\(\int \frac {e^{2 x} (4+16 x)+e^{2 x} (x+2 x^2) \log ^2(3 x+6 x^2)+(e^{2 x} (-4-16 x-16 x^2) \log (3 x+6 x^2)+e^{2 x} (4+17 x+20 x^2+4 x^3) \log ^2(3 x+6 x^2)) \log (\frac {-4+(4+x) \log (3 x+6 x^2)}{\log (3 x+6 x^2)})}{(-4-8 x) \log (3 x+6 x^2)+(4+9 x+2 x^2) \log ^2(3 x+6 x^2)} \, dx\) [5810]

Optimal result

Integrand size = 171, antiderivative size = 24 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=\int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx \]

[In]

Int[(E^(2*x)*(4 + 16*x) + E^(2*x)*(x + 2*x^2)*Log[3*x + 6*x^2]^2 + (E^(2*x)*(-4 - 16*x - 16*x^2)*Log[3*x + 6*x
^2] + E^(2*x)*(4 + 17*x + 20*x^2 + 4*x^3)*Log[3*x + 6*x^2]^2)*Log[(-4 + (4 + x)*Log[3*x + 6*x^2])/Log[3*x + 6*
x^2]])/((-4 - 8*x)*Log[3*x + 6*x^2] + (4 + 9*x + 2*x^2)*Log[3*x + 6*x^2]^2),x]

[Out]

E^(2*x)/2 - (4*ExpIntegralEi[2*(4 + x)])/E^8 - 2*Defer[Int][E^(2*x)/Log[3*x*(1 + 2*x)], x] + Defer[Int][E^(2*x
)/((1 + 2*x)*Log[3*x*(1 + 2*x)]), x] - 16*Defer[Int][E^(2*x)/((4 + x)*(-4 + 4*Log[3*x*(1 + 2*x)] + x*Log[3*x*(
1 + 2*x)])), x] - (7*Defer[Int][E^(2*x)/((1 + 2*x)*(-4 + 4*Log[3*x*(1 + 2*x)] + x*Log[3*x*(1 + 2*x)])), x])/2
+ (23*Defer[Int][E^(2*x)/(-4 + (4 + x)*Log[3*x*(1 + 2*x)]), x])/2 + 2*Defer[Int][(E^(2*x)*x)/(-4 + (4 + x)*Log
[3*x*(1 + 2*x)]), x] + Defer[Int][E^(2*x)*Log[4 + x - 4/Log[3*x*(1 + 2*x)]], x] + 2*Defer[Int][E^(2*x)*x*Log[4
+ x - 4/Log[3*x*(1 + 2*x)]], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2 x} \left (-4-16 x-x (1+2 x) \log ^2(3 x (1+2 x))-(1+2 x)^2 \log (3 x (1+2 x)) (-4+(4+x) \log (3 x (1+2 x))) \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right )\right )}{(1+2 x) \log (3 x (1+2 x)) (4-(4+x) \log (3 x (1+2 x)))} \, dx \\ & = \int \left (\frac {e^{2 x} \left (4+16 x+x \log ^2(3 x (1+2 x))+2 x^2 \log ^2(3 x (1+2 x))\right )}{(1+2 x) \log (3 x (1+2 x)) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))}+e^{2 x} (1+2 x) \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right )\right ) \, dx \\ & = \int \frac {e^{2 x} \left (4+16 x+x \log ^2(3 x (1+2 x))+2 x^2 \log ^2(3 x (1+2 x))\right )}{(1+2 x) \log (3 x (1+2 x)) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))} \, dx+\int e^{2 x} (1+2 x) \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx \\ & = \int \frac {e^{2 x} \left (-4-16 x-x (1+2 x) \log ^2(3 x (1+2 x))\right )}{(1+2 x) \log (3 x (1+2 x)) (4-(4+x) \log (3 x (1+2 x)))} \, dx+\int \left (e^{2 x} \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right )+2 e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right )\right ) \, dx \\ & = 2 \int e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx+\int \left (\frac {e^{2 x} x}{4+x}+\frac {e^{2 x} (-1-4 x)}{(1+2 x) \log (3 x (1+2 x))}+\frac {4 e^{2 x} x}{(4+x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))}+\frac {e^{2 x} (4+x) (1+4 x)}{(1+2 x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))}\right ) \, dx+\int e^{2 x} \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx \\ & = 2 \int e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx+4 \int \frac {e^{2 x} x}{(4+x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))} \, dx+\int \frac {e^{2 x} x}{4+x} \, dx+\int \frac {e^{2 x} (-1-4 x)}{(1+2 x) \log (3 x (1+2 x))} \, dx+\int \frac {e^{2 x} (4+x) (1+4 x)}{(1+2 x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))} \, dx+\int e^{2 x} \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx \\ & = 2 \int e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx+4 \int \left (\frac {e^{2 x}}{-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x))}-\frac {4 e^{2 x}}{(4+x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))}\right ) \, dx+\int \left (e^{2 x}-\frac {4 e^{2 x}}{4+x}\right ) \, dx+\int \left (-\frac {2 e^{2 x}}{\log (3 x (1+2 x))}+\frac {e^{2 x}}{(1+2 x) \log (3 x (1+2 x))}\right ) \, dx+\int \left (\frac {15 e^{2 x}}{2 (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))}+\frac {2 e^{2 x} x}{-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x))}-\frac {7 e^{2 x}}{2 (1+2 x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))}\right ) \, dx+\int e^{2 x} \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx \\ & = -\left (2 \int \frac {e^{2 x}}{\log (3 x (1+2 x))} \, dx\right )+2 \int \frac {e^{2 x} x}{-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x))} \, dx+2 \int e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx-\frac {7}{2} \int \frac {e^{2 x}}{(1+2 x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))} \, dx-4 \int \frac {e^{2 x}}{4+x} \, dx+4 \int \frac {e^{2 x}}{-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x))} \, dx+\frac {15}{2} \int \frac {e^{2 x}}{-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x))} \, dx-16 \int \frac {e^{2 x}}{(4+x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))} \, dx+\int e^{2 x} \, dx+\int \frac {e^{2 x}}{(1+2 x) \log (3 x (1+2 x))} \, dx+\int e^{2 x} \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx \\ & = \frac {e^{2 x}}{2}-\frac {4 \text {Ei}(2 (4+x))}{e^8}-2 \int \frac {e^{2 x}}{\log (3 x (1+2 x))} \, dx+2 \int \frac {e^{2 x} x}{-4+(4+x) \log (3 x (1+2 x))} \, dx+2 \int e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx-\frac {7}{2} \int \frac {e^{2 x}}{(1+2 x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))} \, dx+4 \int \frac {e^{2 x}}{-4+(4+x) \log (3 x (1+2 x))} \, dx+\frac {15}{2} \int \frac {e^{2 x}}{-4+(4+x) \log (3 x (1+2 x))} \, dx-16 \int \frac {e^{2 x}}{(4+x) (-4+4 \log (3 x (1+2 x))+x \log (3 x (1+2 x)))} \, dx+\int \frac {e^{2 x}}{(1+2 x) \log (3 x (1+2 x))} \, dx+\int e^{2 x} \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \]

[In]

Integrate[(E^(2*x)*(4 + 16*x) + E^(2*x)*(x + 2*x^2)*Log[3*x + 6*x^2]^2 + (E^(2*x)*(-4 - 16*x - 16*x^2)*Log[3*x
+ 6*x^2] + E^(2*x)*(4 + 17*x + 20*x^2 + 4*x^3)*Log[3*x + 6*x^2]^2)*Log[(-4 + (4 + x)*Log[3*x + 6*x^2])/Log[3*
x + 6*x^2]])/((-4 - 8*x)*Log[3*x + 6*x^2] + (4 + 9*x + 2*x^2)*Log[3*x + 6*x^2]^2),x]

[Out]

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

Maple [A] (verified)

Time = 94.42 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54

method	result	size
parallelrisch	\(\ln \left (\frac {\left (4+x \right ) \ln \left (6 x^{2}+3 x \right )-4}{\ln \left (6 x^{2}+3 x \right )}\right ) {\mathrm e}^{2 x} x\)	\(37\)
risch	\(\text {Expression too large to display}\)	\(2063\)

[In]

int((((4*x^3+20*x^2+17*x+4)*exp(x)^2*ln(6*x^2+3*x)^2+(-16*x^2-16*x-4)*exp(x)^2*ln(6*x^2+3*x))*ln(((4+x)*ln(6*x
^2+3*x)-4)/ln(6*x^2+3*x))+(2*x^2+x)*exp(x)^2*ln(6*x^2+3*x)^2+(16*x+4)*exp(x)^2)/((2*x^2+9*x+4)*ln(6*x^2+3*x)^2
+(-8*x-4)*ln(6*x^2+3*x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=x e^{\left (2 \, x\right )} \log \left (\frac {{\left (x + 4\right )} \log \left (6 \, x^{2} + 3 \, x\right ) - 4}{\log \left (6 \, x^{2} + 3 \, x\right )}\right ) \]

[In]

integrate((((4*x^3+20*x^2+17*x+4)*exp(x)^2*log(6*x^2+3*x)^2+(-16*x^2-16*x-4)*exp(x)^2*log(6*x^2+3*x))*log(((4+
x)*log(6*x^2+3*x)-4)/log(6*x^2+3*x))+(2*x^2+x)*exp(x)^2*log(6*x^2+3*x)^2+(16*x+4)*exp(x)^2)/((2*x^2+9*x+4)*log
(6*x^2+3*x)^2+(-8*x-4)*log(6*x^2+3*x)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=\text {Timed out} \]

[In]

integrate((((4*x**3+20*x**2+17*x+4)*exp(x)**2*ln(6*x**2+3*x)**2+(-16*x**2-16*x-4)*exp(x)**2*ln(6*x**2+3*x))*ln
(((4+x)*ln(6*x**2+3*x)-4)/ln(6*x**2+3*x))+(2*x**2+x)*exp(x)**2*ln(6*x**2+3*x)**2+(16*x+4)*exp(x)**2)/((2*x**2+
9*x+4)*ln(6*x**2+3*x)**2+(-8*x-4)*ln(6*x**2+3*x)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (23) = 46\).

Time = 0.34 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.42 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=x e^{\left (2 \, x\right )} \log \left (x {\left (\log \left (3\right ) + \log \left (2 \, x + 1\right )\right )} + {\left (x + 4\right )} \log \left (x\right ) + 4 \, \log \left (3\right ) + 4 \, \log \left (2 \, x + 1\right ) - 4\right ) - x e^{\left (2 \, x\right )} \log \left (\log \left (3\right ) + \log \left (2 \, x + 1\right ) + \log \left (x\right )\right ) \]

[In]

integrate((((4*x^3+20*x^2+17*x+4)*exp(x)^2*log(6*x^2+3*x)^2+(-16*x^2-16*x-4)*exp(x)^2*log(6*x^2+3*x))*log(((4+
x)*log(6*x^2+3*x)-4)/log(6*x^2+3*x))+(2*x^2+x)*exp(x)^2*log(6*x^2+3*x)^2+(16*x+4)*exp(x)^2)/((2*x^2+9*x+4)*log
(6*x^2+3*x)^2+(-8*x-4)*log(6*x^2+3*x)),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (23) = 46\).

Time = 0.91 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=x e^{\left (2 \, x\right )} \log \left (x \log \left (6 \, x^{2} + 3 \, x\right ) + 4 \, \log \left (6 \, x^{2} + 3 \, x\right ) - 4\right ) - x e^{\left (2 \, x\right )} \log \left (\log \left (6 \, x^{2} + 3 \, x\right )\right ) \]

[In]

integrate((((4*x^3+20*x^2+17*x+4)*exp(x)^2*log(6*x^2+3*x)^2+(-16*x^2-16*x-4)*exp(x)^2*log(6*x^2+3*x))*log(((4+
x)*log(6*x^2+3*x)-4)/log(6*x^2+3*x))+(2*x^2+x)*exp(x)^2*log(6*x^2+3*x)^2+(16*x+4)*exp(x)^2)/((2*x^2+9*x+4)*log
(6*x^2+3*x)^2+(-8*x-4)*log(6*x^2+3*x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.68 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=x\,{\mathrm {e}}^{2\,x}\,\ln \left (\frac {\ln \left (6\,x^2+3\,x\right )\,\left (x+4\right )-4}{\ln \left (6\,x^2+3\,x\right )}\right ) \]

[In]

int((log((log(3*x + 6*x^2)*(x + 4) - 4)/log(3*x + 6*x^2))*(exp(2*x)*log(3*x + 6*x^2)^2*(17*x + 20*x^2 + 4*x^3
+ 4) - exp(2*x)*log(3*x + 6*x^2)*(16*x + 16*x^2 + 4)) + exp(2*x)*(16*x + 4) + exp(2*x)*log(3*x + 6*x^2)^2*(x +
2*x^2))/(log(3*x + 6*x^2)^2*(9*x + 2*x^2 + 4) - log(3*x + 6*x^2)*(8*x + 4)),x)

[Out]

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