - (1 + E^3 - 3*E^6)*Defer[Int][1/(x*(E^6 + (1 + E^3)*x + x^2*Log[x])), x] + E^3*Defer[Int][x/(E
^6 + (1 + E^3)*x + x^2*Log[x]), x] - (1 + E^3)*Defer[Int][x/(E^6 + (1 + E^3)*x + x^2*Log[x]), x] + Defer[Int][
Log[E^6 + (1 + E^3)*x + x^2*Log[x]]/x^2, x] - Defer[Int][(Log[x]*Log[E^6 + (1 + E^3)*x + x^2*Log[x]])/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+\left (1+e^3\right ) x^3+x^4 \log (x)} \, dx\\ &=\int \left (\frac {2}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)}+\frac {e^3 (-2+x)}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}+\frac {e^6 (-1+x)}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {\left (1+e^3+x-3 x^2+x^3\right ) \log (x)}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {2 \log ^2(x)}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}+\frac {(1-\log (x)) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+2 \int \frac {\log ^2(x)}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+e^3 \int \frac {-2+x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+e^6 \int \frac {-1+x}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\int \frac {\left (1+e^3+x-3 x^2+x^3\right ) \log (x)}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\int \frac {(1-\log (x)) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx\\ &=2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+2 \int \left (\frac {-e^6-\left (1+e^3\right ) x}{x^4}+\frac {\log (x)}{x^2}+\frac {\left (e^6+\left (1+e^3\right ) x\right )^2}{x^4 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}\right ) \, dx+e^3 \int \left (\frac {2}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)}+\frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}\right ) \, dx+e^6 \int \left (\frac {1}{x \left (-e^6-\left (1+e^3\right ) x-x^2 \log (x)\right )}+\frac {1}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}\right ) \, dx+\int \left (\frac {1+e^3+x-3 x^2+x^3}{x^3}+\frac {-e^6 \left (1+e^3\right )-\left (1+2 e^3+2 e^6\right ) x-\left (1+e^3-3 e^6\right ) x^2+\left (3+3 e^3-e^6\right ) x^3-\left (1+e^3\right ) x^4}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}\right ) \, dx+\int \left (\frac {\log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2}-\frac {\log (x) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {-e^6-\left (1+e^3\right ) x}{x^4} \, dx+2 \int \frac {\log (x)}{x^2} \, dx+2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+2 \int \frac {\left (e^6+\left (1+e^3\right ) x\right )^2}{x^4 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+e^3 \int \frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\left (2 e^3\right ) \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+e^6 \int \frac {1}{x \left (-e^6-\left (1+e^3\right ) x-x^2 \log (x)\right )} \, dx+e^6 \int \frac {1}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\int \frac {1+e^3+x-3 x^2+x^3}{x^3} \, dx+\int \frac {-e^6 \left (1+e^3\right )-\left (1+2 e^3+2 e^6\right ) x-\left (1+e^3-3 e^6\right ) x^2+\left (3+3 e^3-e^6\right ) x^3-\left (1+e^3\right ) x^4}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\int \frac {\log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx-\int \frac {\log (x) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx\\ &=-\frac {2}{x}-\frac {2 \log (x)}{x}+2 \int \left (-\frac {e^6}{x^4}+\frac {-1-e^3}{x^3}\right ) \, dx+2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+2 \int \left (\frac {e^{12}}{x^4 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {2 \left (e^6+e^9\right )}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {\left (1+e^3\right )^2}{x^2 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}\right ) \, dx+e^3 \int \frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\left (2 e^3\right ) \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+e^6 \int \frac {1}{x \left (-e^6-\left (1+e^3\right ) x-x^2 \log (x)\right )} \, dx+e^6 \int \frac {1}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\int \left (1+\frac {1+e^3}{x^3}+\frac {1}{x^2}-\frac {3}{x}\right ) \, dx+\int \left (\frac {e^6 \left (1-\frac {3 \left (1+e^3\right )}{e^6}\right )}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)}+\frac {e^6 \left (-1-e^3\right )}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {-1-2 e^3-2 e^6}{x^2 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {-1-e^3+3 e^6}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {\left (-1-e^3\right ) x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}\right ) \, dx+\int \frac {\log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx-\int \frac {\log (x) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx\\ &=\frac {2 e^6}{3 x^3}+\frac {1+e^3}{2 x^2}-\frac {3}{x}+x-3 \log (x)-\frac {2 \log (x)}{x}+2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+e^3 \int \frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\left (2 e^3\right ) \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+e^6 \int \frac {1}{x \left (-e^6-\left (1+e^3\right ) x-x^2 \log (x)\right )} \, dx+e^6 \int \frac {1}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\left (2 e^{12}\right ) \int \frac {1}{x^4 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (-1-e^3\right ) \int \frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx-\left (e^6 \left (1+e^3\right )\right ) \int \frac {1}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (4 e^6 \left (1+e^3\right )\right ) \int \frac {1}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (2 \left (1+e^3\right )^2\right ) \int \frac {1}{x^2 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (-1-2 e^3-2 e^6\right ) \int \frac {1}{x^2 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (-3-3 e^3+e^6\right ) \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+\left (-1-e^3+3 e^6\right ) \int \frac {1}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\int \frac {\log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx-\int \frac {\log (x) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 48, normalized size = 1.60 \begin {gather*} x-\log (x)-\log \left (e^6+x+e^3 x+x^2 \log (x)\right )+\frac {\log (x) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - Log[x] - Log[E^6 + x + E^3*x + x^2*Log[x]] + (Log[x]*Log[E^6 + x + E^3*x + x^2*Log[x]])/x

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fricas [A]  time = 0.78, size = 36, normalized size = 1.20 \begin {gather*} \frac {x^{2} - {\left (x - \log \relax (x)\right )} \log \left (x^{2} \log \relax (x) + x e^{3} + x + e^{6}\right ) - x \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.49, size = 55, normalized size = 1.83 \begin {gather*} \frac {x^{2} - x \log \left (-x^{2} \log \relax (x) - x e^{3} - x - e^{6}\right ) - x \log \relax (x) + \log \left (x^{2} \log \relax (x) + x e^{3} + x + e^{6}\right ) \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 46, normalized size = 1.53

method	result	size
risch	\(\frac {\ln \relax (x ) \ln \left (x^{2} \ln \relax (x )+{\mathrm e}^{6}+x \,{\mathrm e}^{3}+x \right )}{x}+x -3 \ln \relax (x )-\ln \left (\ln \relax (x )+\frac {{\mathrm e}^{6}+x \,{\mathrm e}^{3}+x}{x^{2}}\right )\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 54, normalized size = 1.80 \begin {gather*} \frac {x^{2} + \log \left (x^{2} \log \relax (x) + x {\left (e^{3} + 1\right )} + e^{6}\right ) \log \relax (x)}{x} - 3 \, \log \relax (x) - \log \left (\frac {x^{2} \log \relax (x) + x {\left (e^{3} + 1\right )} + e^{6}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.16, size = 50, normalized size = 1.67 \begin {gather*} x-\ln \left (x+{\mathrm {e}}^6+x^2\,\ln \relax (x)+x\,{\mathrm {e}}^3\right )-\ln \left (\frac {1}{x^2}\right )-3\,\ln \relax (x)+\frac {\ln \left (x+{\mathrm {e}}^6+x^2\,\ln \relax (x)+x\,{\mathrm {e}}^3\right )\,\ln \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.70, size = 48, normalized size = 1.60 \begin {gather*} x - 3 \log {\relax (x )} - \log {\left (\log {\relax (x )} + \frac {x + x e^{3} + e^{6}}{x^{2}} \right )} + \frac {\log {\relax (x )} \log {\left (x^{2} \log {\relax (x )} + x + x e^{3} + e^{6} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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