3.34.78 \(\int \frac {-8 x+2 e^{2 x} x^3+e^x (-4 x+4 x^3)+(8 x+6 e^{2 x} x^2+e^x (-4+4 x+8 x^2)) \log (3 x)+(6 e^{2 x} x+e^x (4+4 x)) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+(-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2) \log (3 x)+(-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x) \log ^2(3 x)+(-4-e^4+e^{2 x}) \log ^3(3 x)} \, dx\)

Optimal. Leaf size=25 \[ \log \left (4+e^4-\left (e^x+\frac {2 x}{x+\log (3 x)}\right )^2\right ) \]

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Rubi [F]  time = 39.59, 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 {-8 x+2 e^{2 x} x^3+e^x \left (-4 x+4 x^3\right )+\left (8 x+6 e^{2 x} x^2+e^x \left (-4+4 x+8 x^2\right )\right ) \log (3 x)+\left (6 e^{2 x} x+e^x (4+4 x)\right ) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+\left (-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2\right ) \log (3 x)+\left (-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x\right ) \log ^2(3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^3(3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x + 2*E^(2*x)*x^3 + E^x*(-4*x + 4*x^3) + (8*x + 6*E^(2*x)*x^2 + E^x*(-4 + 4*x + 8*x^2))*Log[3*x] + (6*
E^(2*x)*x + E^x*(4 + 4*x))*Log[3*x]^2 + 2*E^(2*x)*Log[3*x]^3)/(-(E^4*x^3) + 4*E^x*x^3 + E^(2*x)*x^3 + (-8*x^2
- 3*E^4*x^2 + 8*E^x*x^2 + 3*E^(2*x)*x^2)*Log[3*x] + (-12*x - 3*E^4*x + 4*E^x*x + 3*E^(2*x)*x)*Log[3*x]^2 + (-4
- E^4 + E^(2*x))*Log[3*x]^3),x]

[Out]

2*x + 8*Defer[Int][x/((x + Log[3*x])*(E^4*x^2 - 4*E^x*x^2 - E^(2*x)*x^2 - 4*E^x*x*Log[3*x] - 2*E^(2*x)*x*Log[3
*x] + 8*(1 + E^4/4)*x*Log[3*x] - E^(2*x)*Log[3*x]^2 + 4*(1 + E^4/4)*Log[3*x]^2)), x] + 4*Defer[Int][(E^x*x)/((
x + Log[3*x])*(E^4*x^2 - 4*E^x*x^2 - E^(2*x)*x^2 - 4*E^x*x*Log[3*x] - 2*E^(2*x)*x*Log[3*x] + 8*(1 + E^4/4)*x*L
og[3*x] - E^(2*x)*Log[3*x]^2 + 4*(1 + E^4/4)*Log[3*x]^2)), x] + 4*Defer[Int][(E^x*x^3)/((x + Log[3*x])*(E^4*x^
2 - 4*E^x*x^2 - E^(2*x)*x^2 - 4*E^x*x*Log[3*x] - 2*E^(2*x)*x*Log[3*x] + 8*(1 + E^4/4)*x*Log[3*x] - E^(2*x)*Log
[3*x]^2 + 4*(1 + E^4/4)*Log[3*x]^2)), x] + 4*Defer[Int][(E^x*Log[3*x])/((x + Log[3*x])*(E^4*x^2 - 4*E^x*x^2 -
E^(2*x)*x^2 - 4*E^x*x*Log[3*x] - 2*E^(2*x)*x*Log[3*x] + 8*(1 + E^4/4)*x*Log[3*x] - E^(2*x)*Log[3*x]^2 + 4*(1 +
E^4/4)*Log[3*x]^2)), x] + 8*Defer[Int][(E^x*x^2*Log[3*x])/((x + Log[3*x])*(E^4*x^2 - 4*E^x*x^2 - E^(2*x)*x^2
- 4*E^x*x*Log[3*x] - 2*E^(2*x)*x*Log[3*x] + 8*(1 + E^4/4)*x*Log[3*x] - E^(2*x)*Log[3*x]^2 + 4*(1 + E^4/4)*Log[
3*x]^2)), x] + 4*Defer[Int][(E^x*x*Log[3*x]^2)/((x + Log[3*x])*(E^4*x^2 - 4*E^x*x^2 - E^(2*x)*x^2 - 4*E^x*x*Lo
g[3*x] - 2*E^(2*x)*x*Log[3*x] + 8*(1 + E^4/4)*x*Log[3*x] - E^(2*x)*Log[3*x]^2 + 4*(1 + E^4/4)*Log[3*x]^2)), x]
- 2*(4 + E^4)*Defer[Int][Log[3*x]^3/((x + Log[3*x])*(E^4*x^2 - 4*E^x*x^2 - E^(2*x)*x^2 - 4*E^x*x*Log[3*x] - 2
*E^(2*x)*x*Log[3*x] + 8*(1 + E^4/4)*x*Log[3*x] - E^(2*x)*Log[3*x]^2 + 4*(1 + E^4/4)*Log[3*x]^2)), x] + 2*E^4*D
efer[Int][x^3/((x + Log[3*x])*((-E^4 + 4*E^x + E^(2*x))*x^2 + 2*(-4 - E^4 + 2*E^x + E^(2*x))*x*Log[3*x] + (-4
- E^4 + E^(2*x))*Log[3*x]^2)), x] + 8*Defer[Int][(x*Log[3*x])/((x + Log[3*x])*((-E^4 + 4*E^x + E^(2*x))*x^2 +
2*(-4 - E^4 + 2*E^x + E^(2*x))*x*Log[3*x] + (-4 - E^4 + E^(2*x))*Log[3*x]^2)), x] + 4*Defer[Int][(E^x*x*Log[3*
x])/((x + Log[3*x])*((-E^4 + 4*E^x + E^(2*x))*x^2 + 2*(-4 - E^4 + 2*E^x + E^(2*x))*x*Log[3*x] + (-4 - E^4 + E^
(2*x))*Log[3*x]^2)), x] + 2*(8 + 3*E^4)*Defer[Int][(x^2*Log[3*x])/((x + Log[3*x])*((-E^4 + 4*E^x + E^(2*x))*x^
2 + 2*(-4 - E^4 + 2*E^x + E^(2*x))*x*Log[3*x] + (-4 - E^4 + E^(2*x))*Log[3*x]^2)), x] + 4*Defer[Int][(E^x*Log[
3*x]^2)/((x + Log[3*x])*((-E^4 + 4*E^x + E^(2*x))*x^2 + 2*(-4 - E^4 + 2*E^x + E^(2*x))*x*Log[3*x] + (-4 - E^4
+ E^(2*x))*Log[3*x]^2)), x] + 6*(4 + E^4)*Defer[Int][(x*Log[3*x]^2)/((x + Log[3*x])*((-E^4 + 4*E^x + E^(2*x))*
x^2 + 2*(-4 - E^4 + 2*E^x + E^(2*x))*x*Log[3*x] + (-4 - E^4 + E^(2*x))*Log[3*x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (\left (2+e^x\right ) x \left (-2+e^x x^2\right )+\left (4 x+3 e^{2 x} x^2+2 e^x \left (-1+x+2 x^2\right )\right ) \log (3 x)+e^x \left (2+\left (2+3 e^x\right ) x\right ) \log ^2(3 x)+e^{2 x} \log ^3(3 x)\right )}{(x+\log (3 x)) \left (\left (-e^4+4 e^x+e^{2 x}\right ) x^2+2 \left (-4-e^4+2 e^x+e^{2 x}\right ) x \log (3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^2(3 x)\right )} \, dx\\ &=2 \int \frac {\left (2+e^x\right ) x \left (-2+e^x x^2\right )+\left (4 x+3 e^{2 x} x^2+2 e^x \left (-1+x+2 x^2\right )\right ) \log (3 x)+e^x \left (2+\left (2+3 e^x\right ) x\right ) \log ^2(3 x)+e^{2 x} \log ^3(3 x)}{(x+\log (3 x)) \left (\left (-e^4+4 e^x+e^{2 x}\right ) x^2+2 \left (-4-e^4+2 e^x+e^{2 x}\right ) x \log (3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^2(3 x)\right )} \, dx\\ &=2 \int \left (1+\frac {4 x+2 e^x x-e^4 x^3+2 e^x x^3+2 e^x \log (3 x)-4 x \log (3 x)-2 e^x x \log (3 x)+4 e^x x^2 \log (3 x)-8 \left (1+\frac {3 e^4}{8}\right ) x^2 \log (3 x)-2 e^x \log ^2(3 x)+2 e^x x \log ^2(3 x)-12 \left (1+\frac {e^4}{4}\right ) x \log ^2(3 x)-4 \left (1+\frac {e^4}{4}\right ) \log ^3(3 x)}{(x+\log (3 x)) \left (e^4 x^2-4 e^x x^2-e^{2 x} x^2-4 e^x x \log (3 x)-2 e^{2 x} x \log (3 x)+8 \left (1+\frac {e^4}{4}\right ) x \log (3 x)-e^{2 x} \log ^2(3 x)+4 \left (1+\frac {e^4}{4}\right ) \log ^2(3 x)\right )}\right ) \, dx\\ &=2 x+2 \int \frac {4 x+2 e^x x-e^4 x^3+2 e^x x^3+2 e^x \log (3 x)-4 x \log (3 x)-2 e^x x \log (3 x)+4 e^x x^2 \log (3 x)-8 \left (1+\frac {3 e^4}{8}\right ) x^2 \log (3 x)-2 e^x \log ^2(3 x)+2 e^x x \log ^2(3 x)-12 \left (1+\frac {e^4}{4}\right ) x \log ^2(3 x)-4 \left (1+\frac {e^4}{4}\right ) \log ^3(3 x)}{(x+\log (3 x)) \left (e^4 x^2-4 e^x x^2-e^{2 x} x^2-4 e^x x \log (3 x)-2 e^{2 x} x \log (3 x)+8 \left (1+\frac {e^4}{4}\right ) x \log (3 x)-e^{2 x} \log ^2(3 x)+4 \left (1+\frac {e^4}{4}\right ) \log ^2(3 x)\right )} \, dx\\ &=2 x+2 \int \frac {x \left (-4+e^4 x^2-2 e^x \left (1+x^2\right )\right )+\left (3 e^4 x^2+4 x (1+2 x)+e^x \left (-2+2 x-4 x^2\right )\right ) \log (3 x)+\left (-2 e^x (-1+x)+12 x+3 e^4 x\right ) \log ^2(3 x)+\left (4+e^4\right ) \log ^3(3 x)}{(x+\log (3 x)) \left (\left (-e^4+4 e^x+e^{2 x}\right ) x^2+2 \left (-4-e^4+2 e^x+e^{2 x}\right ) x \log (3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^2(3 x)\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.46, size = 113, normalized size = 4.52 \begin {gather*} 2 \left (-\log (x+\log (3 x))+\frac {1}{2} \log \left (-e^4 x^2+4 e^x x^2+e^{2 x} x^2-8 x \log (3 x)-2 e^4 x \log (3 x)+4 e^x x \log (3 x)+2 e^{2 x} x \log (3 x)-4 \log ^2(3 x)-e^4 \log ^2(3 x)+e^{2 x} \log ^2(3 x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x + 2*E^(2*x)*x^3 + E^x*(-4*x + 4*x^3) + (8*x + 6*E^(2*x)*x^2 + E^x*(-4 + 4*x + 8*x^2))*Log[3*x]
+ (6*E^(2*x)*x + E^x*(4 + 4*x))*Log[3*x]^2 + 2*E^(2*x)*Log[3*x]^3)/(-(E^4*x^3) + 4*E^x*x^3 + E^(2*x)*x^3 + (-
8*x^2 - 3*E^4*x^2 + 8*E^x*x^2 + 3*E^(2*x)*x^2)*Log[3*x] + (-12*x - 3*E^4*x + 4*E^x*x + 3*E^(2*x)*x)*Log[3*x]^2
+ (-4 - E^4 + E^(2*x))*Log[3*x]^3),x]

[Out]

2*(-Log[x + Log[3*x]] + Log[-(E^4*x^2) + 4*E^x*x^2 + E^(2*x)*x^2 - 8*x*Log[3*x] - 2*E^4*x*Log[3*x] + 4*E^x*x*L
og[3*x] + 2*E^(2*x)*x*Log[3*x] - 4*Log[3*x]^2 - E^4*Log[3*x]^2 + E^(2*x)*Log[3*x]^2]/2)

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fricas [B]  time = 0.75, size = 101, normalized size = 4.04 \begin {gather*} -2 \, \log \left (x + \log \left (3 \, x\right )\right ) + \log \left (\frac {x^{2} e^{4} - x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{x} + {\left (e^{4} - e^{\left (2 \, x\right )} + 4\right )} \log \left (3 \, x\right )^{2} + 2 \, {\left (x e^{4} - x e^{\left (2 \, x\right )} - 2 \, x e^{x} + 4 \, x\right )} \log \left (3 \, x\right )}{e^{4} - e^{\left (2 \, x\right )} + 4}\right ) + \log \left (-e^{4} + e^{\left (2 \, x\right )} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^2*log(3*x)^3+(6*x*exp(x)^2+(4*x+4)*exp(x))*log(3*x)^2+(6*exp(x)^2*x^2+(8*x^2+4*x-4)*exp(x)
+8*x)*log(3*x)+2*exp(x)^2*x^3+(4*x^3-4*x)*exp(x)-8*x)/((exp(x)^2-exp(4)-4)*log(3*x)^3+(3*x*exp(x)^2+4*exp(x)*x
-3*x*exp(4)-12*x)*log(3*x)^2+(3*exp(x)^2*x^2+8*exp(x)*x^2-3*x^2*exp(4)-8*x^2)*log(3*x)+exp(x)^2*x^3+4*exp(x)*x
^3-x^3*exp(4)),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 3.93, size = 184, normalized size = 7.36 \begin {gather*} \log \left (x^{2} e^{4} - x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{x} + 2 \, x e^{4} \log \relax (3) - 2 \, x e^{\left (2 \, x\right )} \log \relax (3) - 4 \, x e^{x} \log \relax (3) + e^{4} \log \relax (3)^{2} - e^{\left (2 \, x\right )} \log \relax (3)^{2} + 2 \, x e^{4} \log \relax (x) - 2 \, x e^{\left (2 \, x\right )} \log \relax (x) - 4 \, x e^{x} \log \relax (x) + 2 \, e^{4} \log \relax (3) \log \relax (x) - 2 \, e^{\left (2 \, x\right )} \log \relax (3) \log \relax (x) + e^{4} \log \relax (x)^{2} - e^{\left (2 \, x\right )} \log \relax (x)^{2} + 8 \, x \log \relax (3) + 4 \, \log \relax (3)^{2} + 8 \, x \log \relax (x) + 8 \, \log \relax (3) \log \relax (x) + 4 \, \log \relax (x)^{2}\right ) - 2 \, \log \left (x + \log \relax (3) + \log \relax (x)\right ) + \log \left (e^{4} - e^{\left (2 \, x\right )} + 4\right ) - \log \left (-e^{4} + e^{\left (2 \, x\right )} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^2*log(3*x)^3+(6*x*exp(x)^2+(4*x+4)*exp(x))*log(3*x)^2+(6*exp(x)^2*x^2+(8*x^2+4*x-4)*exp(x)
+8*x)*log(3*x)+2*exp(x)^2*x^3+(4*x^3-4*x)*exp(x)-8*x)/((exp(x)^2-exp(4)-4)*log(3*x)^3+(3*x*exp(x)^2+4*exp(x)*x
-3*x*exp(4)-12*x)*log(3*x)^2+(3*exp(x)^2*x^2+8*exp(x)*x^2-3*x^2*exp(4)-8*x^2)*log(3*x)+exp(x)^2*x^3+4*exp(x)*x
^3-x^3*exp(4)),x, algorithm="giac")

[Out]

log(x^2*e^4 - x^2*e^(2*x) - 4*x^2*e^x + 2*x*e^4*log(3) - 2*x*e^(2*x)*log(3) - 4*x*e^x*log(3) + e^4*log(3)^2 -
e^(2*x)*log(3)^2 + 2*x*e^4*log(x) - 2*x*e^(2*x)*log(x) - 4*x*e^x*log(x) + 2*e^4*log(3)*log(x) - 2*e^(2*x)*log(
3)*log(x) + e^4*log(x)^2 - e^(2*x)*log(x)^2 + 8*x*log(3) + 4*log(3)^2 + 8*x*log(x) + 8*log(3)*log(x) + 4*log(x
)^2) - 2*log(x + log(3) + log(x)) + log(e^4 - e^(2*x) + 4) - log(-e^4 + e^(2*x) - 4)

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maple [B]  time = 0.07, size = 92, normalized size = 3.68

method	result	size
risch	\(\ln \left ({\mathrm e}^{2 x}-{\mathrm e}^{4}-4\right )-2 \ln \left (x +\ln \left (3 x \right )\right )+\ln \left (\ln \left (3 x \right )^{2}+\frac {2 x \left (-{\mathrm e}^{2 x}+{\mathrm e}^{4}-2 \,{\mathrm e}^{x}+4\right ) \ln \left (3 x \right )}{-{\mathrm e}^{2 x}+{\mathrm e}^{4}+4}+\frac {\left (-{\mathrm e}^{2 x}+{\mathrm e}^{4}-4 \,{\mathrm e}^{x}\right ) x^{2}}{-{\mathrm e}^{2 x}+{\mathrm e}^{4}+4}\right )\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x)^2*ln(3*x)^3+(6*x*exp(x)^2+(4*x+4)*exp(x))*ln(3*x)^2+(6*exp(x)^2*x^2+(8*x^2+4*x-4)*exp(x)+8*x)*ln
(3*x)+2*exp(x)^2*x^3+(4*x^3-4*x)*exp(x)-8*x)/((exp(x)^2-exp(4)-4)*ln(3*x)^3+(3*x*exp(x)^2+4*exp(x)*x-3*x*exp(4
)-12*x)*ln(3*x)^2+(3*exp(x)^2*x^2+8*exp(x)*x^2-3*x^2*exp(4)-8*x^2)*ln(3*x)+exp(x)^2*x^3+4*exp(x)*x^3-x^3*exp(4
)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.74, size = 139, normalized size = 5.56 \begin {gather*} \log \left (-\frac {x^{2} e^{4} + e^{4} \log \relax (3)^{2} + {\left (e^{4} + 4\right )} \log \relax (x)^{2} + 2 \, {\left (e^{4} \log \relax (3) + 4 \, \log \relax (3)\right )} x - {\left (x^{2} + 2 \, x \log \relax (3) + \log \relax (3)^{2} + 2 \, {\left (x + \log \relax (3)\right )} \log \relax (x) + \log \relax (x)^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{2} + x \log \relax (3) + x \log \relax (x)\right )} e^{x} + 4 \, \log \relax (3)^{2} + 2 \, {\left (x {\left (e^{4} + 4\right )} + e^{4} \log \relax (3) + 4 \, \log \relax (3)\right )} \log \relax (x)}{x^{2} + 2 \, x \log \relax (3) + \log \relax (3)^{2} + 2 \, {\left (x + \log \relax (3)\right )} \log \relax (x) + \log \relax (x)^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^2*log(3*x)^3+(6*x*exp(x)^2+(4*x+4)*exp(x))*log(3*x)^2+(6*exp(x)^2*x^2+(8*x^2+4*x-4)*exp(x)
+8*x)*log(3*x)+2*exp(x)^2*x^3+(4*x^3-4*x)*exp(x)-8*x)/((exp(x)^2-exp(4)-4)*log(3*x)^3+(3*x*exp(x)^2+4*exp(x)*x
-3*x*exp(4)-12*x)*log(3*x)^2+(3*exp(x)^2*x^2+8*exp(x)*x^2-3*x^2*exp(4)-8*x^2)*log(3*x)+exp(x)^2*x^3+4*exp(x)*x
^3-x^3*exp(4)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,{\ln \left (3\,x\right )}^3\,{\mathrm {e}}^{2\,x}-8\,x+2\,x^3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (4\,x-4\,x^3\right )+{\ln \left (3\,x\right )}^2\,\left (6\,x\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (4\,x+4\right )\right )+\ln \left (3\,x\right )\,\left (8\,x+6\,x^2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (8\,x^2+4\,x-4\right )\right )}{4\,x^3\,{\mathrm {e}}^x-{\ln \left (3\,x\right )}^2\,\left (12\,x-3\,x\,{\mathrm {e}}^{2\,x}+3\,x\,{\mathrm {e}}^4-4\,x\,{\mathrm {e}}^x\right )+\ln \left (3\,x\right )\,\left (8\,x^2\,{\mathrm {e}}^x+3\,x^2\,{\mathrm {e}}^{2\,x}-3\,x^2\,{\mathrm {e}}^4-8\,x^2\right )-{\ln \left (3\,x\right )}^3\,\left ({\mathrm {e}}^4-{\mathrm {e}}^{2\,x}+4\right )+x^3\,{\mathrm {e}}^{2\,x}-x^3\,{\mathrm {e}}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(3*x)^3*exp(2*x) - 8*x + 2*x^3*exp(2*x) - exp(x)*(4*x - 4*x^3) + log(3*x)^2*(6*x*exp(2*x) + exp(x)*(
4*x + 4)) + log(3*x)*(8*x + 6*x^2*exp(2*x) + exp(x)*(4*x + 8*x^2 - 4)))/(4*x^3*exp(x) - log(3*x)^2*(12*x - 3*x
*exp(2*x) + 3*x*exp(4) - 4*x*exp(x)) + log(3*x)*(8*x^2*exp(x) + 3*x^2*exp(2*x) - 3*x^2*exp(4) - 8*x^2) - log(3
*x)^3*(exp(4) - exp(2*x) + 4) + x^3*exp(2*x) - x^3*exp(4)),x)

[Out]

int((2*log(3*x)^3*exp(2*x) - 8*x + 2*x^3*exp(2*x) - exp(x)*(4*x - 4*x^3) + log(3*x)^2*(6*x*exp(2*x) + exp(x)*(
4*x + 4)) + log(3*x)*(8*x + 6*x^2*exp(2*x) + exp(x)*(4*x + 8*x^2 - 4)))/(4*x^3*exp(x) - log(3*x)^2*(12*x - 3*x
*exp(2*x) + 3*x*exp(4) - 4*x*exp(x)) + log(3*x)*(8*x^2*exp(x) + 3*x^2*exp(2*x) - 3*x^2*exp(4) - 8*x^2) - log(3
*x)^3*(exp(4) - exp(2*x) + 4) + x^3*exp(2*x) - x^3*exp(4)), x)

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sympy [B]  time = 2.95, size = 85, normalized size = 3.40 \begin {gather*} \log {\left (\frac {4 x e^{x}}{x + \log {\left (3 x \right )}} + e^{2 x} + \frac {- x^{2} e^{4} - 2 x e^{4} \log {\left (3 x \right )} - 8 x \log {\left (3 x \right )} - e^{4} \log {\left (3 x \right )}^{2} - 4 \log {\left (3 x \right )}^{2}}{x^{2} + 2 x \log {\left (3 x \right )} + \log {\left (3 x \right )}^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)**2*ln(3*x)**3+(6*x*exp(x)**2+(4*x+4)*exp(x))*ln(3*x)**2+(6*exp(x)**2*x**2+(8*x**2+4*x-4)*e
xp(x)+8*x)*ln(3*x)+2*exp(x)**2*x**3+(4*x**3-4*x)*exp(x)-8*x)/((exp(x)**2-exp(4)-4)*ln(3*x)**3+(3*x*exp(x)**2+4
*exp(x)*x-3*x*exp(4)-12*x)*ln(3*x)**2+(3*exp(x)**2*x**2+8*exp(x)*x**2-3*x**2*exp(4)-8*x**2)*ln(3*x)+exp(x)**2*
x**3+4*exp(x)*x**3-x**3*exp(4)),x)

[Out]

log(4*x*exp(x)/(x + log(3*x)) + exp(2*x) + (-x**2*exp(4) - 2*x*exp(4)*log(3*x) - 8*x*log(3*x) - exp(4)*log(3*x
)**2 - 4*log(3*x)**2)/(x**2 + 2*x*log(3*x) + log(3*x)**2))

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