Integrand size = 249, antiderivative size = 33 \[ \int \frac {-12 x^4-24 x^5+12 x^2 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )+\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{-12 x^4+3 e^{\frac {1}{3} (-16+x)} x^4+\left (-24 x^2+6 e^{\frac {1}{3} (-16+x)} x^2\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-12+3 e^{\frac {1}{3} (-16+x)}\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx=x^2+\frac {(x-\log (3))^2}{x+\frac {\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )}{x}} \] Output:
x^2+(-ln(3)+x)^2/(ln(exp(1/3*x-16/3)-4)/x+x)
Leaf count is larger than twice the leaf count of optimal. \(150\) vs. \(2(33)=66\).
Time = 0.21 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.55 \[ \int \frac {-12 x^4-24 x^5+12 x^2 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )+\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{-12 x^4+3 e^{\frac {1}{3} (-16+x)} x^4+\left (-24 x^2+6 e^{\frac {1}{3} (-16+x)} x^2\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-12+3 e^{\frac {1}{3} (-16+x)}\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx=\frac {x \left (-24 e^{16/3} x \left (x^2+x^3+\log ^2(3)-x \log (9)\right )+e^{x/3} \left (x^2+7 x^3+6 x^4-12 x^2 \log (3)+\log ^2(3)+6 x \log ^2(3)-x \log (9)\right )+x \left (-24 e^{16/3} x+e^{x/3} (1+6 x)\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )}{\left (-24 e^{16/3} x+e^{x/3} (1+6 x)\right ) \left (x^2+\log \left (-4+e^{\frac {1}{3} (-16+x)}\right )\right )} \] Input:
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)*L og[-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]
Output:
(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)]))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-24 x^5-12 x^4+12 x^2 \log ^2(3)+\left (-48 x^3-36 x^2+e^{\frac {x-16}{3}} \left (12 x^3+9 x^2-12 x \log (3)+3 \log ^2(3)\right )+48 x \log (3)-12 \log ^2(3)\right ) \log \left (e^{\frac {x-16}{3}}-4\right )+e^{\frac {x-16}{3}} \left (6 x^5+3 x^4-x^3+\left (-3 x^2-x\right ) \log ^2(3)+2 x^2 \log (3)\right )+\left (6 e^{\frac {x-16}{3}} x-24 x\right ) \log ^2\left (e^{\frac {x-16}{3}}-4\right )}{3 e^{\frac {x-16}{3}} x^4-12 x^4+\left (6 e^{\frac {x-16}{3}} x^2-24 x^2\right ) \log \left (e^{\frac {x-16}{3}}-4\right )+\left (3 e^{\frac {x-16}{3}}-12\right ) \log ^2\left (e^{\frac {x-16}{3}}-4\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{16/3} \left (24 x^5+12 x^4-12 x^2 \log ^2(3)-\left (-48 x^3-36 x^2+e^{\frac {x-16}{3}} \left (12 x^3+9 x^2-12 x \log (3)+3 \log ^2(3)\right )+48 x \log (3)-12 \log ^2(3)\right ) \log \left (e^{\frac {x-16}{3}}-4\right )-e^{\frac {x-16}{3}} \left (6 x^5+3 x^4-x^3+\left (-3 x^2-x\right ) \log ^2(3)+2 x^2 \log (3)\right )-\left (6 e^{\frac {x-16}{3}} x-24 x\right ) \log ^2\left (e^{\frac {x-16}{3}}-4\right )\right )}{3 \left (4 e^{16/3}-e^{x/3}\right ) \left (x^2+\log \left (e^{\frac {x-16}{3}}-4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} e^{16/3} \int \frac {24 x^5+12 x^4-12 \log ^2(3) x^2+6 \left (4 x-e^{\frac {x-16}{3}} x\right ) \log ^2\left (-4+e^{\frac {x-16}{3}}\right )+e^{\frac {x-16}{3}} \left (-6 x^5-3 x^4+x^3-2 \log (3) x^2+\left (3 x^2+x\right ) \log ^2(3)\right )+3 \left (16 x^3+12 x^2-16 \log (3) x-e^{\frac {x-16}{3}} \left (4 x^3+3 x^2-4 \log (3) x+\log ^2(3)\right )+4 \log ^2(3)\right ) \log \left (-4+e^{\frac {x-16}{3}}\right )}{\left (4 e^{16/3}-e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} e^{16/3} \int \left (\frac {6 x^5+3 x^4+12 \log \left (-4+e^{\frac {x-16}{3}}\right ) x^3-x^3+9 \log \left (-4+e^{\frac {x-16}{3}}\right ) x^2-3 \log ^2(3) \left (1-\frac {\log (9)}{3 \log ^2(3)}\right ) x^2+6 \log ^2\left (-4+e^{\frac {x-16}{3}}\right ) x-12 \log (3) \log \left (-4+e^{\frac {x-16}{3}}\right ) x-\log ^2(3) x+3 \log ^2(3) \log \left (-4+e^{\frac {x-16}{3}}\right )}{e^{16/3} \left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}-\frac {4 x (x-\log (3))^2}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} e^{16/3} \left (-\frac {\log ^2(3) \int \frac {x}{\left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}dx}{e^{16/3}}-4 \log ^2(3) \int \frac {x}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}dx-\frac {\left (6 \log ^2(3)-\log (9)\right ) \int \frac {x^2}{\left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}dx}{e^{16/3}}+\frac {3 \log ^2(3) \int \frac {1}{x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )}dx}{e^{16/3}}+8 \log (3) \int \frac {x^2}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}dx-\frac {12 \log (3) \int \frac {x}{x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )}dx}{e^{16/3}}+\frac {9 \int \frac {x^2}{x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )}dx}{e^{16/3}}-\frac {6 \int \frac {x^4}{\left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}dx}{e^{16/3}}-\frac {(1-12 \log (3)) \int \frac {x^3}{\left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}dx}{e^{16/3}}-4 \int \frac {x^3}{\left (-4 e^{16/3}+e^{x/3}\right ) \left (x^2+\log \left (-4+e^{\frac {x-16}{3}}\right )\right )^2}dx+\frac {3 x^2}{e^{16/3}}\right )\) |
Input:
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*L og[3] - 12*Log[3]^2 + E^((-16 + x)/3)*(9*x^2 + 12*x^3 - 12*x*Log[3] + 3*Lo g[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]
Output:
$Aborted
Time = 0.78 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(x^{2}+\frac {\left (\ln \left (3\right )^{2}-2 x \ln \left (3\right )+x^{2}\right ) x}{x^{2}+\ln \left ({\mathrm e}^{\frac {x}{3}-\frac {16}{3}}-4\right )}\) | \(35\) |
| parallelrisch | \(\frac {3 x^{4}+3 x \ln \left (3\right )^{2}-6 x^{2} \ln \left (3\right )+3 x^{3}+3 \ln \left ({\mathrm e}^{\frac {x}{3}-\frac {16}{3}}-4\right ) x^{2}}{3 x^{2}+3 \ln \left ({\mathrm e}^{\frac {x}{3}-\frac {16}{3}}-4\right )}\) | \(57\) |
Input:
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*l n(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,method=_RETURNVERBOSE)
Output:
x^2+(ln(3)^2-2*x*ln(3)+x^2)*x/(x^2+ln(exp(1/3*x-16/3)-4))
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {-12 x^4-24 x^5+12 x^2 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )+\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{-12 x^4+3 e^{\frac {1}{3} (-16+x)} x^4+\left (-24 x^2+6 e^{\frac {1}{3} (-16+x)} x^2\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-12+3 e^{\frac {1}{3} (-16+x)}\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx=\frac {x^{4} + x^{3} - 2 \, x^{2} \log \left (3\right ) + x \log \left (3\right )^{2} + x^{2} \log \left (e^{\left (\frac {1}{3} \, x - \frac {16}{3}\right )} - 4\right )}{x^{2} + \log \left (e^{\left (\frac {1}{3} \, x - \frac {16}{3}\right )} - 4\right )} \] Input:
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")
Output:
(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))
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-12 x^4-24 x^5+12 x^2 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )+\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{-12 x^4+3 e^{\frac {1}{3} (-16+x)} x^4+\left (-24 x^2+6 e^{\frac {1}{3} (-16+x)} x^2\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-12+3 e^{\frac {1}{3} (-16+x)}\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx=x^{2} + \frac {x^{3} - 2 x^{2} \log {\left (3 \right )} + x \log {\left (3 \right )}^{2}}{x^{2} + \log {\left (e^{\frac {x}{3} - \frac {16}{3}} - 4 \right )}} \] Input:
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)
Output:
x**2 + (x**3 - 2*x**2*log(3) + x*log(3)**2)/(x**2 + log(exp(x/3 - 16/3) - 4))
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (30) = 60\).
Time = 0.58 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \frac {-12 x^4-24 x^5+12 x^2 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )+\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{-12 x^4+3 e^{\frac {1}{3} (-16+x)} x^4+\left (-24 x^2+6 e^{\frac {1}{3} (-16+x)} x^2\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-12+3 e^{\frac {1}{3} (-16+x)}\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx=\frac {3 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} {\left (3 \, \log \left (3\right ) + 8\right )} + 3 \, x \log \left (3\right )^{2} + 3 \, x^{2} \log \left (-4 \, e^{\frac {16}{3}} + e^{\left (\frac {1}{3} \, x\right )}\right )}{3 \, x^{2} + 3 \, \log \left (-4 \, e^{\frac {16}{3}} + e^{\left (\frac {1}{3} \, x\right )}\right ) - 16} \] Input:
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")
Output:
(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)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (30) = 60\).
Time = 0.39 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {-12 x^4-24 x^5+12 x^2 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )+\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{-12 x^4+3 e^{\frac {1}{3} (-16+x)} x^4+\left (-24 x^2+6 e^{\frac {1}{3} (-16+x)} x^2\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-12+3 e^{\frac {1}{3} (-16+x)}\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx=\frac {x^{4} + x^{3} - 2 \, x^{2} \log \left (3\right ) + x \log \left (3\right )^{2} + x^{2} \log \left (-{\left (4 \, e^{\frac {16}{3}} - e^{\left (\frac {1}{3} \, x\right )}\right )} e^{\left (-\frac {16}{3}\right )}\right )}{x^{2} + \log \left (-{\left (4 \, e^{\frac {16}{3}} - e^{\left (\frac {1}{3} \, x\right )}\right )} e^{\left (-\frac {16}{3}\right )}\right )} \] Input:
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")
Output:
(x^4 + x^3 - 2*x^2*log(3) + x*log(3)^2 + x^2*log(-(4*e^(16/3) - e^(1/3*x)) *e^(-16/3)))/(x^2 + log(-(4*e^(16/3) - e^(1/3*x))*e^(-16/3)))
Time = 0.60 (sec) , antiderivative size = 293, normalized size of antiderivative = 8.88 \[ \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=\frac {3\,x}{2}+\frac {4\,\ln \left (3\right )+6\,{\ln \left (3\right )}^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 \left (3\right )}^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 \left (3\right )}^2-2\,x^2\,{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}\,\ln \left (3\right )+3\,x^2\,{\mathrm {e}}^{\frac {x}{3}-\frac {16}{3}}\,{\ln \left (3\right )}^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 \left (3\right )\,x+{\ln \left (3\right )}^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 \left (3\right )}^2+12\,x\,\ln \left (3\right )+x\,{\ln \left (3\right )}^2-4\,x^2\,\ln \left (3\right )-24\,x^3\,\ln \left (3\right )-3\,{\ln \left (3\right )}^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 )} \] Input:
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 - 1 6/3) - 12*x^4),x)
Output:
(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))
\[ \int \frac {-12 x^4-24 x^5+12 x^2 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (-x^3+3 x^4+6 x^5+2 x^2 \log (3)+\left (-x-3 x^2\right ) \log ^2(3)\right )+\left (-36 x^2-48 x^3+48 x \log (3)-12 \log ^2(3)+e^{\frac {1}{3} (-16+x)} \left (9 x^2+12 x^3-12 x \log (3)+3 \log ^2(3)\right )\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-24 x+6 e^{\frac {1}{3} (-16+x)} x\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )}{-12 x^4+3 e^{\frac {1}{3} (-16+x)} x^4+\left (-24 x^2+6 e^{\frac {1}{3} (-16+x)} x^2\right ) \log \left (-4+e^{\frac {1}{3} (-16+x)}\right )+\left (-12+3 e^{\frac {1}{3} (-16+x)}\right ) \log ^2\left (-4+e^{\frac {1}{3} (-16+x)}\right )} \, dx=\text {too large to display} \] Input:
int(((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)
Output:
( - 24*e**(1/3)*int(x**5/(e**(x/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1 /3)*e**5))**2 + 2*e**(x/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5 ))*x**2 + e**(x/3)*x**4 - 4*e**(1/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e** (1/3)*e**5))**2*e**5 - 8*e**(1/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/ 3)*e**5))*e**5*x**2 - 4*e**(1/3)*e**5*x**4),x)*e**5 - 12*e**(1/3)*int(x**4 /(e**(x/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))**2 + 2*e**(x/ 3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))*x**2 + e**(x/3)*x**4 - 4*e**(1/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))**2*e**5 - 8 *e**(1/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))*e**5*x**2 - 4* e**(1/3)*e**5*x**4),x)*e**5 + 12*e**(1/3)*int(x**2/(e**(x/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))**2 + 2*e**(x/3)*log((e**(x/3) - 4*e** (1/3)*e**5)/(e**(1/3)*e**5))*x**2 + e**(x/3)*x**4 - 4*e**(1/3)*log((e**(x/ 3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))**2*e**5 - 8*e**(1/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))*e**5*x**2 - 4*e**(1/3)*e**5*x**4),x)*l og(3)**2*e**5 - 12*e**(1/3)*int(log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3) *e**5))/(e**(x/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))**2 + 2 *e**(x/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))*x**2 + e**(x/3 )*x**4 - 4*e**(1/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))**2*e **5 - 8*e**(1/3)*log((e**(x/3) - 4*e**(1/3)*e**5)/(e**(1/3)*e**5))*e**5*x* *2 - 4*e**(1/3)*e**5*x**4),x)*log(3)**2*e**5 - 24*e**(1/3)*int((log((e*...