Integrand size = 106, antiderivative size = 24 \[ \int \frac {-24 e^{5 x}+e^{2+e^{2-5 x} x^3} \left (-24 x^2+40 x^3\right )}{e^{5 x+2 e^{2-5 x} x^3}+e^{5 x+e^{2-5 x} x^3} (6 x+6 \log (2))+e^{5 x} \left (9 x^2+18 x \log (2)+9 \log ^2(2)\right )} \, dx=\frac {8}{e^{e^{2-5 x} x^3}+3 (x+\log (2))} \] Output:
8/(3*ln(2)+3*x+exp(x^3*exp(1)^2/exp(5*x)))
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-24 e^{5 x}+e^{2+e^{2-5 x} x^3} \left (-24 x^2+40 x^3\right )}{e^{5 x+2 e^{2-5 x} x^3}+e^{5 x+e^{2-5 x} x^3} (6 x+6 \log (2))+e^{5 x} \left (9 x^2+18 x \log (2)+9 \log ^2(2)\right )} \, dx=\frac {8}{e^{e^{2-5 x} x^3}+3 x+\log (8)} \] Input:
Integrate[(-24*E^(5*x) + E^(2 + E^(2 - 5*x)*x^3)*(-24*x^2 + 40*x^3))/(E^(5 *x + 2*E^(2 - 5*x)*x^3) + E^(5*x + E^(2 - 5*x)*x^3)*(6*x + 6*Log[2]) + E^( 5*x)*(9*x^2 + 18*x*Log[2] + 9*Log[2]^2)),x]
Output:
8/(E^(E^(2 - 5*x)*x^3) + 3*x + Log[8])
Time = 1.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {7239, 7237}
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.
\(\displaystyle \int \frac {e^{e^{2-5 x} x^3+2} \left (40 x^3-24 x^2\right )-24 e^{5 x}}{e^{2 e^{2-5 x} x^3+5 x}+e^{e^{2-5 x} x^3+5 x} (6 x+6 \log (2))+e^{5 x} \left (9 x^2+18 x \log (2)+9 \log ^2(2)\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{-5 x} \left (8 e^{e^{2-5 x} x^3+2} x^2 (5 x-3)-24 e^{5 x}\right )}{\left (e^{e^{2-5 x} x^3}+3 x+\log (8)\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {8}{e^{e^{2-5 x} x^3}+3 x+\log (8)}\) |
Input:
Int[(-24*E^(5*x) + E^(2 + E^(2 - 5*x)*x^3)*(-24*x^2 + 40*x^3))/(E^(5*x + 2 *E^(2 - 5*x)*x^3) + E^(5*x + E^(2 - 5*x)*x^3)*(6*x + 6*Log[2]) + E^(5*x)*( 9*x^2 + 18*x*Log[2] + 9*Log[2]^2)),x]
Output:
8/(E^(E^(2 - 5*x)*x^3) + 3*x + Log[8])
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.68 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {8}{3 \ln \left (2\right )+3 x +{\mathrm e}^{x^{3} {\mathrm e}^{-5 x +2}}}\) | \(24\) |
norman | \(\frac {8}{3 \ln \left (2\right )+3 x +{\mathrm e}^{x^{3} {\mathrm e}^{2} {\mathrm e}^{-5 x}}}\) | \(28\) |
parallelrisch | \(\frac {8}{3 \ln \left (2\right )+3 x +{\mathrm e}^{x^{3} {\mathrm e}^{2} {\mathrm e}^{-5 x}}}\) | \(28\) |
Input:
int(((40*x^3-24*x^2)*exp(1)^2*exp(x^3*exp(1)^2/exp(5*x))-24*exp(5*x))/(exp (5*x)*exp(x^3*exp(1)^2/exp(5*x))^2+(6*ln(2)+6*x)*exp(5*x)*exp(x^3*exp(1)^2 /exp(5*x))+(9*ln(2)^2+18*x*ln(2)+9*x^2)*exp(5*x)),x,method=_RETURNVERBOSE)
Output:
8/(3*ln(2)+3*x+exp(x^3*exp(-5*x+2)))
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {-24 e^{5 x}+e^{2+e^{2-5 x} x^3} \left (-24 x^2+40 x^3\right )}{e^{5 x+2 e^{2-5 x} x^3}+e^{5 x+e^{2-5 x} x^3} (6 x+6 \log (2))+e^{5 x} \left (9 x^2+18 x \log (2)+9 \log ^2(2)\right )} \, dx=\frac {8 \, e^{\left (5 \, x\right )}}{3 \, {\left (x + \log \left (2\right )\right )} e^{\left (5 \, x\right )} + e^{\left ({\left (x^{3} e^{2} + 5 \, x e^{\left (5 \, x\right )}\right )} e^{\left (-5 \, x\right )}\right )}} \] Input:
integrate(((40*x^3-24*x^2)*exp(1)^2*exp(x^3*exp(1)^2/exp(5*x))-24*exp(5*x) )/(exp(5*x)*exp(x^3*exp(1)^2/exp(5*x))^2+(6*log(2)+6*x)*exp(5*x)*exp(x^3*e xp(1)^2/exp(5*x))+(9*log(2)^2+18*x*log(2)+9*x^2)*exp(5*x)),x, algorithm="f ricas")
Output:
8*e^(5*x)/(3*(x + log(2))*e^(5*x) + e^((x^3*e^2 + 5*x*e^(5*x))*e^(-5*x)))
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-24 e^{5 x}+e^{2+e^{2-5 x} x^3} \left (-24 x^2+40 x^3\right )}{e^{5 x+2 e^{2-5 x} x^3}+e^{5 x+e^{2-5 x} x^3} (6 x+6 \log (2))+e^{5 x} \left (9 x^2+18 x \log (2)+9 \log ^2(2)\right )} \, dx=\frac {8}{3 x + e^{x^{3} e^{2} e^{- 5 x}} + 3 \log {\left (2 \right )}} \] Input:
integrate(((40*x**3-24*x**2)*exp(1)**2*exp(x**3*exp(1)**2/exp(5*x))-24*exp (5*x))/(exp(5*x)*exp(x**3*exp(1)**2/exp(5*x))**2+(6*ln(2)+6*x)*exp(5*x)*ex p(x**3*exp(1)**2/exp(5*x))+(9*ln(2)**2+18*x*ln(2)+9*x**2)*exp(5*x)),x)
Output:
8/(3*x + exp(x**3*exp(2)*exp(-5*x)) + 3*log(2))
Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-24 e^{5 x}+e^{2+e^{2-5 x} x^3} \left (-24 x^2+40 x^3\right )}{e^{5 x+2 e^{2-5 x} x^3}+e^{5 x+e^{2-5 x} x^3} (6 x+6 \log (2))+e^{5 x} \left (9 x^2+18 x \log (2)+9 \log ^2(2)\right )} \, dx=\frac {8}{3 \, x + e^{\left (x^{3} e^{\left (-5 \, x + 2\right )}\right )} + 3 \, \log \left (2\right )} \] Input:
integrate(((40*x^3-24*x^2)*exp(1)^2*exp(x^3*exp(1)^2/exp(5*x))-24*exp(5*x) )/(exp(5*x)*exp(x^3*exp(1)^2/exp(5*x))^2+(6*log(2)+6*x)*exp(5*x)*exp(x^3*e xp(1)^2/exp(5*x))+(9*log(2)^2+18*x*log(2)+9*x^2)*exp(5*x)),x, algorithm="m axima")
Output:
8/(3*x + e^(x^3*e^(-5*x + 2)) + 3*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (23) = 46\).
Time = 0.15 (sec) , antiderivative size = 621, normalized size of antiderivative = 25.88 \[ \int \frac {-24 e^{5 x}+e^{2+e^{2-5 x} x^3} \left (-24 x^2+40 x^3\right )}{e^{5 x+2 e^{2-5 x} x^3}+e^{5 x+e^{2-5 x} x^3} (6 x+6 \log (2))+e^{5 x} \left (9 x^2+18 x \log (2)+9 \log ^2(2)\right )} \, dx =\text {Too large to display} \] Input:
integrate(((40*x^3-24*x^2)*exp(1)^2*exp(x^3*exp(1)^2/exp(5*x))-24*exp(5*x) )/(exp(5*x)*exp(x^3*exp(1)^2/exp(5*x))^2+(6*log(2)+6*x)*exp(5*x)*exp(x^3*e xp(1)^2/exp(5*x))+(9*log(2)^2+18*x*log(2)+9*x^2)*exp(5*x)),x, algorithm="g iac")
Output:
8*(15*x^5*e^(5*x + 2) + 30*x^4*e^(5*x + 2)*log(2) + 15*x^3*e^(5*x + 2)*log (2)^2 + 5*x^4*e^(x^3*e^(-5*x + 2) + 5*x + 2) - 9*x^4*e^(5*x + 2) + 5*x^3*e ^(x^3*e^(-5*x + 2) + 5*x + 2)*log(2) - 18*x^3*e^(5*x + 2)*log(2) - 9*x^2*e ^(5*x + 2)*log(2)^2 - 3*x^3*e^(x^3*e^(-5*x + 2) + 5*x + 2) - 3*x^2*e^(x^3* e^(-5*x + 2) + 5*x + 2)*log(2) + 3*x*e^(10*x) + 3*e^(10*x)*log(2) + e^(x^3 *e^(-5*x + 2) + 10*x))/(45*x^6*e^(5*x + 2) + 135*x^5*e^(5*x + 2)*log(2) + 135*x^4*e^(5*x + 2)*log(2)^2 + 45*x^3*e^(5*x + 2)*log(2)^3 + 30*x^5*e^(x^3 *e^(-5*x + 2) + 5*x + 2) - 27*x^5*e^(5*x + 2) + 60*x^4*e^(x^3*e^(-5*x + 2) + 5*x + 2)*log(2) - 81*x^4*e^(5*x + 2)*log(2) + 30*x^3*e^(x^3*e^(-5*x + 2 ) + 5*x + 2)*log(2)^2 - 81*x^3*e^(5*x + 2)*log(2)^2 - 27*x^2*e^(5*x + 2)*l og(2)^3 + 5*x^4*e^(2*x^3*e^(-5*x + 2) + 5*x + 2) - 18*x^4*e^(x^3*e^(-5*x + 2) + 5*x + 2) + 5*x^3*e^(2*x^3*e^(-5*x + 2) + 5*x + 2)*log(2) - 36*x^3*e^ (x^3*e^(-5*x + 2) + 5*x + 2)*log(2) - 18*x^2*e^(x^3*e^(-5*x + 2) + 5*x + 2 )*log(2)^2 - 3*x^3*e^(2*x^3*e^(-5*x + 2) + 5*x + 2) - 3*x^2*e^(2*x^3*e^(-5 *x + 2) + 5*x + 2)*log(2) + 9*x^2*e^(10*x) + 18*x*e^(10*x)*log(2) + 9*e^(1 0*x)*log(2)^2 + 6*x*e^(x^3*e^(-5*x + 2) + 10*x) + 6*e^(x^3*e^(-5*x + 2) + 10*x)*log(2) + e^(2*x^3*e^(-5*x + 2) + 10*x))
Timed out. \[ \int \frac {-24 e^{5 x}+e^{2+e^{2-5 x} x^3} \left (-24 x^2+40 x^3\right )}{e^{5 x+2 e^{2-5 x} x^3}+e^{5 x+e^{2-5 x} x^3} (6 x+6 \log (2))+e^{5 x} \left (9 x^2+18 x \log (2)+9 \log ^2(2)\right )} \, dx=\int -\frac {24\,{\mathrm {e}}^{5\,x}+{\mathrm {e}}^{x^3\,{\mathrm {e}}^{-5\,x}\,{\mathrm {e}}^2}\,{\mathrm {e}}^2\,\left (24\,x^2-40\,x^3\right )}{{\mathrm {e}}^{5\,x}\,\left (9\,x^2+18\,\ln \left (2\right )\,x+9\,{\ln \left (2\right )}^2\right )+{\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{2\,x^3\,{\mathrm {e}}^{-5\,x}\,{\mathrm {e}}^2}+{\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{x^3\,{\mathrm {e}}^{-5\,x}\,{\mathrm {e}}^2}\,\left (6\,x+6\,\ln \left (2\right )\right )} \,d x \] Input:
int(-(24*exp(5*x) + exp(x^3*exp(-5*x)*exp(2))*exp(2)*(24*x^2 - 40*x^3))/(e xp(5*x)*(18*x*log(2) + 9*log(2)^2 + 9*x^2) + exp(5*x)*exp(2*x^3*exp(-5*x)* exp(2)) + exp(5*x)*exp(x^3*exp(-5*x)*exp(2))*(6*x + 6*log(2))),x)
Output:
int(-(24*exp(5*x) + exp(x^3*exp(-5*x)*exp(2))*exp(2)*(24*x^2 - 40*x^3))/(e xp(5*x)*(18*x*log(2) + 9*log(2)^2 + 9*x^2) + exp(5*x)*exp(2*x^3*exp(-5*x)* exp(2)) + exp(5*x)*exp(x^3*exp(-5*x)*exp(2))*(6*x + 6*log(2))), x)
\[ \int \frac {-24 e^{5 x}+e^{2+e^{2-5 x} x^3} \left (-24 x^2+40 x^3\right )}{e^{5 x+2 e^{2-5 x} x^3}+e^{5 x+e^{2-5 x} x^3} (6 x+6 \log (2))+e^{5 x} \left (9 x^2+18 x \log (2)+9 \log ^2(2)\right )} \, dx=40 \left (\int \frac {e^{\frac {e^{2} x^{3}}{e^{5 x}}} x^{3}}{e^{\frac {5 e^{5 x} x +2 e^{2} x^{3}}{e^{5 x}}}+6 e^{\frac {5 e^{5 x} x +e^{2} x^{3}}{e^{5 x}}} \mathrm {log}\left (2\right )+6 e^{\frac {5 e^{5 x} x +e^{2} x^{3}}{e^{5 x}}} x +9 e^{5 x} \mathrm {log}\left (2\right )^{2}+18 e^{5 x} \mathrm {log}\left (2\right ) x +9 e^{5 x} x^{2}}d x \right ) e^{2}-24 \left (\int \frac {e^{\frac {e^{2} x^{3}}{e^{5 x}}} x^{2}}{e^{\frac {5 e^{5 x} x +2 e^{2} x^{3}}{e^{5 x}}}+6 e^{\frac {5 e^{5 x} x +e^{2} x^{3}}{e^{5 x}}} \mathrm {log}\left (2\right )+6 e^{\frac {5 e^{5 x} x +e^{2} x^{3}}{e^{5 x}}} x +9 e^{5 x} \mathrm {log}\left (2\right )^{2}+18 e^{5 x} \mathrm {log}\left (2\right ) x +9 e^{5 x} x^{2}}d x \right ) e^{2}-24 \left (\int \frac {1}{e^{\frac {2 e^{2} x^{3}}{e^{5 x}}}+6 e^{\frac {e^{2} x^{3}}{e^{5 x}}} \mathrm {log}\left (2\right )+6 e^{\frac {e^{2} x^{3}}{e^{5 x}}} x +9 \mathrm {log}\left (2\right )^{2}+18 \,\mathrm {log}\left (2\right ) x +9 x^{2}}d x \right ) \] Input:
int(((40*x^3-24*x^2)*exp(1)^2*exp(x^3*exp(1)^2/exp(5*x))-24*exp(5*x))/(exp (5*x)*exp(x^3*exp(1)^2/exp(5*x))^2+(6*log(2)+6*x)*exp(5*x)*exp(x^3*exp(1)^ 2/exp(5*x))+(9*log(2)^2+18*x*log(2)+9*x^2)*exp(5*x)),x)
Output:
8*(5*int((e**((e**2*x**3)/e**(5*x))*x**3)/(e**((5*e**(5*x)*x + 2*e**2*x**3 )/e**(5*x)) + 6*e**((5*e**(5*x)*x + e**2*x**3)/e**(5*x))*log(2) + 6*e**((5 *e**(5*x)*x + e**2*x**3)/e**(5*x))*x + 9*e**(5*x)*log(2)**2 + 18*e**(5*x)* log(2)*x + 9*e**(5*x)*x**2),x)*e**2 - 3*int((e**((e**2*x**3)/e**(5*x))*x** 2)/(e**((5*e**(5*x)*x + 2*e**2*x**3)/e**(5*x)) + 6*e**((5*e**(5*x)*x + e** 2*x**3)/e**(5*x))*log(2) + 6*e**((5*e**(5*x)*x + e**2*x**3)/e**(5*x))*x + 9*e**(5*x)*log(2)**2 + 18*e**(5*x)*log(2)*x + 9*e**(5*x)*x**2),x)*e**2 - 3 *int(1/(e**((2*e**2*x**3)/e**(5*x)) + 6*e**((e**2*x**3)/e**(5*x))*log(2) + 6*e**((e**2*x**3)/e**(5*x))*x + 9*log(2)**2 + 18*log(2)*x + 9*x**2),x))