Integrand size = 33, antiderivative size = 20 \[ \int e^{4 e^{1+\log ^2(5)} x} \left (-4 x^3-4 e^{1+\log ^2(5)} x^4\right ) \, dx=2-e^{4 e^{1+\log ^2(5)} x} x^4 \] Output:
2-exp(x*exp(1)*exp(ln(5)^2))^4*x^4
Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int e^{4 e^{1+\log ^2(5)} x} \left (-4 x^3-4 e^{1+\log ^2(5)} x^4\right ) \, dx=-e^{4 e^{1+\log ^2(5)} x} x^4 \] Input:
Integrate[E^(4*E^(1 + Log[5]^2)*x)*(-4*x^3 - 4*E^(1 + Log[5]^2)*x^4),x]
Output:
-(E^(4*E^(1 + Log[5]^2)*x)*x^4)
Time = 0.55 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2027, 2626, 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.
\(\displaystyle \int e^{4 x e^{1+\log ^2(5)}} \left (-4 x^4 e^{1+\log ^2(5)}-4 x^3\right ) \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int x^3 e^{4 x e^{1+\log ^2(5)}} \left (-4 x e^{1+\log ^2(5)}-4\right )dx\) |
\(\Big \downarrow \) 2626 |
\(\displaystyle \int \left (-4 x^4 e^{4 x e^{1+\log ^2(5)}+1+\log ^2(5)}-4 x^3 e^{4 x e^{1+\log ^2(5)}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^4 \left (-e^{4 x e^{1+\log ^2(5)}}\right )\) |
Input:
Int[E^(4*E^(1 + Log[5]^2)*x)*(-4*x^3 - 4*E^(1 + Log[5]^2)*x^4),x]
Output:
-(E^(4*E^(1 + Log[5]^2)*x)*x^4)
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
risch | \(-{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{4}\) | \(17\) |
gosper | \(-{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{4}\) | \(18\) |
norman | \(-{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{4}\) | \(18\) |
parallelrisch | \(-{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{4}\) | \(18\) |
meijerg | \(\frac {{\mathrm e}^{-4-4 \ln \left (5\right )^{2}} \left (24-\frac {{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} \left (1280 x^{4} {\mathrm e}^{4+4 \ln \left (5\right )^{2}}-1280 x^{3} {\mathrm e}^{3+3 \ln \left (5\right )^{2}}+960 x^{2} {\mathrm e}^{2+2 \ln \left (5\right )^{2}}-480 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}+120\right )}{5}\right )}{256}-\frac {{\mathrm e}^{-4-4 \ln \left (5\right )^{2}} \left (6-\frac {{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} \left (-256 x^{3} {\mathrm e}^{3+3 \ln \left (5\right )^{2}}+192 x^{2} {\mathrm e}^{2+2 \ln \left (5\right )^{2}}-96 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}+24\right )}{4}\right )}{64}\) | \(148\) |
derivativedivides | \(4 \,{\mathrm e}^{-1} {\mathrm e}^{-\ln \left (5\right )^{2}} \left (-{\mathrm e}^{-3} {\mathrm e}^{-3 \ln \left (5\right )^{2}} \left (\frac {{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{3} {\mathrm e}^{3} {\mathrm e}^{3 \ln \left (5\right )^{2}}}{4}-\frac {3 x^{2} {\mathrm e}^{2} {\mathrm e}^{2 \ln \left (5\right )^{2}} {\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{16}+\frac {3 x \,{\mathrm e} \,{\mathrm e}^{\ln \left (5\right )^{2}} {\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{32}-\frac {3 \,{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{128}\right )-{\mathrm e}^{-3} {\mathrm e}^{-3 \ln \left (5\right )^{2}} \left (\frac {{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{4} {\mathrm e}^{4} {\mathrm e}^{4 \ln \left (5\right )^{2}}}{4}-\frac {{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{3} {\mathrm e}^{3} {\mathrm e}^{3 \ln \left (5\right )^{2}}}{4}+\frac {3 x^{2} {\mathrm e}^{2} {\mathrm e}^{2 \ln \left (5\right )^{2}} {\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{16}-\frac {3 x \,{\mathrm e} \,{\mathrm e}^{\ln \left (5\right )^{2}} {\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{32}+\frac {3 \,{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{128}\right )\right )\) | \(255\) |
default | \(4 \,{\mathrm e}^{-1} {\mathrm e}^{-\ln \left (5\right )^{2}} \left (-{\mathrm e}^{-3} {\mathrm e}^{-3 \ln \left (5\right )^{2}} \left (\frac {{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{3} {\mathrm e}^{3} {\mathrm e}^{3 \ln \left (5\right )^{2}}}{4}-\frac {3 x^{2} {\mathrm e}^{2} {\mathrm e}^{2 \ln \left (5\right )^{2}} {\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{16}+\frac {3 x \,{\mathrm e} \,{\mathrm e}^{\ln \left (5\right )^{2}} {\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{32}-\frac {3 \,{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{128}\right )-{\mathrm e}^{-3} {\mathrm e}^{-3 \ln \left (5\right )^{2}} \left (\frac {{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{4} {\mathrm e}^{4} {\mathrm e}^{4 \ln \left (5\right )^{2}}}{4}-\frac {{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}} x^{3} {\mathrm e}^{3} {\mathrm e}^{3 \ln \left (5\right )^{2}}}{4}+\frac {3 x^{2} {\mathrm e}^{2} {\mathrm e}^{2 \ln \left (5\right )^{2}} {\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{16}-\frac {3 x \,{\mathrm e} \,{\mathrm e}^{\ln \left (5\right )^{2}} {\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{32}+\frac {3 \,{\mathrm e}^{4 x \,{\mathrm e}^{1+\ln \left (5\right )^{2}}}}{128}\right )\right )\) | \(255\) |
Input:
int((-4*x^4*exp(1)*exp(ln(5)^2)-4*x^3)*exp(x*exp(1)*exp(ln(5)^2))^4,x,meth od=_RETURNVERBOSE)
Output:
-exp(4*x*exp(1+ln(5)^2))*x^4
Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int e^{4 e^{1+\log ^2(5)} x} \left (-4 x^3-4 e^{1+\log ^2(5)} x^4\right ) \, dx=-x^{4} e^{\left (4 \, x e^{\left (\log \left (5\right )^{2} + 1\right )}\right )} \] Input:
integrate((-4*x^4*exp(1)*exp(log(5)^2)-4*x^3)*exp(x*exp(1)*exp(log(5)^2))^ 4,x, algorithm="fricas")
Output:
-x^4*e^(4*x*e^(log(5)^2 + 1))
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int e^{4 e^{1+\log ^2(5)} x} \left (-4 x^3-4 e^{1+\log ^2(5)} x^4\right ) \, dx=- x^{4} e^{4 e x e^{\log {\left (5 \right )}^{2}}} \] Input:
integrate((-4*x**4*exp(1)*exp(ln(5)**2)-4*x**3)*exp(x*exp(1)*exp(ln(5)**2) )**4,x)
Output:
-x**4*exp(4*E*x*exp(log(5)**2))
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (18) = 36\).
Time = 0.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 6.85 \[ \int e^{4 e^{1+\log ^2(5)} x} \left (-4 x^3-4 e^{1+\log ^2(5)} x^4\right ) \, dx=-\frac {1}{32} \, {\left (32 \, x^{4} e^{\left (4 \, \log \left (5\right )^{2} + 4\right )} - 32 \, x^{3} e^{\left (3 \, \log \left (5\right )^{2} + 3\right )} + 24 \, x^{2} e^{\left (2 \, \log \left (5\right )^{2} + 2\right )} - 12 \, x e^{\left (\log \left (5\right )^{2} + 1\right )} + 3\right )} e^{\left (4 \, x e^{\left (\log \left (5\right )^{2} + 1\right )} - 4 \, \log \left (5\right )^{2} - 4\right )} - \frac {1}{32} \, {\left (32 \, x^{3} e^{\left (3 \, \log \left (5\right )^{2} + 3\right )} - 24 \, x^{2} e^{\left (2 \, \log \left (5\right )^{2} + 2\right )} + 12 \, x e^{\left (\log \left (5\right )^{2} + 1\right )} - 3\right )} e^{\left (4 \, x e^{\left (\log \left (5\right )^{2} + 1\right )} - 4 \, \log \left (5\right )^{2} - 4\right )} \] Input:
integrate((-4*x^4*exp(1)*exp(log(5)^2)-4*x^3)*exp(x*exp(1)*exp(log(5)^2))^ 4,x, algorithm="maxima")
Output:
-1/32*(32*x^4*e^(4*log(5)^2 + 4) - 32*x^3*e^(3*log(5)^2 + 3) + 24*x^2*e^(2 *log(5)^2 + 2) - 12*x*e^(log(5)^2 + 1) + 3)*e^(4*x*e^(log(5)^2 + 1) - 4*lo g(5)^2 - 4) - 1/32*(32*x^3*e^(3*log(5)^2 + 3) - 24*x^2*e^(2*log(5)^2 + 2) + 12*x*e^(log(5)^2 + 1) - 3)*e^(4*x*e^(log(5)^2 + 1) - 4*log(5)^2 - 4)
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (18) = 36\).
Time = 0.11 (sec) , antiderivative size = 137, normalized size of antiderivative = 6.85 \[ \int e^{4 e^{1+\log ^2(5)} x} \left (-4 x^3-4 e^{1+\log ^2(5)} x^4\right ) \, dx=-\frac {1}{32} \, {\left (32 \, x^{4} e^{\left (4 \, \log \left (5\right )^{2} + 4\right )} - 32 \, x^{3} e^{\left (3 \, \log \left (5\right )^{2} + 3\right )} + 24 \, x^{2} e^{\left (2 \, \log \left (5\right )^{2} + 2\right )} - 12 \, x e^{\left (\log \left (5\right )^{2} + 1\right )} + 3\right )} e^{\left (4 \, x e^{\left (\log \left (5\right )^{2} + 1\right )} - 4 \, \log \left (5\right )^{2} - 4\right )} - \frac {1}{32} \, {\left (32 \, x^{3} e^{\left (3 \, \log \left (5\right )^{2} + 3\right )} - 24 \, x^{2} e^{\left (2 \, \log \left (5\right )^{2} + 2\right )} + 12 \, x e^{\left (\log \left (5\right )^{2} + 1\right )} - 3\right )} e^{\left (4 \, x e^{\left (\log \left (5\right )^{2} + 1\right )} - 4 \, \log \left (5\right )^{2} - 4\right )} \] Input:
integrate((-4*x^4*exp(1)*exp(log(5)^2)-4*x^3)*exp(x*exp(1)*exp(log(5)^2))^ 4,x, algorithm="giac")
Output:
-1/32*(32*x^4*e^(4*log(5)^2 + 4) - 32*x^3*e^(3*log(5)^2 + 3) + 24*x^2*e^(2 *log(5)^2 + 2) - 12*x*e^(log(5)^2 + 1) + 3)*e^(4*x*e^(log(5)^2 + 1) - 4*lo g(5)^2 - 4) - 1/32*(32*x^3*e^(3*log(5)^2 + 3) - 24*x^2*e^(2*log(5)^2 + 2) + 12*x*e^(log(5)^2 + 1) - 3)*e^(4*x*e^(log(5)^2 + 1) - 4*log(5)^2 - 4)
Time = 4.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int e^{4 e^{1+\log ^2(5)} x} \left (-4 x^3-4 e^{1+\log ^2(5)} x^4\right ) \, dx=-x^4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{{\ln \left (5\right )}^2}\,\mathrm {e}} \] Input:
int(-exp(4*x*exp(log(5)^2)*exp(1))*(4*x^3 + 4*x^4*exp(log(5)^2)*exp(1)),x)
Output:
-x^4*exp(4*x*exp(log(5)^2)*exp(1))
Time = 0.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int e^{4 e^{1+\log ^2(5)} x} \left (-4 x^3-4 e^{1+\log ^2(5)} x^4\right ) \, dx=-e^{4 e^{\mathrm {log}\left (5\right )^{2}} e x} x^{4} \] Input:
int((-4*x^4*exp(1)*exp(log(5)^2)-4*x^3)*exp(x*exp(1)*exp(log(5)^2))^4,x)
Output:
- e**(4*e**(log(5)**2)*e*x)*x**4