Integrand size = 159, antiderivative size = 25 \[ \int \frac {e^{18} \left (96+32 e^3\right )+e^{36} \left (9 x+6 e^3 x+e^6 x\right )+e^{18} \left (6+2 e^3\right ) \log (x)}{65536 x+e^{18} \left (1536 x^2+512 e^3 x^2\right )+e^{36} \left (9 x^3+6 e^3 x^3+e^6 x^3\right )+\left (16384 x+e^{18} \left (192 x^2+64 e^3 x^2\right )\right ) \log (x)+\left (1536 x+e^{18} \left (6 x^2+2 e^3 x^2\right )\right ) \log ^2(x)+64 x \log ^3(x)+x \log ^4(x)} \, dx=5-\frac {1}{x+\frac {(16+\log (x))^2}{e^{18} \left (3+e^3\right )}} \]
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^{18} \left (96+32 e^3\right )+e^{36} \left (9 x+6 e^3 x+e^6 x\right )+e^{18} \left (6+2 e^3\right ) \log (x)}{65536 x+e^{18} \left (1536 x^2+512 e^3 x^2\right )+e^{36} \left (9 x^3+6 e^3 x^3+e^6 x^3\right )+\left (16384 x+e^{18} \left (192 x^2+64 e^3 x^2\right )\right ) \log (x)+\left (1536 x+e^{18} \left (6 x^2+2 e^3 x^2\right )\right ) \log ^2(x)+64 x \log ^3(x)+x \log ^4(x)} \, dx=-\frac {e^{18} \left (3+e^3\right )}{256+3 e^{18} x+e^{21} x+32 \log (x)+\log ^2(x)} \]
Integrate[(E^18*(96 + 32*E^3) + E^36*(9*x + 6*E^3*x + E^6*x) + E^18*(6 + 2 *E^3)*Log[x])/(65536*x + E^18*(1536*x^2 + 512*E^3*x^2) + E^36*(9*x^3 + 6*E ^3*x^3 + E^6*x^3) + (16384*x + E^18*(192*x^2 + 64*E^3*x^2))*Log[x] + (1536 *x + E^18*(6*x^2 + 2*E^3*x^2))*Log[x]^2 + 64*x*Log[x]^3 + x*Log[x]^4),x]
Time = 0.51 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {7239, 27, 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.
\(\displaystyle \int \frac {e^{36} \left (e^6 x+6 e^3 x+9 x\right )+e^{18} \left (6+2 e^3\right ) \log (x)+e^{18} \left (96+32 e^3\right )}{e^{36} \left (e^6 x^3+6 e^3 x^3+9 x^3\right )+e^{18} \left (512 e^3 x^2+1536 x^2\right )+\left (e^{18} \left (2 e^3 x^2+6 x^2\right )+1536 x\right ) \log ^2(x)+\left (e^{18} \left (64 e^3 x^2+192 x^2\right )+16384 x\right ) \log (x)+65536 x+x \log ^4(x)+64 x \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{18} \left (3+e^3\right ) \left (3 e^{18} \left (1+\frac {e^3}{3}\right ) x+2 \log (x)+32\right )}{x \left (3 e^{18} \left (1+\frac {e^3}{3}\right ) x+\log ^2(x)+32 \log (x)+256\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^{18} \left (3+e^3\right ) \int \frac {e^{18} \left (3+e^3\right ) x+2 \log (x)+32}{x \left (\log ^2(x)+32 \log (x)+e^{18} \left (3+e^3\right ) x+256\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle -\frac {e^{18} \left (3+e^3\right )}{e^{18} \left (3+e^3\right ) x+\log ^2(x)+32 \log (x)+256}\) |
Int[(E^18*(96 + 32*E^3) + E^36*(9*x + 6*E^3*x + E^6*x) + E^18*(6 + 2*E^3)* Log[x])/(65536*x + E^18*(1536*x^2 + 512*E^3*x^2) + E^36*(9*x^3 + 6*E^3*x^3 + E^6*x^3) + (16384*x + E^18*(192*x^2 + 64*E^3*x^2))*Log[x] + (1536*x + E ^18*(6*x^2 + 2*E^3*x^2))*Log[x]^2 + 64*x*Log[x]^3 + x*Log[x]^4),x]
3.22.91.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 1.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32
method | result | size |
default | \(\frac {{\mathrm e}^{18} \left (-{\mathrm e}^{3}-3\right )}{{\mathrm e}^{18} {\mathrm e}^{3} x +3 \,{\mathrm e}^{18} x +\ln \left (x \right )^{2}+32 \ln \left (x \right )+256}\) | \(33\) |
norman | \(\frac {-{\mathrm e}^{18} {\mathrm e}^{3}-3 \,{\mathrm e}^{18}}{{\mathrm e}^{18} {\mathrm e}^{3} x +3 \,{\mathrm e}^{18} x +\ln \left (x \right )^{2}+32 \ln \left (x \right )+256}\) | \(36\) |
parallelrisch | \(\frac {-{\mathrm e}^{18} {\mathrm e}^{3}-3 \,{\mathrm e}^{18}}{{\mathrm e}^{18} {\mathrm e}^{3} x +3 \,{\mathrm e}^{18} x +\ln \left (x \right )^{2}+32 \ln \left (x \right )+256}\) | \(36\) |
risch | \(-\frac {{\mathrm e}^{21}}{x \,{\mathrm e}^{21}+3 \,{\mathrm e}^{18} x +\ln \left (x \right )^{2}+32 \ln \left (x \right )+256}-\frac {3 \,{\mathrm e}^{18}}{x \,{\mathrm e}^{21}+3 \,{\mathrm e}^{18} x +\ln \left (x \right )^{2}+32 \ln \left (x \right )+256}\) | \(64\) |
int(((2*exp(3)+6)*exp(18)*ln(x)+(x*exp(3)^2+6*x*exp(3)+9*x)*exp(18)^2+(32* exp(3)+96)*exp(18))/(x*ln(x)^4+64*x*ln(x)^3+((2*x^2*exp(3)+6*x^2)*exp(18)+ 1536*x)*ln(x)^2+((64*x^2*exp(3)+192*x^2)*exp(18)+16384*x)*ln(x)+(x^3*exp(3 )^2+6*x^3*exp(3)+9*x^3)*exp(18)^2+(512*x^2*exp(3)+1536*x^2)*exp(18)+65536* x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{18} \left (96+32 e^3\right )+e^{36} \left (9 x+6 e^3 x+e^6 x\right )+e^{18} \left (6+2 e^3\right ) \log (x)}{65536 x+e^{18} \left (1536 x^2+512 e^3 x^2\right )+e^{36} \left (9 x^3+6 e^3 x^3+e^6 x^3\right )+\left (16384 x+e^{18} \left (192 x^2+64 e^3 x^2\right )\right ) \log (x)+\left (1536 x+e^{18} \left (6 x^2+2 e^3 x^2\right )\right ) \log ^2(x)+64 x \log ^3(x)+x \log ^4(x)} \, dx=-\frac {e^{21} + 3 \, e^{18}}{x e^{21} + 3 \, x e^{18} + \log \left (x\right )^{2} + 32 \, \log \left (x\right ) + 256} \]
integrate(((2*exp(3)+6)*exp(18)*log(x)+(x*exp(3)^2+6*x*exp(3)+9*x)*exp(18) ^2+(32*exp(3)+96)*exp(18))/(x*log(x)^4+64*x*log(x)^3+((2*x^2*exp(3)+6*x^2) *exp(18)+1536*x)*log(x)^2+((64*x^2*exp(3)+192*x^2)*exp(18)+16384*x)*log(x) +(x^3*exp(3)^2+6*x^3*exp(3)+9*x^3)*exp(18)^2+(512*x^2*exp(3)+1536*x^2)*exp (18)+65536*x),x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {e^{18} \left (96+32 e^3\right )+e^{36} \left (9 x+6 e^3 x+e^6 x\right )+e^{18} \left (6+2 e^3\right ) \log (x)}{65536 x+e^{18} \left (1536 x^2+512 e^3 x^2\right )+e^{36} \left (9 x^3+6 e^3 x^3+e^6 x^3\right )+\left (16384 x+e^{18} \left (192 x^2+64 e^3 x^2\right )\right ) \log (x)+\left (1536 x+e^{18} \left (6 x^2+2 e^3 x^2\right )\right ) \log ^2(x)+64 x \log ^3(x)+x \log ^4(x)} \, dx=\frac {- e^{21} - 3 e^{18}}{3 x e^{18} + x e^{21} + \log {\left (x \right )}^{2} + 32 \log {\left (x \right )} + 256} \]
integrate(((2*exp(3)+6)*exp(18)*ln(x)+(x*exp(3)**2+6*x*exp(3)+9*x)*exp(18) **2+(32*exp(3)+96)*exp(18))/(x*ln(x)**4+64*x*ln(x)**3+((2*x**2*exp(3)+6*x* *2)*exp(18)+1536*x)*ln(x)**2+((64*x**2*exp(3)+192*x**2)*exp(18)+16384*x)*l n(x)+(x**3*exp(3)**2+6*x**3*exp(3)+9*x**3)*exp(18)**2+(512*x**2*exp(3)+153 6*x**2)*exp(18)+65536*x),x)
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{18} \left (96+32 e^3\right )+e^{36} \left (9 x+6 e^3 x+e^6 x\right )+e^{18} \left (6+2 e^3\right ) \log (x)}{65536 x+e^{18} \left (1536 x^2+512 e^3 x^2\right )+e^{36} \left (9 x^3+6 e^3 x^3+e^6 x^3\right )+\left (16384 x+e^{18} \left (192 x^2+64 e^3 x^2\right )\right ) \log (x)+\left (1536 x+e^{18} \left (6 x^2+2 e^3 x^2\right )\right ) \log ^2(x)+64 x \log ^3(x)+x \log ^4(x)} \, dx=-\frac {e^{21} + 3 \, e^{18}}{x {\left (e^{21} + 3 \, e^{18}\right )} + \log \left (x\right )^{2} + 32 \, \log \left (x\right ) + 256} \]
integrate(((2*exp(3)+6)*exp(18)*log(x)+(x*exp(3)^2+6*x*exp(3)+9*x)*exp(18) ^2+(32*exp(3)+96)*exp(18))/(x*log(x)^4+64*x*log(x)^3+((2*x^2*exp(3)+6*x^2) *exp(18)+1536*x)*log(x)^2+((64*x^2*exp(3)+192*x^2)*exp(18)+16384*x)*log(x) +(x^3*exp(3)^2+6*x^3*exp(3)+9*x^3)*exp(18)^2+(512*x^2*exp(3)+1536*x^2)*exp (18)+65536*x),x, algorithm=\
Time = 0.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{18} \left (96+32 e^3\right )+e^{36} \left (9 x+6 e^3 x+e^6 x\right )+e^{18} \left (6+2 e^3\right ) \log (x)}{65536 x+e^{18} \left (1536 x^2+512 e^3 x^2\right )+e^{36} \left (9 x^3+6 e^3 x^3+e^6 x^3\right )+\left (16384 x+e^{18} \left (192 x^2+64 e^3 x^2\right )\right ) \log (x)+\left (1536 x+e^{18} \left (6 x^2+2 e^3 x^2\right )\right ) \log ^2(x)+64 x \log ^3(x)+x \log ^4(x)} \, dx=-\frac {2 \, {\left (e^{21} + 3 \, e^{18}\right )}}{x e^{21} + 3 \, x e^{18} + \log \left (x\right )^{2} + 32 \, \log \left (x\right ) + 256} \]
integrate(((2*exp(3)+6)*exp(18)*log(x)+(x*exp(3)^2+6*x*exp(3)+9*x)*exp(18) ^2+(32*exp(3)+96)*exp(18))/(x*log(x)^4+64*x*log(x)^3+((2*x^2*exp(3)+6*x^2) *exp(18)+1536*x)*log(x)^2+((64*x^2*exp(3)+192*x^2)*exp(18)+16384*x)*log(x) +(x^3*exp(3)^2+6*x^3*exp(3)+9*x^3)*exp(18)^2+(512*x^2*exp(3)+1536*x^2)*exp (18)+65536*x),x, algorithm=\
Time = 19.85 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.36 \[ \int \frac {e^{18} \left (96+32 e^3\right )+e^{36} \left (9 x+6 e^3 x+e^6 x\right )+e^{18} \left (6+2 e^3\right ) \log (x)}{65536 x+e^{18} \left (1536 x^2+512 e^3 x^2\right )+e^{36} \left (9 x^3+6 e^3 x^3+e^6 x^3\right )+\left (16384 x+e^{18} \left (192 x^2+64 e^3 x^2\right )\right ) \log (x)+\left (1536 x+e^{18} \left (6 x^2+2 e^3 x^2\right )\right ) \log ^2(x)+64 x \log ^3(x)+x \log ^4(x)} \, dx=\frac {\frac {{\left (3\,{\mathrm {e}}^{18}+{\mathrm {e}}^{21}\right )}^2\,x^3}{256}+\left (\frac {3\,{\mathrm {e}}^{18}}{256}+\frac {{\mathrm {e}}^{21}}{256}\right )\,x^2\,{\ln \left (x\right )}^2+\left (\frac {3\,{\mathrm {e}}^{18}}{8}+\frac {{\mathrm {e}}^{21}}{8}\right )\,x^2\,\ln \left (x\right )}{32\,x^2\,\ln \left (x\right )+x^2\,{\ln \left (x\right )}^2+3\,x^3\,{\mathrm {e}}^{18}+x^3\,{\mathrm {e}}^{21}+256\,x^2} \]
int((exp(36)*(9*x + 6*x*exp(3) + x*exp(6)) + exp(18)*(32*exp(3) + 96) + ex p(18)*log(x)*(2*exp(3) + 6))/(65536*x + 64*x*log(x)^3 + x*log(x)^4 + log(x )*(16384*x + exp(18)*(64*x^2*exp(3) + 192*x^2)) + exp(36)*(6*x^3*exp(3) + x^3*exp(6) + 9*x^3) + log(x)^2*(1536*x + exp(18)*(2*x^2*exp(3) + 6*x^2)) + exp(18)*(512*x^2*exp(3) + 1536*x^2)),x)