Integrand size = 80, antiderivative size = 29 \[ \int \frac {e^{\frac {e^3+3 x \log \left (\frac {-3 x+(11+x) \log (2)}{\log (2)}\right )}{x}} \left (24 e^3 x-72 x^2+\left (e^3 (-88-8 x)+24 x^2\right ) \log (2)\right )}{-3 x^3+\left (11 x^2+x^3\right ) \log (2)} \, dx=8 \left (1+e^{\frac {e^3}{x}} \left (-4+x-3 \left (-5+\frac {x}{\log (2)}\right )\right )^3\right ) \] Output:
8+8*exp(3*ln(x-3*x/ln(2)+11)+exp(3)/x)
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {e^3+3 x \log \left (\frac {-3 x+(11+x) \log (2)}{\log (2)}\right )}{x}} \left (24 e^3 x-72 x^2+\left (e^3 (-88-8 x)+24 x^2\right ) \log (2)\right )}{-3 x^3+\left (11 x^2+x^3\right ) \log (2)} \, dx=\frac {8 e^{\frac {e^3}{x}} (x (-3+\log (2))+\log (2048))^3}{\log ^3(2)} \] Input:
Integrate[(E^((E^3 + 3*x*Log[(-3*x + (11 + x)*Log[2])/Log[2]])/x)*(24*E^3* x - 72*x^2 + (E^3*(-88 - 8*x) + 24*x^2)*Log[2]))/(-3*x^3 + (11*x^2 + x^3)* Log[2]),x]
Output:
(8*E^(E^3/x)*(x*(-3 + Log[2]) + Log[2048])^3)/Log[2]^3
Time = 1.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2026, 7257}
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.
\(\displaystyle \int \frac {\left (-72 x^2+\left (24 x^2+e^3 (-8 x-88)\right ) \log (2)+24 e^3 x\right ) \exp \left (\frac {3 x \log \left (\frac {(x+11) \log (2)-3 x}{\log (2)}\right )+e^3}{x}\right )}{\left (x^3+11 x^2\right ) \log (2)-3 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (-72 x^2+\left (24 x^2+e^3 (-8 x-88)\right ) \log (2)+24 e^3 x\right ) \exp \left (\frac {3 x \log \left (\frac {(x+11) \log (2)-3 x}{\log (2)}\right )+e^3}{x}\right )}{x^2 (\log (2048)-x (3-\log (2)))}dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle -\frac {8 e^{\frac {e^3}{x}} (3 x-(x+11) \log (2))^3}{\log ^3(2)}\) |
Input:
Int[(E^((E^3 + 3*x*Log[(-3*x + (11 + x)*Log[2])/Log[2]])/x)*(24*E^3*x - 72 *x^2 + (E^3*(-88 - 8*x) + 24*x^2)*Log[2]))/(-3*x^3 + (11*x^2 + x^3)*Log[2] ),x]
Output:
(-8*E^(E^3/x)*(3*x - (11 + x)*Log[2])^3)/Log[2]^3
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 1.73 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
method | result | size |
norman | \(8 \,{\mathrm e}^{\frac {3 x \ln \left (\frac {\left (11+x \right ) \ln \left (2\right )-3 x}{\ln \left (2\right )}\right )+{\mathrm e}^{3}}{x}}\) | \(30\) |
parallelrisch | \(8 \,{\mathrm e}^{\frac {3 x \ln \left (\frac {\left (11+x \right ) \ln \left (2\right )-3 x}{\ln \left (2\right )}\right )+{\mathrm e}^{3}}{x}}\) | \(30\) |
gosper | \(8 \,{\mathrm e}^{\frac {3 x \ln \left (\frac {x \ln \left (2\right )+11 \ln \left (2\right )-3 x}{\ln \left (2\right )}\right )+{\mathrm e}^{3}}{x}}\) | \(32\) |
risch | \(\frac {8 \left (\ln \left (2\right )^{3} x^{3}+33 x^{2} \ln \left (2\right )^{3}-9 x^{3} \ln \left (2\right )^{2}+363 x \ln \left (2\right )^{3}-198 x^{2} \ln \left (2\right )^{2}+27 x^{3} \ln \left (2\right )+1331 \ln \left (2\right )^{3}-1089 x \ln \left (2\right )^{2}+297 x^{2} \ln \left (2\right )-27 x^{3}\right ) \left (\left (11+x \right ) \ln \left (2\right )-3 x \right )^{3} {\mathrm e}^{\frac {{\mathrm e}^{3}}{x}}}{\left (x \ln \left (2\right )+11 \ln \left (2\right )-3 x \right )^{3} \ln \left (2\right )^{3}}\) | \(115\) |
default | \(\text {Expression too large to display}\) | \(4677\) |
Input:
int((((-8*x-88)*exp(3)+24*x^2)*ln(2)+24*x*exp(3)-72*x^2)*exp((3*x*ln(((11+ x)*ln(2)-3*x)/ln(2))+exp(3))/x)/((x^3+11*x^2)*ln(2)-3*x^3),x,method=_RETUR NVERBOSE)
Output:
8*exp((3*x*ln(((11+x)*ln(2)-3*x)/ln(2))+exp(3))/x)
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {e^3+3 x \log \left (\frac {-3 x+(11+x) \log (2)}{\log (2)}\right )}{x}} \left (24 e^3 x-72 x^2+\left (e^3 (-88-8 x)+24 x^2\right ) \log (2)\right )}{-3 x^3+\left (11 x^2+x^3\right ) \log (2)} \, dx=8 \, e^{\left (\frac {3 \, x \log \left (\frac {{\left (x + 11\right )} \log \left (2\right ) - 3 \, x}{\log \left (2\right )}\right ) + e^{3}}{x}\right )} \] Input:
integrate((((-8*x-88)*exp(3)+24*x^2)*log(2)+24*x*exp(3)-72*x^2)*exp((3*x*l og(((11+x)*log(2)-3*x)/log(2))+exp(3))/x)/((x^3+11*x^2)*log(2)-3*x^3),x, a lgorithm="fricas")
Output:
8*e^((3*x*log(((x + 11)*log(2) - 3*x)/log(2)) + e^3)/x)
Time = 51.98 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {e^3+3 x \log \left (\frac {-3 x+(11+x) \log (2)}{\log (2)}\right )}{x}} \left (24 e^3 x-72 x^2+\left (e^3 (-88-8 x)+24 x^2\right ) \log (2)\right )}{-3 x^3+\left (11 x^2+x^3\right ) \log (2)} \, dx=8 e^{\frac {3 x \log {\left (\frac {- 3 x + \left (x + 11\right ) \log {\left (2 \right )}}{\log {\left (2 \right )}} \right )} + e^{3}}{x}} \] Input:
integrate((((-8*x-88)*exp(3)+24*x**2)*ln(2)+24*x*exp(3)-72*x**2)*exp((3*x* ln(((11+x)*ln(2)-3*x)/ln(2))+exp(3))/x)/((x**3+11*x**2)*ln(2)-3*x**3),x)
Output:
8*exp((3*x*log((-3*x + (x + 11)*log(2))/log(2)) + exp(3))/x)
\[ \int \frac {e^{\frac {e^3+3 x \log \left (\frac {-3 x+(11+x) \log (2)}{\log (2)}\right )}{x}} \left (24 e^3 x-72 x^2+\left (e^3 (-88-8 x)+24 x^2\right ) \log (2)\right )}{-3 x^3+\left (11 x^2+x^3\right ) \log (2)} \, dx=\int { \frac {8 \, {\left (9 \, x^{2} - 3 \, x e^{3} - {\left (3 \, x^{2} - {\left (x + 11\right )} e^{3}\right )} \log \left (2\right )\right )} e^{\left (\frac {3 \, x \log \left (\frac {{\left (x + 11\right )} \log \left (2\right ) - 3 \, x}{\log \left (2\right )}\right ) + e^{3}}{x}\right )}}{3 \, x^{3} - {\left (x^{3} + 11 \, x^{2}\right )} \log \left (2\right )} \,d x } \] Input:
integrate((((-8*x-88)*exp(3)+24*x^2)*log(2)+24*x*exp(3)-72*x^2)*exp((3*x*l og(((11+x)*log(2)-3*x)/log(2))+exp(3))/x)/((x^3+11*x^2)*log(2)-3*x^3),x, a lgorithm="maxima")
Output:
8*((log(2)^3 - 9*log(2)^2 + 27*log(2) - 27)*x^3 + 33*(log(2)^3 - 6*log(2)^ 2 + 9*log(2))*x^2 + 363*(log(2)^3 - 3*log(2)^2)*x)*e^(e^3/x)/log(2)^3 - 10 648*integrate(e^(e^3/x + 3)/x^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (25) = 50\).
Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 5.90 \[ \int \frac {e^{\frac {e^3+3 x \log \left (\frac {-3 x+(11+x) \log (2)}{\log (2)}\right )}{x}} \left (24 e^3 x-72 x^2+\left (e^3 (-88-8 x)+24 x^2\right ) \log (2)\right )}{-3 x^3+\left (11 x^2+x^3\right ) \log (2)} \, dx=\frac {8 \, {\left (e^{\left (\frac {e^{3}}{x} + 12\right )} \log \left (2\right )^{3} - 9 \, e^{\left (\frac {e^{3}}{x} + 12\right )} \log \left (2\right )^{2} + \frac {33 \, e^{\left (\frac {e^{3}}{x} + 12\right )} \log \left (2\right )^{3}}{x} + 27 \, e^{\left (\frac {e^{3}}{x} + 12\right )} \log \left (2\right ) - \frac {198 \, e^{\left (\frac {e^{3}}{x} + 12\right )} \log \left (2\right )^{2}}{x} + \frac {363 \, e^{\left (\frac {e^{3}}{x} + 12\right )} \log \left (2\right )^{3}}{x^{2}} + \frac {297 \, e^{\left (\frac {e^{3}}{x} + 12\right )} \log \left (2\right )}{x} - \frac {1089 \, e^{\left (\frac {e^{3}}{x} + 12\right )} \log \left (2\right )^{2}}{x^{2}} + \frac {1331 \, e^{\left (\frac {e^{3}}{x} + 12\right )} \log \left (2\right )^{3}}{x^{3}} - 27 \, e^{\left (\frac {e^{3}}{x} + 12\right )}\right )} x^{3} e^{\left (-12\right )}}{\log \left (2\right )^{3}} \] Input:
integrate((((-8*x-88)*exp(3)+24*x^2)*log(2)+24*x*exp(3)-72*x^2)*exp((3*x*l og(((11+x)*log(2)-3*x)/log(2))+exp(3))/x)/((x^3+11*x^2)*log(2)-3*x^3),x, a lgorithm="giac")
Output:
8*(e^(e^3/x + 12)*log(2)^3 - 9*e^(e^3/x + 12)*log(2)^2 + 33*e^(e^3/x + 12) *log(2)^3/x + 27*e^(e^3/x + 12)*log(2) - 198*e^(e^3/x + 12)*log(2)^2/x + 3 63*e^(e^3/x + 12)*log(2)^3/x^2 + 297*e^(e^3/x + 12)*log(2)/x - 1089*e^(e^3 /x + 12)*log(2)^2/x^2 + 1331*e^(e^3/x + 12)*log(2)^3/x^3 - 27*e^(e^3/x + 1 2))*x^3*e^(-12)/log(2)^3
Timed out. \[ \int \frac {e^{\frac {e^3+3 x \log \left (\frac {-3 x+(11+x) \log (2)}{\log (2)}\right )}{x}} \left (24 e^3 x-72 x^2+\left (e^3 (-88-8 x)+24 x^2\right ) \log (2)\right )}{-3 x^3+\left (11 x^2+x^3\right ) \log (2)} \, dx=\int \frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^3+3\,x\,\ln \left (-\frac {3\,x-\ln \left (2\right )\,\left (x+11\right )}{\ln \left (2\right )}\right )}{x}}\,\left (24\,x\,{\mathrm {e}}^3+\ln \left (2\right )\,\left (24\,x^2-{\mathrm {e}}^3\,\left (8\,x+88\right )\right )-72\,x^2\right )}{\ln \left (2\right )\,\left (x^3+11\,x^2\right )-3\,x^3} \,d x \] Input:
int((exp((exp(3) + 3*x*log(-(3*x - log(2)*(x + 11))/log(2)))/x)*(24*x*exp( 3) + log(2)*(24*x^2 - exp(3)*(8*x + 88)) - 72*x^2))/(log(2)*(11*x^2 + x^3) - 3*x^3),x)
Output:
int((exp((exp(3) + 3*x*log(-(3*x - log(2)*(x + 11))/log(2)))/x)*(24*x*exp( 3) + log(2)*(24*x^2 - exp(3)*(8*x + 88)) - 72*x^2))/(log(2)*(11*x^2 + x^3) - 3*x^3), x)
Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {e^{\frac {e^3+3 x \log \left (\frac {-3 x+(11+x) \log (2)}{\log (2)}\right )}{x}} \left (24 e^3 x-72 x^2+\left (e^3 (-88-8 x)+24 x^2\right ) \log (2)\right )}{-3 x^3+\left (11 x^2+x^3\right ) \log (2)} \, dx=\frac {8 e^{\frac {e^{3}}{x}} \left (\mathrm {log}\left (2\right )^{3} x^{3}+33 \mathrm {log}\left (2\right )^{3} x^{2}+363 \mathrm {log}\left (2\right )^{3} x +1331 \mathrm {log}\left (2\right )^{3}-9 \mathrm {log}\left (2\right )^{2} x^{3}-198 \mathrm {log}\left (2\right )^{2} x^{2}-1089 \mathrm {log}\left (2\right )^{2} x +27 \,\mathrm {log}\left (2\right ) x^{3}+297 \,\mathrm {log}\left (2\right ) x^{2}-27 x^{3}\right )}{\mathrm {log}\left (2\right )^{3}} \] Input:
int((((-8*x-88)*exp(3)+24*x^2)*log(2)+24*x*exp(3)-72*x^2)*exp((3*x*log(((1 1+x)*log(2)-3*x)/log(2))+exp(3))/x)/((x^3+11*x^2)*log(2)-3*x^3),x)
Output:
(8*e**(e**3/x)*(log(2)**3*x**3 + 33*log(2)**3*x**2 + 363*log(2)**3*x + 133 1*log(2)**3 - 9*log(2)**2*x**3 - 198*log(2)**2*x**2 - 1089*log(2)**2*x + 2 7*log(2)*x**3 + 297*log(2)*x**2 - 27*x**3))/log(2)**3