3.15.28 \(\int \frac {-12 x^4-24 x^5+12 x^2 \log ^2(3)+e^{\frac {1}{3} (-16+x)} (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+(-x-3 x^2) \log ^2(3))+(-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3))) \log (-4+e^{\frac {1}{3} (-16+x)})+(-24 x+6 e^{\frac {1}{3} (-16+x)} x) \log ^2(-4+e^{\frac {1}{3} (-16+x)})}{-12 x^4+3 e^{\frac {1}{3} (-16+x)} x^4+(-24 x^2+6 e^{\frac {1}{3} (-16+x)} x^2) \log (-4+e^{\frac {1}{3} (-16+x)})+(-12+3 e^{\frac {1}{3} (-16+x)}) \log ^2(-4+e^{\frac {1}{3} (-16+x)})} \, dx\)

Optimal. Leaf size=33 \[ x^2+\frac {(x-\log (3))^2}{x+\frac {\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )}{x}} \]

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Rubi [F]  time = 7.25, 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 {-12 x^4-24 x^5+12 x^2 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )+\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{-12 x^4+3 e^{\frac {1}{3} (-16+x)} x^4+\left (-24 x^2+6 e^{\frac {1}{3} (-16+x)} x^2\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-12+3 e^{\frac {1}{3} (-16+x)}\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12*x^4 - 24*x^5 + 12*x^2*Log[3]^2 + E^((-16 + x)/3)*(-x^3 + 3*x^4 + 6*x^5 + 2*x^2*Log[3] + (-x - 3*x^2)*
Log[3]^2) + (-36*x^2 - 48*x^3 + 48*x*Log[3] - 12*Log[3]^2 + E^((-16 + x)/3)*(9*x^2 + 12*x^3 - 12*x*Log[3] + 3*
Log[3]^2))*Log[-4 + E^((-16 + x)/3)] + (-24*x + 6*E^((-16 + x)/3)*x)*Log[-4 + E^((-16 + x)/3)]^2)/(-12*x^4 + 3
*E^((-16 + x)/3)*x^4 + (-24*x^2 + 6*E^((-16 + x)/3)*x^2)*Log[-4 + E^((-16 + x)/3)] + (-12 + 3*E^((-16 + x)/3))
*Log[-4 + E^((-16 + x)/3)]^2),x]

[Out]

x^2 - (Log[3]^2*Defer[Int][x/(x^2 + Log[-4 + E^((-16 + x)/3)])^2, x])/3 - (4*E^(16/3)*Log[3]^2*Defer[Int][x/((
-4*E^(16/3) + E^(x/3))*(x^2 + Log[-4 + E^((-16 + x)/3)])^2), x])/3 - ((6*Log[3]^2 - Log[9])*Defer[Int][x^2/(x^
2 + Log[-4 + E^((-16 + x)/3)])^2, x])/3 + (8*E^(16/3)*Log[3]*Defer[Int][x^2/((-4*E^(16/3) + E^(x/3))*(x^2 + Lo
g[-4 + E^((-16 + x)/3)])^2), x])/3 - ((1 - 12*Log[3])*Defer[Int][x^3/(x^2 + Log[-4 + E^((-16 + x)/3)])^2, x])/
3 - (4*E^(16/3)*Defer[Int][x^3/((-4*E^(16/3) + E^(x/3))*(x^2 + Log[-4 + E^((-16 + x)/3)])^2), x])/3 - 2*Defer[
Int][x^4/(x^2 + Log[-4 + E^((-16 + x)/3)])^2, x] + Log[3]^2*Defer[Int][(x^2 + Log[-4 + E^((-16 + x)/3)])^(-1),
x] - 4*Log[3]*Defer[Int][x/(x^2 + Log[-4 + E^((-16 + x)/3)]), x] + 3*Defer[Int][x^2/(x^2 + Log[-4 + E^((-16 +
x)/3)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{16/3} \left (12 x^4+24 x^5-12 x^2 \log ^2(3)-e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )-\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )-\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )}{3 \left (4 e^{16/3}-e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} e^{16/3} \int \frac {12 x^4+24 x^5-12 x^2 \log ^2(3)-e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )-\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )-\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{\left (4 e^{16/3}-e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} e^{16/3} \int \left (-\frac {4 x (x-\log (3))^2}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}+\frac {-x^3+3 x^4+6 x^5-x \log ^2(3)-3 x^2 \log ^2(3) \left (1-\frac {\log (9)}{3 \log ^2(3)}\right )+9 x^2 \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+12 x^3 \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )-12 x \log (3) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+3 \log ^2(3) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+6 x \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{e^{16/3} \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {-x^3+3 x^4+6 x^5-x \log ^2(3)-3 x^2 \log ^2(3) \left (1-\frac {\log (9)}{3 \log ^2(3)}\right )+9 x^2 \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+12 x^3 \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )-12 x \log (3) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+3 \log ^2(3) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+6 x \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx-\frac {1}{3} \left (4 e^{16/3}\right ) \int \frac {x (x-\log (3))^2}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {x \left (-x^2+3 x^3+6 x^4-\log ^2(3)+x \left (-3 \log ^2(3)+\log (9)\right )\right )+3 \left (3 x^2+4 x^3-4 x \log (3)+\log ^2(3)\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+6 x \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx-\frac {1}{3} \left (4 e^{16/3}\right ) \int \left (\frac {x^3}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}-\frac {2 x^2 \log (3)}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}+\frac {x \log ^2(3)}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}\right ) \, dx\\ &=\frac {1}{3} \int \left (6 x+\frac {x \left (-6 x^3-x^2 (1-12 \log (3))-\log ^2(3)-x \left (6 \log ^2(3)-\log (9)\right )\right )}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}+\frac {3 \left (3 x^2-4 x \log (3)+\log ^2(3)\right )}{x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )}\right ) \, dx-\frac {1}{3} \left (4 e^{16/3}\right ) \int \frac {x^3}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx+\frac {1}{3} \left (8 e^{16/3} \log (3)\right ) \int \frac {x^2}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx-\frac {1}{3} \left (4 e^{16/3} \log ^2(3)\right ) \int \frac {x}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx\\ &=x^2+\frac {1}{3} \int \frac {x \left (-6 x^3-x^2 (1-12 \log (3))-\log ^2(3)-x \left (6 \log ^2(3)-\log (9)\right )\right )}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx-\frac {1}{3} \left (4 e^{16/3}\right ) \int \frac {x^3}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx+\frac {1}{3} \left (8 e^{16/3} \log (3)\right ) \int \frac {x^2}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx-\frac {1}{3} \left (4 e^{16/3} \log ^2(3)\right ) \int \frac {x}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx+\int \frac {3 x^2-4 x \log (3)+\log ^2(3)}{x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx\\ &=x^2+\frac {1}{3} \int \left (-\frac {6 x^4}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}-\frac {x \log ^2(3)}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}+\frac {x^3 (-1+12 \log (3))}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}-\frac {x^2 \left (6 \log ^2(3)-\log (9)\right )}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2}\right ) \, dx-\frac {1}{3} \left (4 e^{16/3}\right ) \int \frac {x^3}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx+\frac {1}{3} \left (8 e^{16/3} \log (3)\right ) \int \frac {x^2}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx-\frac {1}{3} \left (4 e^{16/3} \log ^2(3)\right ) \int \frac {x}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx+\int \left (\frac {3 x^2}{x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )}-\frac {4 x \log (3)}{x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )}+\frac {\log ^2(3)}{x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )}\right ) \, dx\\ &=x^2-2 \int \frac {x^4}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx+3 \int \frac {x^2}{x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx-\frac {1}{3} \left (4 e^{16/3}\right ) \int \frac {x^3}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx-(4 \log (3)) \int \frac {x}{x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx+\frac {1}{3} \left (8 e^{16/3} \log (3)\right ) \int \frac {x^2}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx-\frac {1}{3} \log ^2(3) \int \frac {x}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx+\log ^2(3) \int \frac {1}{x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx-\frac {1}{3} \left (4 e^{16/3} \log ^2(3)\right ) \int \frac {x}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx+\frac {1}{3} (-1+12 \log (3)) \int \frac {x^3}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx+\frac {1}{3} \left (-6 \log ^2(3)+\log (9)\right ) \int \frac {x^2}{\left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.42, size = 150, normalized size = 4.55 \begin {gather*} \frac {x \left (-24 e^{16/3} x \left (x^2+x^3+\log ^2(3)-x \log (9)\right )+e^{x/3} \left (x^2+7 x^3+6 x^4-12 x^2 \log (3)+\log ^2(3)+6 x \log ^2(3)-x \log (9)\right )+x \left (-24 e^{16/3} x+e^{x/3} (1+6 x)\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )}{\left (-24 e^{16/3} x+e^{x/3} (1+6 x)\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12*x^4 - 24*x^5 + 12*x^2*Log[3]^2 + E^((-16 + x)/3)*(-x^3 + 3*x^4 + 6*x^5 + 2*x^2*Log[3] + (-x - 3
*x^2)*Log[3]^2) + (-36*x^2 - 48*x^3 + 48*x*Log[3] - 12*Log[3]^2 + E^((-16 + x)/3)*(9*x^2 + 12*x^3 - 12*x*Log[3
] + 3*Log[3]^2))*Log[-4 + E^((-16 + x)/3)] + (-24*x + 6*E^((-16 + x)/3)*x)*Log[-4 + E^((-16 + x)/3)]^2)/(-12*x
^4 + 3*E^((-16 + x)/3)*x^4 + (-24*x^2 + 6*E^((-16 + x)/3)*x^2)*Log[-4 + E^((-16 + x)/3)] + (-12 + 3*E^((-16 +
x)/3))*Log[-4 + E^((-16 + x)/3)]^2),x]

[Out]

(x*(-24*E^(16/3)*x*(x^2 + x^3 + Log[3]^2 - x*Log[9]) + E^(x/3)*(x^2 + 7*x^3 + 6*x^4 - 12*x^2*Log[3] + Log[3]^2
+ 6*x*Log[3]^2 - x*Log[9]) + x*(-24*E^(16/3)*x + E^(x/3)*(1 + 6*x))*Log[-4 + E^((-16 + x)/3)]))/((-24*E^(16/3
)*x + E^(x/3)*(1 + 6*x))*(x^2 + Log[-4 + E^((-16 + x)/3)]))

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fricas [A]  time = 1.46, size = 49, normalized size = 1.48 \begin {gather*} \frac {x^{4} + x^{3} - 2 \, x^{2} \log \relax (3) + x \log \relax (3)^{2} + x^{2} \log \left (e^{\left (\frac {1}{3} \, x - \frac {16}{3}\right )} - 4\right )}{x^{2} + \log \left (e^{\left (\frac {1}{3} \, x - \frac {16}{3}\right )} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*exp(1/3*x-16/3)-24*x)*log(exp(1/3*x-16/3)-4)^2+((3*log(3)^2-12*x*log(3)+12*x^3+9*x^2)*exp(1/3*
x-16/3)-12*log(3)^2+48*x*log(3)-48*x^3-36*x^2)*log(exp(1/3*x-16/3)-4)+((-3*x^2-x)*log(3)^2+2*x^2*log(3)+6*x^5+
3*x^4-x^3)*exp(1/3*x-16/3)+12*x^2*log(3)^2-24*x^5-12*x^4)/((3*exp(1/3*x-16/3)-12)*log(exp(1/3*x-16/3)-4)^2+(6*
x^2*exp(1/3*x-16/3)-24*x^2)*log(exp(1/3*x-16/3)-4)+3*x^4*exp(1/3*x-16/3)-12*x^4),x, algorithm="fricas")

[Out]

(x^4 + x^3 - 2*x^2*log(3) + x*log(3)^2 + x^2*log(e^(1/3*x - 16/3) - 4))/(x^2 + log(e^(1/3*x - 16/3) - 4))

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giac [A]  time = 1.49, size = 49, normalized size = 1.48 \begin {gather*} \frac {x^{4} + x^{3} - 2 \, x^{2} \log \relax (3) + x \log \relax (3)^{2} + x^{2} \log \left (e^{\left (\frac {1}{3} \, x - \frac {16}{3}\right )} - 4\right )}{x^{2} + \log \left (e^{\left (\frac {1}{3} \, x - \frac {16}{3}\right )} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*exp(1/3*x-16/3)-24*x)*log(exp(1/3*x-16/3)-4)^2+((3*log(3)^2-12*x*log(3)+12*x^3+9*x^2)*exp(1/3*
x-16/3)-12*log(3)^2+48*x*log(3)-48*x^3-36*x^2)*log(exp(1/3*x-16/3)-4)+((-3*x^2-x)*log(3)^2+2*x^2*log(3)+6*x^5+
3*x^4-x^3)*exp(1/3*x-16/3)+12*x^2*log(3)^2-24*x^5-12*x^4)/((3*exp(1/3*x-16/3)-12)*log(exp(1/3*x-16/3)-4)^2+(6*
x^2*exp(1/3*x-16/3)-24*x^2)*log(exp(1/3*x-16/3)-4)+3*x^4*exp(1/3*x-16/3)-12*x^4),x, algorithm="giac")

[Out]

(x^4 + x^3 - 2*x^2*log(3) + x*log(3)^2 + x^2*log(e^(1/3*x - 16/3) - 4))/(x^2 + log(e^(1/3*x - 16/3) - 4))

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maple [A]  time = 0.04, size = 35, normalized size = 1.06

method	result	size
risch	\(x^{2}+\frac {\left (\ln \relax (3)^{2}-2 x \ln \relax (3)+x^{2}\right ) x}{x^{2}+\ln \left ({\mathrm e}^{\frac {x}{3}-\frac {16}{3}}-4\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x*exp(1/3*x-16/3)-24*x)*ln(exp(1/3*x-16/3)-4)^2+((3*ln(3)^2-12*x*ln(3)+12*x^3+9*x^2)*exp(1/3*x-16/3)-1
2*ln(3)^2+48*x*ln(3)-48*x^3-36*x^2)*ln(exp(1/3*x-16/3)-4)+((-3*x^2-x)*ln(3)^2+2*x^2*ln(3)+6*x^5+3*x^4-x^3)*exp
(1/3*x-16/3)+12*x^2*ln(3)^2-24*x^5-12*x^4)/((3*exp(1/3*x-16/3)-12)*ln(exp(1/3*x-16/3)-4)^2+(6*x^2*exp(1/3*x-16
/3)-24*x^2)*ln(exp(1/3*x-16/3)-4)+3*x^4*exp(1/3*x-16/3)-12*x^4),x,method=_RETURNVERBOSE)

[Out]

x^2+(ln(3)^2-2*x*ln(3)+x^2)*x/(x^2+ln(exp(1/3*x-16/3)-4))

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maxima [B]  time = 1.07, size = 66, normalized size = 2.00 \begin {gather*} \frac {3 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} {\left (3 \, \log \relax (3) + 8\right )} + 3 \, x \log \relax (3)^{2} + 3 \, x^{2} \log \left (-4 \, e^{\frac {16}{3}} + e^{\left (\frac {1}{3} \, x\right )}\right )}{3 \, x^{2} + 3 \, \log \left (-4 \, e^{\frac {16}{3}} + e^{\left (\frac {1}{3} \, x\right )}\right ) - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*exp(1/3*x-16/3)-24*x)*log(exp(1/3*x-16/3)-4)^2+((3*log(3)^2-12*x*log(3)+12*x^3+9*x^2)*exp(1/3*
x-16/3)-12*log(3)^2+48*x*log(3)-48*x^3-36*x^2)*log(exp(1/3*x-16/3)-4)+((-3*x^2-x)*log(3)^2+2*x^2*log(3)+6*x^5+
3*x^4-x^3)*exp(1/3*x-16/3)+12*x^2*log(3)^2-24*x^5-12*x^4)/((3*exp(1/3*x-16/3)-12)*log(exp(1/3*x-16/3)-4)^2+(6*
x^2*exp(1/3*x-16/3)-24*x^2)*log(exp(1/3*x-16/3)-4)+3*x^4*exp(1/3*x-16/3)-12*x^4),x, algorithm="maxima")

[Out]

(3*x^4 + 3*x^3 - 2*x^2*(3*log(3) + 8) + 3*x*log(3)^2 + 3*x^2*log(-4*e^(16/3) + e^(1/3*x)))/(3*x^2 + 3*log(-4*e
^(16/3) + e^(1/3*x)) - 16)

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mupad [B]  time = 0.55, size = 293, normalized size = 8.88 \begin {gather*} \frac {3\,x}{2}+\frac {4\,\ln \relax (3)+6\,{\ln \relax (3)}^2+\frac {1}{2}}{12\,x+2}+x^2+\frac {\frac {x^3\,{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}-12\,x^2\,{\ln \relax (3)}^2-3\,x^4\,{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}+12\,x^4+x\,{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}\,{\ln \relax (3)}^2-2\,x^2\,{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}\,\ln \relax (3)+3\,x^2\,{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}\,{\ln \relax (3)}^2}{{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}-24\,x+6\,x\,{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}}-\frac {3\,\ln \left ({\mathrm {e}}^{-\frac {16}{3}}\,{\left ({\mathrm {e}}^x\right )}^{1/3}-4\right )\,\left ({\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}-4\right )\,\left (3\,x^2-4\,\ln \relax (3)\,x+{\ln \relax (3)}^2\right )}{{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}-24\,x+6\,x\,{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}}}{\ln \left ({\mathrm {e}}^{-\frac {16}{3}}\,{\left ({\mathrm {e}}^x\right )}^{1/3}-4\right )+x^2}+\frac {12\,\left (6\,x^2\,{\ln \relax (3)}^2+12\,x\,\ln \relax (3)+x\,{\ln \relax (3)}^2-4\,x^2\,\ln \relax (3)-24\,x^3\,\ln \relax (3)-3\,{\ln \relax (3)}^2-9\,x^2+3\,x^3+18\,x^4\right )}{\left (6\,x+1\right )\,\left (24\,x-{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}\,\left (6\,x+1\right )\right )\,\left (6\,x^2+x-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(exp(x/3 - 16/3) - 4)*(12*log(3)^2 - 48*x*log(3) - exp(x/3 - 16/3)*(3*log(3)^2 - 12*x*log(3) + 9*x^2
+ 12*x^3) + 36*x^2 + 48*x^3) - 12*x^2*log(3)^2 + log(exp(x/3 - 16/3) - 4)^2*(24*x - 6*x*exp(x/3 - 16/3)) + 12*
x^4 + 24*x^5 - exp(x/3 - 16/3)*(2*x^2*log(3) - log(3)^2*(x + 3*x^2) - x^3 + 3*x^4 + 6*x^5))/(log(exp(x/3 - 16/
3) - 4)*(6*x^2*exp(x/3 - 16/3) - 24*x^2) + log(exp(x/3 - 16/3) - 4)^2*(3*exp(x/3 - 16/3) - 12) + 3*x^4*exp(x/3
- 16/3) - 12*x^4),x)

[Out]

(3*x)/2 + (4*log(3) + 6*log(3)^2 + 1/2)/(12*x + 2) + x^2 + ((x^3*exp(x/3 - 16/3) - 12*x^2*log(3)^2 - 3*x^4*exp
(x/3 - 16/3) + 12*x^4 + x*exp(x/3 - 16/3)*log(3)^2 - 2*x^2*exp(x/3 - 16/3)*log(3) + 3*x^2*exp(x/3 - 16/3)*log(
3)^2)/(exp(x/3 - 16/3) - 24*x + 6*x*exp(x/3 - 16/3)) - (3*log(exp(-16/3)*exp(x)^(1/3) - 4)*(exp(x/3 - 16/3) -
4)*(log(3)^2 - 4*x*log(3) + 3*x^2))/(exp(x/3 - 16/3) - 24*x + 6*x*exp(x/3 - 16/3)))/(log(exp(-16/3)*exp(x)^(1/
3) - 4) + x^2) + (12*(6*x^2*log(3)^2 + 12*x*log(3) + x*log(3)^2 - 4*x^2*log(3) - 24*x^3*log(3) - 3*log(3)^2 -
9*x^2 + 3*x^3 + 18*x^4))/((6*x + 1)*(24*x - exp(x/3 - 16/3)*(6*x + 1))*(x + 6*x^2 - 3))

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sympy [A]  time = 0.32, size = 36, normalized size = 1.09 \begin {gather*} x^{2} + \frac {x^{3} - 2 x^{2} \log {\relax (3 )} + x \log {\relax (3 )}^{2}}{x^{2} + \log {\left (e^{\frac {x}{3} - \frac {16}{3}} - 4 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*exp(1/3*x-16/3)-24*x)*ln(exp(1/3*x-16/3)-4)**2+((3*ln(3)**2-12*x*ln(3)+12*x**3+9*x**2)*exp(1/3
*x-16/3)-12*ln(3)**2+48*x*ln(3)-48*x**3-36*x**2)*ln(exp(1/3*x-16/3)-4)+((-3*x**2-x)*ln(3)**2+2*x**2*ln(3)+6*x*
*5+3*x**4-x**3)*exp(1/3*x-16/3)+12*x**2*ln(3)**2-24*x**5-12*x**4)/((3*exp(1/3*x-16/3)-12)*ln(exp(1/3*x-16/3)-4
)**2+(6*x**2*exp(1/3*x-16/3)-24*x**2)*ln(exp(1/3*x-16/3)-4)+3*x**4*exp(1/3*x-16/3)-12*x**4),x)

[Out]

x**2 + (x**3 - 2*x**2*log(3) + x*log(3)**2)/(x**2 + log(exp(x/3 - 16/3) - 4))

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