Integrand size = 63, antiderivative size = 28 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=2+e^{x^2} \left (-4+e^x\right ) x^2-\frac {x^3}{e^8 \log ^2(x)} \] Output:
2+x^2*exp(x^2)*(exp(x)-4)-x^3/exp(4)^2/ln(x)^2
Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=\frac {x^2 \left (e^{8+x^2} \left (-4+e^x\right )-\frac {x}{\log ^2(x)}\right )}{e^8} \] Input:
Integrate[(2*x^2 - 3*x^2*Log[x] + E^x^2*(E^8*(-8*x - 8*x^3) + E^(8 + x)*(2 *x + x^2 + 2*x^3))*Log[x]^3)/(E^8*Log[x]^3),x]
Output:
(x^2*(E^(8 + x^2)*(-4 + E^x) - x/Log[x]^2))/E^8
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.69 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {27, 7293, 2009}
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.
\(\displaystyle \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x^3-8 x\right )+e^{x+8} \left (2 x^3+x^2+2 x\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {-e^{x^2} \left (8 e^8 \left (x^3+x\right )-e^{x+8} \left (2 x^3+x^2+2 x\right )\right ) \log ^3(x)-3 x^2 \log (x)+2 x^2}{\log ^3(x)}dx}{e^8}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {\int \left (e^{x^2+8} x \left (2 e^x x^2-8 x^2+e^x x+2 e^x-8\right )-\frac {x^2 (3 \log (x)-2)}{\log ^3(x)}\right )dx}{e^8}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {9}{2} \operatorname {ExpIntegralEi}(3 \log (x))+\frac {9}{2} (2-3 \log (x)) \operatorname {ExpIntegralEi}(3 \log (x))+27 \log (x) \operatorname {ExpIntegralEi}(3 \log (x))-\frac {9}{2} (3 \log (x)+1) \operatorname {ExpIntegralEi}(3 \log (x))-9 x^3-\frac {x^3 (2-3 \log (x))}{2 \log ^2(x)}+\frac {3 x^3 (3 \log (x)+1)}{2 \log (x)}-\frac {3 x^3 (2-3 \log (x))}{2 \log (x)}-e^{x^2+8} \left (4 x^2-e^x x^2\right )}{e^8}\) |
Input:
Int[(2*x^2 - 3*x^2*Log[x] + E^x^2*(E^8*(-8*x - 8*x^3) + E^(8 + x)*(2*x + x ^2 + 2*x^3))*Log[x]^3)/(E^8*Log[x]^3),x]
Output:
(-9*x^3 - E^(8 + x^2)*(4*x^2 - E^x*x^2) - (9*ExpIntegralEi[3*Log[x]])/2 + (9*ExpIntegralEi[3*Log[x]]*(2 - 3*Log[x]))/2 - (x^3*(2 - 3*Log[x]))/(2*Log [x]^2) - (3*x^3*(2 - 3*Log[x]))/(2*Log[x]) + 27*ExpIntegralEi[3*Log[x]]*Lo g[x] - (9*ExpIntegralEi[3*Log[x]]*(1 + 3*Log[x]))/2 + (3*x^3*(1 + 3*Log[x] ))/(2*Log[x]))/E^8
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 12.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
risch | \(x^{2} {\mathrm e}^{x^{2}} \left ({\mathrm e}^{x}-4\right )-\frac {x^{3} {\mathrm e}^{-8}}{\ln \left (x \right )^{2}}\) | \(25\) |
default | \({\mathrm e}^{-8} \left ({\mathrm e}^{8} x^{2} {\mathrm e}^{x^{2}+x}-\frac {x^{3}}{\ln \left (x \right )^{2}}-4 \,{\mathrm e}^{8} x^{2} {\mathrm e}^{x^{2}}\right )\) | \(43\) |
parallelrisch | \(\frac {{\mathrm e}^{-8} \left ({\mathrm e}^{8} x^{2} \ln \left (x \right )^{2} {\mathrm e}^{x} {\mathrm e}^{x^{2}}-4 \ln \left (x \right )^{2} {\mathrm e}^{x^{2}} {\mathrm e}^{8} x^{2}-x^{3}\right )}{\ln \left (x \right )^{2}}\) | \(51\) |
parts | \(-4 x^{2} {\mathrm e}^{x^{2}}+x^{2} {\mathrm e}^{x^{2}+x}+2 \,{\mathrm e}^{-8} \left (-\frac {x^{3}}{2 \ln \left (x \right )^{2}}-\frac {3 x^{3}}{2 \ln \left (x \right )}-\frac {9 \,\operatorname {expIntegral}_{1}\left (-3 \ln \left (x \right )\right )}{2}\right )-3 \,{\mathrm e}^{-8} \left (-\frac {x^{3}}{\ln \left (x \right )}-3 \,\operatorname {expIntegral}_{1}\left (-3 \ln \left (x \right )\right )\right )\) | \(78\) |
Input:
int((((2*x^3+x^2+2*x)*exp(4)^2*exp(x)+(-8*x^3-8*x)*exp(4)^2)*exp(x^2)*ln(x )^3-3*x^2*ln(x)+2*x^2)/exp(4)^2/ln(x)^3,x,method=_RETURNVERBOSE)
Output:
x^2*exp(x^2)*(exp(x)-4)-x^3*exp(-8)/ln(x)^2
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=-\frac {{\left ({\left (4 \, x^{2} e^{8} - x^{2} e^{\left (x + 8\right )}\right )} e^{\left (x^{2}\right )} \log \left (x\right )^{2} + x^{3}\right )} e^{\left (-8\right )}}{\log \left (x\right )^{2}} \] Input:
integrate((((2*x^3+x^2+2*x)*exp(4)^2*exp(x)+(-8*x^3-8*x)*exp(4)^2)*exp(x^2 )*log(x)^3-3*x^2*log(x)+2*x^2)/exp(4)^2/log(x)^3,x, algorithm="fricas")
Output:
-((4*x^2*e^8 - x^2*e^(x + 8))*e^(x^2)*log(x)^2 + x^3)*e^(-8)/log(x)^2
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=- \frac {x^{3}}{e^{8} \log {\left (x \right )}^{2}} + \left (x^{2} e^{x} - 4 x^{2}\right ) e^{x^{2}} \] Input:
integrate((((2*x**3+x**2+2*x)*exp(4)**2*exp(x)+(-8*x**3-8*x)*exp(4)**2)*ex p(x**2)*ln(x)**3-3*x**2*ln(x)+2*x**2)/exp(4)**2/ln(x)**3,x)
Output:
-x**3*exp(-8)/log(x)**2 + (x**2*exp(x) - 4*x**2)*exp(x**2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 281, normalized size of antiderivative = 10.04 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=\frac {1}{8} \, {\left ({\left (\frac {12 \, {\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{\left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {3}{2}}} - \frac {\sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 1\right )}^{2}}} + 6 \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} - 8 \, \Gamma \left (2, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )\right )} e^{\frac {31}{4}} - {\left (\frac {4 \, {\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{\left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {3}{2}}} - \frac {\sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 1\right )}^{2}}} + 4 \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\frac {31}{4}} - 4 \, {\left (\frac {\sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 1\right )}^{2}}} - 2 \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\frac {31}{4}} - 32 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} - 32 \, e^{\left (x^{2} + 8\right )} - 72 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) - 144 \, \Gamma \left (-2, -3 \, \log \left (x\right )\right )\right )} e^{\left (-8\right )} \] Input:
integrate((((2*x^3+x^2+2*x)*exp(4)^2*exp(x)+(-8*x^3-8*x)*exp(4)^2)*exp(x^2 )*log(x)^3-3*x^2*log(x)+2*x^2)/exp(4)^2/log(x)^3,x, algorithm="maxima")
Output:
1/8*((12*(2*x + 1)^3*gamma(3/2, -1/4*(2*x + 1)^2)/(-(2*x + 1)^2)^(3/2) - s qrt(pi)*(2*x + 1)*(erf(1/2*sqrt(-(2*x + 1)^2)) - 1)/sqrt(-(2*x + 1)^2) + 6 *e^(1/4*(2*x + 1)^2) - 8*gamma(2, -1/4*(2*x + 1)^2))*e^(31/4) - (4*(2*x + 1)^3*gamma(3/2, -1/4*(2*x + 1)^2)/(-(2*x + 1)^2)^(3/2) - sqrt(pi)*(2*x + 1 )*(erf(1/2*sqrt(-(2*x + 1)^2)) - 1)/sqrt(-(2*x + 1)^2) + 4*e^(1/4*(2*x + 1 )^2))*e^(31/4) - 4*(sqrt(pi)*(2*x + 1)*(erf(1/2*sqrt(-(2*x + 1)^2)) - 1)/s qrt(-(2*x + 1)^2) - 2*e^(1/4*(2*x + 1)^2))*e^(31/4) - 32*(x^2*e^8 - e^8)*e ^(x^2) - 32*e^(x^2 + 8) - 72*gamma(-1, -3*log(x)) - 144*gamma(-2, -3*log(x )))*e^(-8)
Time = 0.13 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=\frac {{\left (x^{2} e^{\left (x^{2} + x + 8\right )} \log \left (x\right )^{2} - 4 \, x^{2} e^{\left (x^{2} + 8\right )} \log \left (x\right )^{2} - x^{3}\right )} e^{\left (-8\right )}}{\log \left (x\right )^{2}} \] Input:
integrate((((2*x^3+x^2+2*x)*exp(4)^2*exp(x)+(-8*x^3-8*x)*exp(4)^2)*exp(x^2 )*log(x)^3-3*x^2*log(x)+2*x^2)/exp(4)^2/log(x)^3,x, algorithm="giac")
Output:
(x^2*e^(x^2 + x + 8)*log(x)^2 - 4*x^2*e^(x^2 + 8)*log(x)^2 - x^3)*e^(-8)/l og(x)^2
Time = 3.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=x^2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x-4\,x^2\,{\mathrm {e}}^{x^2}-\frac {x^3\,{\mathrm {e}}^{-8}}{{\ln \left (x\right )}^2} \] Input:
int(-(exp(-8)*(3*x^2*log(x) - 2*x^2 + exp(x^2)*log(x)^3*(exp(8)*(8*x + 8*x ^3) - exp(8)*exp(x)*(2*x + x^2 + 2*x^3))))/log(x)^3,x)
Output:
x^2*exp(x^2)*exp(x) - 4*x^2*exp(x^2) - (x^3*exp(-8))/log(x)^2
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=\frac {x^{2} \left (e^{x^{2}+x} \mathrm {log}\left (x \right )^{2} e^{8}-4 e^{x^{2}} \mathrm {log}\left (x \right )^{2} e^{8}-x \right )}{\mathrm {log}\left (x \right )^{2} e^{8}} \] Input:
int((((2*x^3+x^2+2*x)*exp(4)^2*exp(x)+(-8*x^3-8*x)*exp(4)^2)*exp(x^2)*log( x)^3-3*x^2*log(x)+2*x^2)/exp(4)^2/log(x)^3,x)
Output:
(x**2*(e**(x**2 + x)*log(x)**2*e**8 - 4*e**(x**2)*log(x)**2*e**8 - x))/(lo g(x)**2*e**8)