Integrand size = 74, antiderivative size = 31 \[ \int \frac {1}{3} \left (3+2 x+e^{-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4} \left (48 x-144 x^2+96 x^3\right )\right ) \, dx=e^{2 e^{4 \left (-2+e^3+\left (-x+x^2\right )^2\right )}}+x+\frac {x^2}{3} \] Output:
x+1/3*x^2+exp(exp(4*(x^2-x)^2+4*exp(3)-8))^2
Time = 1.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {1}{3} \left (3+2 x+e^{-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4} \left (48 x-144 x^2+96 x^3\right )\right ) \, dx=e^{2 e^{4 \left (-2+e^3\right )+4 x^2-8 x^3+4 x^4}}+x+\frac {x^2}{3} \] Input:
Integrate[(3 + 2*x + E^(-8 + 4*E^3 + 2*E^(-8 + 4*E^3 + 4*x^2 - 8*x^3 + 4*x ^4) + 4*x^2 - 8*x^3 + 4*x^4)*(48*x - 144*x^2 + 96*x^3))/3,x]
Output:
E^(2*E^(4*(-2 + E^3) + 4*x^2 - 8*x^3 + 4*x^4)) + x + x^2/3
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.
\(\displaystyle \int \frac {1}{3} \left (\left (96 x^3-144 x^2+48 x\right ) \exp \left (4 x^4-8 x^3+4 x^2+2 e^{4 x^4-8 x^3+4 x^2+4 e^3-8}+4 e^3-8\right )+2 x+3\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \left (2 x+48 \exp \left (4 x^4-8 x^3+4 x^2+2 e^{4 x^4-8 x^3+4 x^2-4 \left (2-e^3\right )}-4 \left (2-e^3\right )\right ) \left (2 x^3-3 x^2+x\right )+3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (48 \int \exp \left (4 x^4-8 x^3+4 x^2+2 e^{4 x^4-8 x^3+4 x^2-4 \left (2-e^3\right )}-4 \left (2-e^3\right )\right ) xdx-144 \int \exp \left (4 x^4-8 x^3+4 x^2+2 e^{4 x^4-8 x^3+4 x^2-4 \left (2-e^3\right )}-4 \left (2-e^3\right )\right ) x^2dx+96 \int \exp \left (4 x^4-8 x^3+4 x^2+2 e^{4 x^4-8 x^3+4 x^2-4 \left (2-e^3\right )}-4 \left (2-e^3\right )\right ) x^3dx+x^2+3 x\right )\) |
Input:
Int[(3 + 2*x + E^(-8 + 4*E^3 + 2*E^(-8 + 4*E^3 + 4*x^2 - 8*x^3 + 4*x^4) + 4*x^2 - 8*x^3 + 4*x^4)*(48*x - 144*x^2 + 96*x^3))/3,x]
Output:
$Aborted
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06
method | result | size |
default | \(x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{3}+4 x^{4}-8 x^{3}+4 x^{2}-8}}+\frac {x^{2}}{3}\) | \(33\) |
norman | \(x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{3}+4 x^{4}-8 x^{3}+4 x^{2}-8}}+\frac {x^{2}}{3}\) | \(33\) |
risch | \(x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{3}+4 x^{4}-8 x^{3}+4 x^{2}-8}}+\frac {x^{2}}{3}\) | \(33\) |
parallelrisch | \(x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{3}+4 x^{4}-8 x^{3}+4 x^{2}-8}}+\frac {x^{2}}{3}\) | \(33\) |
parts | \(x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{3}+4 x^{4}-8 x^{3}+4 x^{2}-8}}+\frac {x^{2}}{3}\) | \(33\) |
Input:
int(1/3*(96*x^3-144*x^2+48*x)*exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8)*exp(exp(4* exp(3)+4*x^4-8*x^3+4*x^2-8))^2+2/3*x+1,x,method=_RETURNVERBOSE)
Output:
x+exp(exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8))^2+1/3*x^2
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (28) = 56\).
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.32 \[ \int \frac {1}{3} \left (3+2 x+e^{-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4} \left (48 x-144 x^2+96 x^3\right )\right ) \, dx=\frac {1}{3} \, {\left ({\left (x^{2} + 3 \, x\right )} e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} - 8\right )} + 3 \, e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} + 2 \, e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} - 8\right )} - 8\right )}\right )} e^{\left (-4 \, x^{4} + 8 \, x^{3} - 4 \, x^{2} - 4 \, e^{3} + 8\right )} \] Input:
integrate(1/3*(96*x^3-144*x^2+48*x)*exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8)*exp( exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8))^2+2/3*x+1,x, algorithm="fricas")
Output:
1/3*((x^2 + 3*x)*e^(4*x^4 - 8*x^3 + 4*x^2 + 4*e^3 - 8) + 3*e^(4*x^4 - 8*x^ 3 + 4*x^2 + 4*e^3 + 2*e^(4*x^4 - 8*x^3 + 4*x^2 + 4*e^3 - 8) - 8))*e^(-4*x^ 4 + 8*x^3 - 4*x^2 - 4*e^3 + 8)
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {1}{3} \left (3+2 x+e^{-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4} \left (48 x-144 x^2+96 x^3\right )\right ) \, dx=\frac {x^{2}}{3} + x + e^{2 e^{4 x^{4} - 8 x^{3} + 4 x^{2} - 8 + 4 e^{3}}} \] Input:
integrate(1/3*(96*x**3-144*x**2+48*x)*exp(4*exp(3)+4*x**4-8*x**3+4*x**2-8) *exp(exp(4*exp(3)+4*x**4-8*x**3+4*x**2-8))**2+2/3*x+1,x)
Output:
x**2/3 + x + exp(2*exp(4*x**4 - 8*x**3 + 4*x**2 - 8 + 4*exp(3)))
Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {1}{3} \left (3+2 x+e^{-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4} \left (48 x-144 x^2+96 x^3\right )\right ) \, dx=\frac {1}{3} \, x^{2} + x + e^{\left (2 \, e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} - 8\right )}\right )} \] Input:
integrate(1/3*(96*x^3-144*x^2+48*x)*exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8)*exp( exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8))^2+2/3*x+1,x, algorithm="maxima")
Output:
1/3*x^2 + x + e^(2*e^(4*x^4 - 8*x^3 + 4*x^2 + 4*e^3 - 8))
\[ \int \frac {1}{3} \left (3+2 x+e^{-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4} \left (48 x-144 x^2+96 x^3\right )\right ) \, dx=\int { 16 \, {\left (2 \, x^{3} - 3 \, x^{2} + x\right )} e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} + 2 \, e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} - 8\right )} - 8\right )} + \frac {2}{3} \, x + 1 \,d x } \] Input:
integrate(1/3*(96*x^3-144*x^2+48*x)*exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8)*exp( exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8))^2+2/3*x+1,x, algorithm="giac")
Output:
integrate(16*(2*x^3 - 3*x^2 + x)*e^(4*x^4 - 8*x^3 + 4*x^2 + 4*e^3 + 2*e^(4 *x^4 - 8*x^3 + 4*x^2 + 4*e^3 - 8) - 8) + 2/3*x + 1, x)
Time = 0.59 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {1}{3} \left (3+2 x+e^{-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4} \left (48 x-144 x^2+96 x^3\right )\right ) \, dx=x+{\mathrm {e}}^{2\,{\mathrm {e}}^{4\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-8}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{4\,x^4}\,{\mathrm {e}}^{-8\,x^3}}+\frac {x^2}{3} \] Input:
int((2*x)/3 + (exp(4*exp(3) + 4*x^2 - 8*x^3 + 4*x^4 - 8)*exp(2*exp(4*exp(3 ) + 4*x^2 - 8*x^3 + 4*x^4 - 8))*(48*x - 144*x^2 + 96*x^3))/3 + 1,x)
Output:
x + exp(2*exp(4*exp(3))*exp(-8)*exp(4*x^2)*exp(4*x^4)*exp(-8*x^3)) + x^2/3
\[ \int \frac {1}{3} \left (3+2 x+e^{-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4} \left (48 x-144 x^2+96 x^3\right )\right ) \, dx=\frac {96 e^{4 e^{3}} \left (\int \frac {e^{\frac {2 e^{4 x^{4}+4 e^{3}+4 x^{2}}+4 e^{8 x^{3}} e^{8} x^{4}+4 e^{8 x^{3}} e^{8} x^{2}}{e^{8 x^{3}} e^{8}}} x^{3}}{e^{8 x^{3}}}d x \right )-144 e^{4 e^{3}} \left (\int \frac {e^{\frac {2 e^{4 x^{4}+4 e^{3}+4 x^{2}}+4 e^{8 x^{3}} e^{8} x^{4}+4 e^{8 x^{3}} e^{8} x^{2}}{e^{8 x^{3}} e^{8}}} x^{2}}{e^{8 x^{3}}}d x \right )+48 e^{4 e^{3}} \left (\int \frac {e^{\frac {2 e^{4 x^{4}+4 e^{3}+4 x^{2}}+4 e^{8 x^{3}} e^{8} x^{4}+4 e^{8 x^{3}} e^{8} x^{2}}{e^{8 x^{3}} e^{8}}} x}{e^{8 x^{3}}}d x \right )+e^{8} x^{2}+3 e^{8} x}{3 e^{8}} \] Input:
int(1/3*(96*x^3-144*x^2+48*x)*exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8)*exp(exp(4* exp(3)+4*x^4-8*x^3+4*x^2-8))^2+2/3*x+1,x)
Output:
(96*e**(4*e**3)*int((e**((2*e**(4*e**3 + 4*x**4 + 4*x**2) + 4*e**(8*x**3)* e**8*x**4 + 4*e**(8*x**3)*e**8*x**2)/(e**(8*x**3)*e**8))*x**3)/e**(8*x**3) ,x) - 144*e**(4*e**3)*int((e**((2*e**(4*e**3 + 4*x**4 + 4*x**2) + 4*e**(8* x**3)*e**8*x**4 + 4*e**(8*x**3)*e**8*x**2)/(e**(8*x**3)*e**8))*x**2)/e**(8 *x**3),x) + 48*e**(4*e**3)*int((e**((2*e**(4*e**3 + 4*x**4 + 4*x**2) + 4*e **(8*x**3)*e**8*x**4 + 4*e**(8*x**3)*e**8*x**2)/(e**(8*x**3)*e**8))*x)/e** (8*x**3),x) + e**8*x**2 + 3*e**8*x)/(3*e**8)