Integrand size = 156, antiderivative size = 28 \[ \int \frac {-27+3 e^{256}-36 x^3+\left (27-3 e^{256}+9 x-18 x^3\right ) \log (x)+\left (-27+3 e^{256}-9 x+18 x^3\right ) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )}{\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2(x)+\left (18-2 e^{256}+6 x-12 x^3\right ) \log (x) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )+\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2\left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )} \, dx=\frac {3 x}{-\log (x)+\log \left (-3+\frac {e^{256}}{3}-x+2 x^3\right )} \]
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-27+3 e^{256}-36 x^3+\left (27-3 e^{256}+9 x-18 x^3\right ) \log (x)+\left (-27+3 e^{256}-9 x+18 x^3\right ) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )}{\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2(x)+\left (18-2 e^{256}+6 x-12 x^3\right ) \log (x) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )+\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2\left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )} \, dx=-\frac {3 x}{\log (3 x)-\log \left (-9+e^{256}-3 x+6 x^3\right )} \]
Integrate[(-27 + 3*E^256 - 36*x^3 + (27 - 3*E^256 + 9*x - 18*x^3)*Log[x] + (-27 + 3*E^256 - 9*x + 18*x^3)*Log[(-9 + E^256 - 3*x + 6*x^3)/3])/((-9 + E^256 - 3*x + 6*x^3)*Log[x]^2 + (18 - 2*E^256 + 6*x - 12*x^3)*Log[x]*Log[( -9 + E^256 - 3*x + 6*x^3)/3] + (-9 + E^256 - 3*x + 6*x^3)*Log[(-9 + E^256 - 3*x + 6*x^3)/3]^2),x]
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.
| \(\displaystyle \int \frac {-36 x^3+\left (-18 x^3+9 x-3 e^{256}+27\right ) \log (x)+\left (18 x^3-9 x+3 e^{256}-27\right ) \log \left (\frac {1}{3} \left (6 x^3-3 x+e^{256}-9\right )\right )+3 e^{256}-27}{\left (6 x^3-3 x+e^{256}-9\right ) \log ^2(x)+\left (6 x^3-3 x+e^{256}-9\right ) \log ^2\left (\frac {1}{3} \left (6 x^3-3 x+e^{256}-9\right )\right )+\left (-12 x^3+6 x-2 e^{256}+18\right ) \log \left (\frac {1}{3} \left (6 x^3-3 x+e^{256}-9\right )\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {36 x^3-\left (-18 x^3+9 x-3 e^{256}+27\right ) \log (x)-\left (18 x^3-9 x+3 e^{256}-27\right ) \log \left (\frac {1}{3} \left (6 x^3-3 x+e^{256}-9\right )\right )+27 \left (1-\frac {e^{256}}{9}\right )}{\left (-6 x^3+3 x-e^{256}+9\right ) \left (-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (x)+\log (3)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {18 x^3 \log (x)}{\left (6 x^3-3 x+e^{256}-9\right ) \left (-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (x)+\log (3)\right )^2}+\frac {18 x^3 \log \left (\frac {1}{3} \left (6 x^3-3 x+e^{256}-9\right )\right )}{\left (6 x^3-3 x+e^{256}-9\right ) \left (-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (x)+\log (3)\right )^2}-\frac {36 x^3}{\left (6 x^3-3 x+e^{256}-9\right ) \left (-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (x)+\log (3)\right )^2}+\frac {9 x \log (x)}{\left (6 x^3-3 x+e^{256}-9\right ) \left (-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (x)+\log (3)\right )^2}-\frac {9 x \log \left (\frac {1}{3} \left (6 x^3-3 x+e^{256}-9\right )\right )}{\left (6 x^3-3 x+e^{256}-9\right ) \left (-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (x)+\log (3)\right )^2}+\frac {27 \left (1-\frac {e^{256}}{9}\right ) \log (x)}{\left (6 x^3-3 x+e^{256}-9\right ) \left (-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (x)+\log (3)\right )^2}-\frac {27 \left (1-\frac {e^{256}}{9}\right ) \log \left (\frac {1}{3} \left (6 x^3-3 x+e^{256}-9\right )\right )}{\left (6 x^3-3 x+e^{256}-9\right ) \left (-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (x)+\log (3)\right )^2}+\frac {3 \left (e^{256}-9\right )}{\left (6 x^3-3 x+e^{256}-9\right ) \left (-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (x)+\log (3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 \int \frac {1}{\left (\log (x)-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (3)\right )^2}dx-9 \left (9-e^{256}\right ) \int \frac {1}{\left (6 x^3-3 x+e^{256}-9\right ) \left (\log (x)-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (3)\right )^2}dx-18 \int \frac {x}{\left (6 x^3-3 x+e^{256}-9\right ) \left (\log (x)-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (3)\right )^2}dx-3 \int \frac {\log (x)}{\left (\log (x)-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (3)\right )^2}dx+3 \int \frac {\log \left (\frac {1}{3} \left (6 x^3-3 x+e^{256}-9\right )\right )}{\left (\log (x)-\log \left (6 x^3-3 x+e^{256}-9\right )+\log (3)\right )^2}dx\) |
Int[(-27 + 3*E^256 - 36*x^3 + (27 - 3*E^256 + 9*x - 18*x^3)*Log[x] + (-27 + 3*E^256 - 9*x + 18*x^3)*Log[(-9 + E^256 - 3*x + 6*x^3)/3])/((-9 + E^256 - 3*x + 6*x^3)*Log[x]^2 + (18 - 2*E^256 + 6*x - 12*x^3)*Log[x]*Log[(-9 + E ^256 - 3*x + 6*x^3)/3] + (-9 + E^256 - 3*x + 6*x^3)*Log[(-9 + E^256 - 3*x + 6*x^3)/3]^2),x]
3.7.57.3.1 Defintions of rubi rules used
Time = 4.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(-\frac {3 x}{\ln \left (x \right )+\ln \left (3\right )-\ln \left ({\mathrm e}^{256}+6 x^{3}-3 x -9\right )}\) | \(26\) |
| risch | \(-\frac {3 x}{\ln \left (x \right )-\ln \left (\frac {{\mathrm e}^{256}}{3}+2 x^{3}-x -3\right )}\) | \(26\) |
| parallelrisch | \(-\frac {3 x}{\ln \left (x \right )-\ln \left (\frac {{\mathrm e}^{256}}{3}+2 x^{3}-x -3\right )}\) | \(26\) |
int(((3*exp(256)+18*x^3-9*x-27)*ln(1/3*exp(256)+2*x^3-x-3)+(-3*exp(256)-18 *x^3+9*x+27)*ln(x)+3*exp(256)-36*x^3-27)/((exp(256)+6*x^3-3*x-9)*ln(1/3*ex p(256)+2*x^3-x-3)^2+(-2*exp(256)-12*x^3+6*x+18)*ln(x)*ln(1/3*exp(256)+2*x^ 3-x-3)+(exp(256)+6*x^3-3*x-9)*ln(x)^2),x,method=_RETURNVERBOSE)
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-27+3 e^{256}-36 x^3+\left (27-3 e^{256}+9 x-18 x^3\right ) \log (x)+\left (-27+3 e^{256}-9 x+18 x^3\right ) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )}{\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2(x)+\left (18-2 e^{256}+6 x-12 x^3\right ) \log (x) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )+\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2\left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )} \, dx=\frac {3 \, x}{\log \left (2 \, x^{3} - x + \frac {1}{3} \, e^{256} - 3\right ) - \log \left (x\right )} \]
integrate(((3*exp(256)+18*x^3-9*x-27)*log(1/3*exp(256)+2*x^3-x-3)+(-3*exp( 256)-18*x^3+9*x+27)*log(x)+3*exp(256)-36*x^3-27)/((exp(256)+6*x^3-3*x-9)*l og(1/3*exp(256)+2*x^3-x-3)^2+(-2*exp(256)-12*x^3+6*x+18)*log(x)*log(1/3*ex p(256)+2*x^3-x-3)+(exp(256)+6*x^3-3*x-9)*log(x)^2),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {-27+3 e^{256}-36 x^3+\left (27-3 e^{256}+9 x-18 x^3\right ) \log (x)+\left (-27+3 e^{256}-9 x+18 x^3\right ) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )}{\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2(x)+\left (18-2 e^{256}+6 x-12 x^3\right ) \log (x) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )+\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2\left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )} \, dx=\frac {3 x}{- \log {\left (x \right )} + \log {\left (2 x^{3} - x - 3 + \frac {e^{256}}{3} \right )}} \]
integrate(((3*exp(256)+18*x**3-9*x-27)*ln(1/3*exp(256)+2*x**3-x-3)+(-3*exp (256)-18*x**3+9*x+27)*ln(x)+3*exp(256)-36*x**3-27)/((exp(256)+6*x**3-3*x-9 )*ln(1/3*exp(256)+2*x**3-x-3)**2+(-2*exp(256)-12*x**3+6*x+18)*ln(x)*ln(1/3 *exp(256)+2*x**3-x-3)+(exp(256)+6*x**3-3*x-9)*ln(x)**2),x)
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-27+3 e^{256}-36 x^3+\left (27-3 e^{256}+9 x-18 x^3\right ) \log (x)+\left (-27+3 e^{256}-9 x+18 x^3\right ) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )}{\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2(x)+\left (18-2 e^{256}+6 x-12 x^3\right ) \log (x) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )+\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2\left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )} \, dx=-\frac {3 \, x}{\log \left (3\right ) - \log \left (6 \, x^{3} - 3 \, x + e^{256} - 9\right ) + \log \left (x\right )} \]
integrate(((3*exp(256)+18*x^3-9*x-27)*log(1/3*exp(256)+2*x^3-x-3)+(-3*exp( 256)-18*x^3+9*x+27)*log(x)+3*exp(256)-36*x^3-27)/((exp(256)+6*x^3-3*x-9)*l og(1/3*exp(256)+2*x^3-x-3)^2+(-2*exp(256)-12*x^3+6*x+18)*log(x)*log(1/3*ex p(256)+2*x^3-x-3)+(exp(256)+6*x^3-3*x-9)*log(x)^2),x, algorithm=\
Exception generated. \[ \int \frac {-27+3 e^{256}-36 x^3+\left (27-3 e^{256}+9 x-18 x^3\right ) \log (x)+\left (-27+3 e^{256}-9 x+18 x^3\right ) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )}{\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2(x)+\left (18-2 e^{256}+6 x-12 x^3\right ) \log (x) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )+\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2\left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )} \, dx=\text {Exception raised: TypeError} \]
integrate(((3*exp(256)+18*x^3-9*x-27)*log(1/3*exp(256)+2*x^3-x-3)+(-3*exp( 256)-18*x^3+9*x+27)*log(x)+3*exp(256)-36*x^3-27)/((exp(256)+6*x^3-3*x-9)*l og(1/3*exp(256)+2*x^3-x-3)^2+(-2*exp(256)-12*x^3+6*x+18)*log(x)*log(1/3*ex p(256)+2*x^3-x-3)+(exp(256)+6*x^3-3*x-9)*log(x)^2),x, algorithm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[-1.0, 0.0,1.23157876138e+243,undef]proot error [1.0,-0.0,-1.23157876138e+243,und ef]Franci
Time = 12.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-27+3 e^{256}-36 x^3+\left (27-3 e^{256}+9 x-18 x^3\right ) \log (x)+\left (-27+3 e^{256}-9 x+18 x^3\right ) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )}{\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2(x)+\left (18-2 e^{256}+6 x-12 x^3\right ) \log (x) \log \left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )+\left (-9+e^{256}-3 x+6 x^3\right ) \log ^2\left (\frac {1}{3} \left (-9+e^{256}-3 x+6 x^3\right )\right )} \, dx=\frac {3\,x}{\ln \left (2\,x^3-x+\frac {{\mathrm {e}}^{256}}{3}-3\right )-\ln \left (x\right )} \]
int((log(exp(256)/3 - x + 2*x^3 - 3)*(9*x - 3*exp(256) - 18*x^3 + 27) - 3* exp(256) - log(x)*(9*x - 3*exp(256) - 18*x^3 + 27) + 36*x^3 + 27)/(log(exp (256)/3 - x + 2*x^3 - 3)^2*(3*x - exp(256) - 6*x^3 + 9) + log(x)^2*(3*x - exp(256) - 6*x^3 + 9) - log(exp(256)/3 - x + 2*x^3 - 3)*log(x)*(6*x - 2*ex p(256) - 12*x^3 + 18)),x)