3.18.27 \(\int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x (-7+6 x+6 x^2)+(e^2 (1-x)-x+2 x^2+e^x (-1+x+x^2)) \log (x)+(-6 x-6 e^x x+(-x-e^x x) \log (x)) \log (\frac {1}{4} (6 x+x \log (x)))}{6 x+x \log (x)} \, dx\) [1727]

Optimal. Leaf size=30 \[ \left (-e^2+e^x+x\right ) \left (x-\log \left (x+\frac {1}{4} (2 x+x \log (x))\right )\right ) \]

[Out]

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

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Rubi [F]
time = 1.72, 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 {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^2*(7 - 6*x) - 7*x + 12*x^2 + E^x*(-7 + 6*x + 6*x^2) + (E^2*(1 - x) - x + 2*x^2 + E^x*(-1 + x + x^2))*Lo
g[x] + (-6*x - 6*E^x*x + (-x - E^x*x)*Log[x])*Log[(6*x + x*Log[x])/4])/(6*x + x*Log[x]),x]

[Out]

-x - E^2*x + x^2 - ExpIntegralEi[6 + Log[x]]/E^6 + (12*ExpIntegralEi[2*(6 + Log[x])])/E^12 + E^2*Log[x] + (2*E
xpIntegralEi[2*(6 + Log[x])]*Log[x])/E^12 - (2*ExpIntegralEi[2*(6 + Log[x])]*(6 + Log[x]))/E^12 + E^2*Log[6 +
Log[x]] + (E^x*(6*x^2 + x^2*Log[x] - 6*x*Log[(x*(6 + Log[x]))/4] - x*Log[x]*Log[(x*(6 + Log[x]))/4]))/(x*(6 +
Log[x])) - 6*Defer[Int][Log[(x*(6 + Log[x]))/4]/(6 + Log[x]), x] - Defer[Int][(Log[x]*Log[(x*(6 + Log[x]))/4])
/(6 + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{x (6+\log (x))} \, dx\\ &=\int \left (-\frac {7}{6+\log (x)}+\frac {12 x}{6+\log (x)}-\frac {e^2 (-7+6 x)}{x (6+\log (x))}-\frac {\log (x)}{6+\log (x)}-\frac {e^2 (-1+x) \log (x)}{x (6+\log (x))}+\frac {2 x \log (x)}{6+\log (x)}-\frac {6 \log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)}-\frac {\log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)}+\frac {e^x \left (-7+6 x+6 x^2-\log (x)+x \log (x)+x^2 \log (x)-6 x \log \left (\frac {1}{4} x (6+\log (x))\right )-x \log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )\right )}{x (6+\log (x))}\right ) \, dx\\ &=2 \int \frac {x \log (x)}{6+\log (x)} \, dx-6 \int \frac {\log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx-7 \int \frac {1}{6+\log (x)} \, dx+12 \int \frac {x}{6+\log (x)} \, dx-e^2 \int \frac {-7+6 x}{x (6+\log (x))} \, dx-e^2 \int \frac {(-1+x) \log (x)}{x (6+\log (x))} \, dx-\int \frac {\log (x)}{6+\log (x)} \, dx-\int \frac {\log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx+\int \frac {e^x \left (-7+6 x+6 x^2-\log (x)+x \log (x)+x^2 \log (x)-6 x \log \left (\frac {1}{4} x (6+\log (x))\right )-x \log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )\right )}{x (6+\log (x))} \, dx\\ &=\frac {2 \text {Ei}(2 (6+\log (x))) \log (x)}{e^{12}}+\frac {e^x \left (6 x^2+x^2 \log (x)-6 x \log \left (\frac {1}{4} x (6+\log (x))\right )-x \log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )\right )}{x (6+\log (x))}-2 \int \frac {\text {Ei}(2 (6+\log (x)))}{e^{12} x} \, dx-6 \int \frac {\log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx-7 \text {Subst}\left (\int \frac {e^x}{6+x} \, dx,x,\log (x)\right )+12 \text {Subst}\left (\int \frac {e^{2 x}}{6+x} \, dx,x,\log (x)\right )-e^2 \int \left (\frac {6}{6+\log (x)}-\frac {7}{x (6+\log (x))}\right ) \, dx-e^2 \int \left (\frac {-1+x}{x}-\frac {6 (-1+x)}{x (6+\log (x))}\right ) \, dx-\int \left (1-\frac {6}{6+\log (x)}\right ) \, dx-\int \frac {\log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx\\ &=-x-\frac {7 \text {Ei}(6+\log (x))}{e^6}+\frac {12 \text {Ei}(2 (6+\log (x)))}{e^{12}}+\frac {2 \text {Ei}(2 (6+\log (x))) \log (x)}{e^{12}}+\frac {e^x \left (6 x^2+x^2 \log (x)-6 x \log \left (\frac {1}{4} x (6+\log (x))\right )-x \log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )\right )}{x (6+\log (x))}+6 \int \frac {1}{6+\log (x)} \, dx-6 \int \frac {\log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx-\frac {2 \int \frac {\text {Ei}(2 (6+\log (x)))}{x} \, dx}{e^{12}}-e^2 \int \frac {-1+x}{x} \, dx-\left (6 e^2\right ) \int \frac {1}{6+\log (x)} \, dx+\left (6 e^2\right ) \int \frac {-1+x}{x (6+\log (x))} \, dx+\left (7 e^2\right ) \int \frac {1}{x (6+\log (x))} \, dx-\int \frac {\log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx\\ &=-x-\frac {7 \text {Ei}(6+\log (x))}{e^6}+\frac {12 \text {Ei}(2 (6+\log (x)))}{e^{12}}+\frac {2 \text {Ei}(2 (6+\log (x))) \log (x)}{e^{12}}+\frac {e^x \left (6 x^2+x^2 \log (x)-6 x \log \left (\frac {1}{4} x (6+\log (x))\right )-x \log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )\right )}{x (6+\log (x))}-6 \int \frac {\log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx+6 \text {Subst}\left (\int \frac {e^x}{6+x} \, dx,x,\log (x)\right )-\frac {2 \text {Subst}(\int \text {Ei}(2 (6+x)) \, dx,x,\log (x))}{e^{12}}-e^2 \int \left (1-\frac {1}{x}\right ) \, dx+\left (6 e^2\right ) \int \left (\frac {1}{6+\log (x)}-\frac {1}{x (6+\log (x))}\right ) \, dx-\left (6 e^2\right ) \text {Subst}\left (\int \frac {e^x}{6+x} \, dx,x,\log (x)\right )+\left (7 e^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,6+\log (x)\right )-\int \frac {\log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx\\ &=-x-e^2 x-\frac {\text {Ei}(6+\log (x))}{e^6}-\frac {6 \text {Ei}(6+\log (x))}{e^4}+\frac {12 \text {Ei}(2 (6+\log (x)))}{e^{12}}+e^2 \log (x)+\frac {2 \text {Ei}(2 (6+\log (x))) \log (x)}{e^{12}}+7 e^2 \log (6+\log (x))+\frac {e^x \left (6 x^2+x^2 \log (x)-6 x \log \left (\frac {1}{4} x (6+\log (x))\right )-x \log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )\right )}{x (6+\log (x))}-6 \int \frac {\log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx-\frac {\text {Subst}(\int \text {Ei}(x) \, dx,x,12+2 \log (x))}{e^{12}}+\left (6 e^2\right ) \int \frac {1}{6+\log (x)} \, dx-\left (6 e^2\right ) \int \frac {1}{x (6+\log (x))} \, dx-\int \frac {\log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx\\ &=-x-e^2 x+x^2-\frac {\text {Ei}(6+\log (x))}{e^6}-\frac {6 \text {Ei}(6+\log (x))}{e^4}+\frac {12 \text {Ei}(2 (6+\log (x)))}{e^{12}}+e^2 \log (x)+\frac {2 \text {Ei}(2 (6+\log (x))) \log (x)}{e^{12}}-\frac {2 \text {Ei}(12+2 \log (x)) (6+\log (x))}{e^{12}}+7 e^2 \log (6+\log (x))+\frac {e^x \left (6 x^2+x^2 \log (x)-6 x \log \left (\frac {1}{4} x (6+\log (x))\right )-x \log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )\right )}{x (6+\log (x))}-6 \int \frac {\log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx-\left (6 e^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,6+\log (x)\right )+\left (6 e^2\right ) \text {Subst}\left (\int \frac {e^x}{6+x} \, dx,x,\log (x)\right )-\int \frac {\log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx\\ &=-x-e^2 x+x^2-\frac {\text {Ei}(6+\log (x))}{e^6}+\frac {12 \text {Ei}(2 (6+\log (x)))}{e^{12}}+e^2 \log (x)+\frac {2 \text {Ei}(2 (6+\log (x))) \log (x)}{e^{12}}-\frac {2 \text {Ei}(12+2 \log (x)) (6+\log (x))}{e^{12}}+e^2 \log (6+\log (x))+\frac {e^x \left (6 x^2+x^2 \log (x)-6 x \log \left (\frac {1}{4} x (6+\log (x))\right )-x \log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )\right )}{x (6+\log (x))}-6 \int \frac {\log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx-\int \frac {\log (x) \log \left (\frac {1}{4} x (6+\log (x))\right )}{6+\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.13, size = 47, normalized size = 1.57 \begin {gather*} -e^2 x+e^x x+x^2+e^2 \log (x)+e^2 \log (6+\log (x))-\left (e^x+x\right ) \log \left (\frac {1}{4} x (6+\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^2*(7 - 6*x) - 7*x + 12*x^2 + E^x*(-7 + 6*x + 6*x^2) + (E^2*(1 - x) - x + 2*x^2 + E^x*(-1 + x + x^
2))*Log[x] + (-6*x - 6*E^x*x + (-x - E^x*x)*Log[x])*Log[(6*x + x*Log[x])/4])/(6*x + x*Log[x]),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.24, size = 243, normalized size = 8.10

method	result	size
risch	\(2 \,{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{2} \ln \left (x \right )-{\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{x} x -x \ln \left (x \right )+2 x \ln \left (2\right )+\frac {i {\mathrm e}^{x} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \left (x \right )+6\right )\right ) \mathrm {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )}{2}-\frac {i {\mathrm e}^{x} \pi \,\mathrm {csgn}\left (i \left (\ln \left (x \right )+6\right )\right ) \mathrm {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{2}}{2}-{\mathrm e}^{2} x +x^{2}+\frac {i \pi x \mathrm {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{3}}{2}+\frac {i {\mathrm e}^{x} \pi \mathrm {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{3}}{2}-\frac {i {\mathrm e}^{x} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{2}}{2}+{\mathrm e}^{2} \ln \left (\ln \left (x \right )+6\right )-\frac {i \pi x \,\mathrm {csgn}\left (i \left (\ln \left (x \right )+6\right )\right ) \mathrm {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \left (x \right )+6\right )\right ) \mathrm {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )}{2}+\left (-{\mathrm e}^{x}-x \right ) \ln \left (\ln \left (x \right )+6\right )\)	\(243\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-exp(x)*x-x)*ln(x)-6*exp(x)*x-6*x)*ln(1/4*x*ln(x)+3/2*x)+((x^2+x-1)*exp(x)+(1-x)*exp(2)+2*x^2-x)*ln(x)+
(6*x^2+6*x-7)*exp(x)+(-6*x+7)*exp(2)+12*x^2-7*x)/(x*ln(x)+6*x),x,method=_RETURNVERBOSE)

[Out]

2*exp(x)*ln(2)+exp(2)*ln(x)-exp(x)*ln(x)+exp(x)*x-x*ln(x)+2*x*ln(2)+1/2*I*exp(x)*Pi*csgn(I*x)*csgn(I*(ln(x)+6)
)*csgn(I*x*(ln(x)+6))-1/2*I*exp(x)*Pi*csgn(I*(ln(x)+6))*csgn(I*x*(ln(x)+6))^2-exp(2)*x+x^2+1/2*I*Pi*x*csgn(I*x
*(ln(x)+6))^3+1/2*I*exp(x)*Pi*csgn(I*x*(ln(x)+6))^3-1/2*I*exp(x)*Pi*csgn(I*x)*csgn(I*x*(ln(x)+6))^2-1/2*I*Pi*x
*csgn(I*x)*csgn(I*x*(ln(x)+6))^2+exp(2)*ln(ln(x)+6)-1/2*I*Pi*x*csgn(I*(ln(x)+6))*csgn(I*x*(ln(x)+6))^2+1/2*I*P
i*x*csgn(I*x)*csgn(I*(ln(x)+6))*csgn(I*x*(ln(x)+6))+(-exp(x)-x)*ln(ln(x)+6)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).
time = 0.50, size = 61, normalized size = 2.03 \begin {gather*} x^{2} - x {\left (e^{2} - 2 \, \log \left (2\right )\right )} + {\left (x + 2 \, \log \left (2\right ) - \log \left (x\right )\right )} e^{x} - {\left (x - e^{2}\right )} \log \left (x\right ) - {\left (x + 6 \, e^{2} + e^{x}\right )} \log \left (\log \left (x\right ) + 6\right ) + 7 \, e^{2} \log \left (\log \left (x\right ) + 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x-x)*log(x)-6*exp(x)*x-6*x)*log(1/4*x*log(x)+3/2*x)+((x^2+x-1)*exp(x)+(1-x)*exp(2)+2*x^2-
x)*log(x)+(6*x^2+6*x-7)*exp(x)+(-6*x+7)*exp(2)+12*x^2-7*x)/(x*log(x)+6*x),x, algorithm="maxima")

[Out]

x^2 - x*(e^2 - 2*log(2)) + (x + 2*log(2) - log(x))*e^x - (x - e^2)*log(x) - (x + 6*e^2 + e^x)*log(log(x) + 6)
+ 7*e^2*log(log(x) + 6)

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Fricas [A]
time = 0.38, size = 33, normalized size = 1.10 \begin {gather*} x^{2} - x e^{2} + x e^{x} - {\left (x - e^{2} + e^{x}\right )} \log \left (\frac {1}{4} \, x \log \left (x\right ) + \frac {3}{2} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x-x)*log(x)-6*exp(x)*x-6*x)*log(1/4*x*log(x)+3/2*x)+((x^2+x-1)*exp(x)+(1-x)*exp(2)+2*x^2-
x)*log(x)+(6*x^2+6*x-7)*exp(x)+(-6*x+7)*exp(2)+12*x^2-7*x)/(x*log(x)+6*x),x, algorithm="fricas")

[Out]

x^2 - x*e^2 + x*e^x - (x - e^2 + e^x)*log(1/4*x*log(x) + 3/2*x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).
time = 7.13, size = 58, normalized size = 1.93 \begin {gather*} x^{2} - x \log {\left (\frac {x \log {\left (x \right )}}{4} + \frac {3 x}{2} \right )} - x e^{2} + \left (x - \log {\left (\frac {x \log {\left (x \right )}}{4} + \frac {3 x}{2} \right )}\right ) e^{x} + e^{2} \log {\left (x \right )} + e^{2} \log {\left (\log {\left (x \right )} + 6 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x-x)*ln(x)-6*exp(x)*x-6*x)*ln(1/4*x*ln(x)+3/2*x)+((x**2+x-1)*exp(x)+(1-x)*exp(2)+2*x**2-x
)*ln(x)+(6*x**2+6*x-7)*exp(x)+(-6*x+7)*exp(2)+12*x**2-7*x)/(x*ln(x)+6*x),x)

[Out]

x**2 - x*log(x*log(x)/4 + 3*x/2) - x*exp(2) + (x - log(x*log(x)/4 + 3*x/2))*exp(x) + exp(2)*log(x) + exp(2)*lo
g(log(x) + 6)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (23) = 46\).
time = 0.40, size = 65, normalized size = 2.17 \begin {gather*} x^{2} - x e^{2} + x e^{x} + 2 \, x \log \left (2\right ) + 2 \, e^{x} \log \left (2\right ) - x \log \left (x\right ) + e^{2} \log \left (x\right ) - e^{x} \log \left (x\right ) - x \log \left (\log \left (x\right ) + 6\right ) + e^{2} \log \left (\log \left (x\right ) + 6\right ) - e^{x} \log \left (\log \left (x\right ) + 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x-x)*log(x)-6*exp(x)*x-6*x)*log(1/4*x*log(x)+3/2*x)+((x^2+x-1)*exp(x)+(1-x)*exp(2)+2*x^2-
x)*log(x)+(6*x^2+6*x-7)*exp(x)+(-6*x+7)*exp(2)+12*x^2-7*x)/(x*log(x)+6*x),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.35, size = 42, normalized size = 1.40 \begin {gather*} {\mathrm {e}}^2\,\ln \left (\ln \left (x\right )+6\right )-\ln \left (\frac {3\,x}{2}+\frac {x\,\ln \left (x\right )}{4}\right )\,\left (x+{\mathrm {e}}^x\right )-x\,{\mathrm {e}}^2+{\mathrm {e}}^2\,\ln \left (x\right )+x\,{\mathrm {e}}^x+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7*x + log((3*x)/2 + (x*log(x))/4)*(6*x + 6*x*exp(x) + log(x)*(x + x*exp(x))) - exp(x)*(6*x + 6*x^2 - 7)
- 12*x^2 + log(x)*(x + exp(2)*(x - 1) - 2*x^2 - exp(x)*(x + x^2 - 1)) + exp(2)*(6*x - 7))/(6*x + x*log(x)),x)

[Out]

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

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