3.9.41 \(\int \frac {4-17 x+3 e x+e^x (-9 x-3 x^2)+(-6 x+e x+e^x (-3 x-x^2)) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e (-12 x+6 x^2)+e^x (-12 x+6 e x+6 x^2)+(4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e (-4 x+2 x^2)+e^x (-4 x+2 e x+2 x^2)) \log (x)+(12 x-6 e x-6 e^x x-6 x^2+(4 x-2 e x-2 e^x x-2 x^2) \log (x)) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx\)

Optimal. Leaf size=20 \[ \frac {4+x}{-2+e+e^x+x-\log (3+\log (x))} \]

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Rubi [F]  time = 12.87, 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 {4-17 x+3 e x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 - 17*x + 3*E*x + E^x*(-9*x - 3*x^2) + (-6*x + E*x + E^x*(-3*x - x^2))*Log[x] + (-3*x - x*Log[x])*Log[3
+ Log[x]])/(12*x + 3*E^2*x + 3*E^(2*x)*x - 12*x^2 + 3*x^3 + E*(-12*x + 6*x^2) + E^x*(-12*x + 6*E*x + 6*x^2) +
(4*x + E^2*x + E^(2*x)*x - 4*x^2 + x^3 + E*(-4*x + 2*x^2) + E^x*(-4*x + 2*E*x + 2*x^2))*Log[x] + (12*x - 6*E*x
- 6*E^x*x - 6*x^2 + (4*x - 2*E*x - 2*E^x*x - 2*x^2)*Log[x])*Log[3 + Log[x]] + (3*x + x*Log[x])*Log[3 + Log[x]
]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+(-17+3 e) x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{12 x+3 e^2 x+3 e^{2 x} x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx\\ &=\int \frac {4+(-17+3 e) x+e^x \left (-9 x-3 x^2\right )+\left (-6 x+e x+e^x \left (-3 x-x^2\right )\right ) \log (x)+(-3 x-x \log (x)) \log (3+\log (x))}{3 e^{2 x} x+\left (12+3 e^2\right ) x-12 x^2+3 x^3+e \left (-12 x+6 x^2\right )+e^x \left (-12 x+6 e x+6 x^2\right )+\left (4 x+e^2 x+e^{2 x} x-4 x^2+x^3+e \left (-4 x+2 x^2\right )+e^x \left (-4 x+2 e x+2 x^2\right )\right ) \log (x)+\left (12 x-6 e x-6 e^x x-6 x^2+\left (4 x-2 e x-2 e^x x-2 x^2\right ) \log (x)\right ) \log (3+\log (x))+(3 x+x \log (x)) \log ^2(3+\log (x))} \, dx\\ &=\int \frac {4-17 \left (1-\frac {3 e}{17}\right ) x-9 e^x x-3 e^x x^2-3 x \log (3+\log (x))-x \log (x) \left (6-e+e^x (3+x)+\log (3+\log (x))\right )}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx\\ &=\int \left (\frac {3+x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))}+\frac {(4+x) \left (1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))\right )}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}\right ) \, dx\\ &=\int \frac {3+x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+\int \frac {(4+x) \left (1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))\right )}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx\\ &=\int \left (\frac {3}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))}+\frac {x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))}\right ) \, dx+\int \left (\frac {1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {4 \left (1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))\right )}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}\right ) \, dx\\ &=3 \int \frac {1}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+4 \int \frac {1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+\int \frac {1-9 \left (1-\frac {e}{3}\right ) x+3 x^2-3 \left (1-\frac {e}{3}\right ) x \log (x)+x^2 \log (x)-3 x \log (3+\log (x))-x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx\\ &=3 \int \frac {1}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+4 \int \left (\frac {3 (-3+e)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {1}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 x}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {(-3+e) \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {x \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {\log (x) \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}\right ) \, dx+\int \frac {x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+\int \left (\frac {1}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 (-3+e) x}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 x^2}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {(-3+e) x \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {x^2 \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {3 x \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}+\frac {x \log (x) \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2}\right ) \, dx\\ &=3 \int \frac {x^2}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+3 \int \frac {x \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+3 \int \frac {1}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx+4 \int \frac {1}{x (3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+4 \int \frac {x \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+4 \int \frac {\log (x) \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+12 \int \frac {x}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+12 \int \frac {\log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx-(3 (3-e)) \int \frac {x}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx-(4 (3-e)) \int \frac {\log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx-(12 (3-e)) \int \frac {1}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+(-3+e) \int \frac {x \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {1}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {x^2 \log (x)}{(3+\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {x \log (x) \log (3+\log (x))}{(-3-\log (x)) \left (2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))\right )^2} \, dx+\int \frac {x}{2 \left (1-\frac {e}{2}\right )-e^x-x+\log (3+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.40, size = 20, normalized size = 1.00 \begin {gather*} \frac {4+x}{-2+e+e^x+x-\log (3+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 17*x + 3*E*x + E^x*(-9*x - 3*x^2) + (-6*x + E*x + E^x*(-3*x - x^2))*Log[x] + (-3*x - x*Log[x])*
Log[3 + Log[x]])/(12*x + 3*E^2*x + 3*E^(2*x)*x - 12*x^2 + 3*x^3 + E*(-12*x + 6*x^2) + E^x*(-12*x + 6*E*x + 6*x
^2) + (4*x + E^2*x + E^(2*x)*x - 4*x^2 + x^3 + E*(-4*x + 2*x^2) + E^x*(-4*x + 2*E*x + 2*x^2))*Log[x] + (12*x -
6*E*x - 6*E^x*x - 6*x^2 + (4*x - 2*E*x - 2*E^x*x - 2*x^2)*Log[x])*Log[3 + Log[x]] + (3*x + x*Log[x])*Log[3 +
Log[x]]^2),x]

[Out]

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

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fricas [A]  time = 0.82, size = 20, normalized size = 1.00 \begin {gather*} \frac {x + 4}{x + e + e^{x} - \log \left (\log \relax (x) + 3\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*log(x)-3*x)*log(3+log(x))+((-x^2-3*x)*exp(x)+x*exp(1)-6*x)*log(x)+(-3*x^2-9*x)*exp(x)+3*x*exp(1
)-17*x+4)/((x*log(x)+3*x)*log(3+log(x))^2+((-2*exp(x)*x-2*x*exp(1)-2*x^2+4*x)*log(x)-6*exp(x)*x-6*x*exp(1)-6*x
^2+12*x)*log(3+log(x))+(x*exp(x)^2+(2*x*exp(1)+2*x^2-4*x)*exp(x)+x*exp(1)^2+(2*x^2-4*x)*exp(1)+x^3-4*x^2+4*x)*
log(x)+3*x*exp(x)^2+(6*x*exp(1)+6*x^2-12*x)*exp(x)+3*x*exp(1)^2+(6*x^2-12*x)*exp(1)+3*x^3-12*x^2+12*x),x, algo
rithm="fricas")

[Out]

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

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giac [A]  time = 0.92, size = 20, normalized size = 1.00 \begin {gather*} \frac {x + 4}{x + e + e^{x} - \log \left (\log \relax (x) + 3\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*log(x)-3*x)*log(3+log(x))+((-x^2-3*x)*exp(x)+x*exp(1)-6*x)*log(x)+(-3*x^2-9*x)*exp(x)+3*x*exp(1
)-17*x+4)/((x*log(x)+3*x)*log(3+log(x))^2+((-2*exp(x)*x-2*x*exp(1)-2*x^2+4*x)*log(x)-6*exp(x)*x-6*x*exp(1)-6*x
^2+12*x)*log(3+log(x))+(x*exp(x)^2+(2*x*exp(1)+2*x^2-4*x)*exp(x)+x*exp(1)^2+(2*x^2-4*x)*exp(1)+x^3-4*x^2+4*x)*
log(x)+3*x*exp(x)^2+(6*x*exp(1)+6*x^2-12*x)*exp(x)+3*x*exp(1)^2+(6*x^2-12*x)*exp(1)+3*x^3-12*x^2+12*x),x, algo
rithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 21, normalized size = 1.05

method	result	size
risch	\(\frac {4+x}{x -2+{\mathrm e}^{x}+{\mathrm e}-\ln \left (3+\ln \relax (x )\right )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x*ln(x)-3*x)*ln(3+ln(x))+((-x^2-3*x)*exp(x)+x*exp(1)-6*x)*ln(x)+(-3*x^2-9*x)*exp(x)+3*x*exp(1)-17*x+4)/
((x*ln(x)+3*x)*ln(3+ln(x))^2+((-2*exp(x)*x-2*x*exp(1)-2*x^2+4*x)*ln(x)-6*exp(x)*x-6*x*exp(1)-6*x^2+12*x)*ln(3+
ln(x))+(x*exp(x)^2+(2*x*exp(1)+2*x^2-4*x)*exp(x)+x*exp(1)^2+(2*x^2-4*x)*exp(1)+x^3-4*x^2+4*x)*ln(x)+3*x*exp(x)
^2+(6*x*exp(1)+6*x^2-12*x)*exp(x)+3*x*exp(1)^2+(6*x^2-12*x)*exp(1)+3*x^3-12*x^2+12*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.82, size = 20, normalized size = 1.00 \begin {gather*} \frac {x + 4}{x + e + e^{x} - \log \left (\log \relax (x) + 3\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*log(x)-3*x)*log(3+log(x))+((-x^2-3*x)*exp(x)+x*exp(1)-6*x)*log(x)+(-3*x^2-9*x)*exp(x)+3*x*exp(1
)-17*x+4)/((x*log(x)+3*x)*log(3+log(x))^2+((-2*exp(x)*x-2*x*exp(1)-2*x^2+4*x)*log(x)-6*exp(x)*x-6*x*exp(1)-6*x
^2+12*x)*log(3+log(x))+(x*exp(x)^2+(2*x*exp(1)+2*x^2-4*x)*exp(x)+x*exp(1)^2+(2*x^2-4*x)*exp(1)+x^3-4*x^2+4*x)*
log(x)+3*x*exp(x)^2+(6*x*exp(1)+6*x^2-12*x)*exp(x)+3*x*exp(1)^2+(6*x^2-12*x)*exp(1)+3*x^3-12*x^2+12*x),x, algo
rithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {17\,x+\ln \left (\ln \relax (x)+3\right )\,\left (3\,x+x\,\ln \relax (x)\right )+\ln \relax (x)\,\left (6\,x-x\,\mathrm {e}+{\mathrm {e}}^x\,\left (x^2+3\,x\right )\right )-3\,x\,\mathrm {e}+{\mathrm {e}}^x\,\left (3\,x^2+9\,x\right )-4}{12\,x-\ln \left (\ln \relax (x)+3\right )\,\left (6\,x\,\mathrm {e}-12\,x+\ln \relax (x)\,\left (2\,x\,\mathrm {e}-4\,x+2\,x\,{\mathrm {e}}^x+2\,x^2\right )+6\,x\,{\mathrm {e}}^x+6\,x^2\right )+3\,x\,{\mathrm {e}}^{2\,x}-\mathrm {e}\,\left (12\,x-6\,x^2\right )+3\,x\,{\mathrm {e}}^2+{\ln \left (\ln \relax (x)+3\right )}^2\,\left (3\,x+x\,\ln \relax (x)\right )-12\,x^2+3\,x^3+{\mathrm {e}}^x\,\left (6\,x\,\mathrm {e}-12\,x+6\,x^2\right )+\ln \relax (x)\,\left (4\,x+x\,{\mathrm {e}}^{2\,x}-\mathrm {e}\,\left (4\,x-2\,x^2\right )+x\,{\mathrm {e}}^2-4\,x^2+x^3+{\mathrm {e}}^x\,\left (2\,x\,\mathrm {e}-4\,x+2\,x^2\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(17*x + log(log(x) + 3)*(3*x + x*log(x)) + log(x)*(6*x - x*exp(1) + exp(x)*(3*x + x^2)) - 3*x*exp(1) + ex
p(x)*(9*x + 3*x^2) - 4)/(12*x - log(log(x) + 3)*(6*x*exp(1) - 12*x + log(x)*(2*x*exp(1) - 4*x + 2*x*exp(x) + 2
*x^2) + 6*x*exp(x) + 6*x^2) + 3*x*exp(2*x) - exp(1)*(12*x - 6*x^2) + 3*x*exp(2) + log(log(x) + 3)^2*(3*x + x*l
og(x)) - 12*x^2 + 3*x^3 + exp(x)*(6*x*exp(1) - 12*x + 6*x^2) + log(x)*(4*x + x*exp(2*x) - exp(1)*(4*x - 2*x^2)
+ x*exp(2) - 4*x^2 + x^3 + exp(x)*(2*x*exp(1) - 4*x + 2*x^2))),x)

[Out]

int(-(17*x + log(log(x) + 3)*(3*x + x*log(x)) + log(x)*(6*x - x*exp(1) + exp(x)*(3*x + x^2)) - 3*x*exp(1) + ex
p(x)*(9*x + 3*x^2) - 4)/(12*x - log(log(x) + 3)*(6*x*exp(1) - 12*x + log(x)*(2*x*exp(1) - 4*x + 2*x*exp(x) + 2
*x^2) + 6*x*exp(x) + 6*x^2) + 3*x*exp(2*x) - exp(1)*(12*x - 6*x^2) + 3*x*exp(2) + log(log(x) + 3)^2*(3*x + x*l
og(x)) - 12*x^2 + 3*x^3 + exp(x)*(6*x*exp(1) - 12*x + 6*x^2) + log(x)*(4*x + x*exp(2*x) - exp(1)*(4*x - 2*x^2)
+ x*exp(2) - 4*x^2 + x^3 + exp(x)*(2*x*exp(1) - 4*x + 2*x^2))), x)

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sympy [A]  time = 0.51, size = 19, normalized size = 0.95 \begin {gather*} \frac {x + 4}{x + e^{x} - \log {\left (\log {\relax (x )} + 3 \right )} - 2 + e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*ln(x)-3*x)*ln(3+ln(x))+((-x**2-3*x)*exp(x)+x*exp(1)-6*x)*ln(x)+(-3*x**2-9*x)*exp(x)+3*x*exp(1)-
17*x+4)/((x*ln(x)+3*x)*ln(3+ln(x))**2+((-2*exp(x)*x-2*x*exp(1)-2*x**2+4*x)*ln(x)-6*exp(x)*x-6*x*exp(1)-6*x**2+
12*x)*ln(3+ln(x))+(x*exp(x)**2+(2*x*exp(1)+2*x**2-4*x)*exp(x)+x*exp(1)**2+(2*x**2-4*x)*exp(1)+x**3-4*x**2+4*x)
*ln(x)+3*x*exp(x)**2+(6*x*exp(1)+6*x**2-12*x)*exp(x)+3*x*exp(1)**2+(6*x**2-12*x)*exp(1)+3*x**3-12*x**2+12*x),x
)

[Out]

(x + 4)/(x + exp(x) - log(log(x) + 3) - 2 + E)

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