Integrand size = 68, antiderivative size = 25 \[ \int \frac {17-37 x+15 x^2+\left (7-17 x+6 x^2\right ) \log (6)+\left (-7+17 x-6 x^2\right ) \log (-7+3 x)}{-7+17 x-27 x^2+23 x^3-13 x^4+3 x^5} \, dx=\frac {-2-\log (6)+\log (1+2 (-4+x)+x)}{(-1+x)^2+x} \] Output:
(ln(3*x-7)-ln(6)-2)/((-1+x)^2+x)
Time = 0.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {17-37 x+15 x^2+\left (7-17 x+6 x^2\right ) \log (6)+\left (-7+17 x-6 x^2\right ) \log (-7+3 x)}{-7+17 x-27 x^2+23 x^3-13 x^4+3 x^5} \, dx=\frac {-222-257 \log (6)+73 \log (36)+111 \log (-7+3 x)}{111 \left (1-x+x^2\right )} \] Input:
Integrate[(17 - 37*x + 15*x^2 + (7 - 17*x + 6*x^2)*Log[6] + (-7 + 17*x - 6 *x^2)*Log[-7 + 3*x])/(-7 + 17*x - 27*x^2 + 23*x^3 - 13*x^4 + 3*x^5),x]
Output:
(-222 - 257*Log[6] + 73*Log[36] + 111*Log[-7 + 3*x])/(111*(1 - x + x^2))
Result contains complex when optimal does not.
Time = 1.93 (sec) , antiderivative size = 570, normalized size of antiderivative = 22.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2463, 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.
\(\displaystyle \int \frac {15 x^2+\left (-6 x^2+17 x-7\right ) \log (3 x-7)+\left (6 x^2-17 x+7\right ) \log (6)-37 x+17}{3 x^5-13 x^4+23 x^3-27 x^2+17 x-7} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {81 \left (15 x^2+\left (-6 x^2+17 x-7\right ) \log (3 x-7)+\left (6 x^2-17 x+7\right ) \log (6)-37 x+17\right )}{1369 (3 x-7)}-\frac {9 (3 x+4) \left (15 x^2+\left (-6 x^2+17 x-7\right ) \log (3 x-7)+\left (6 x^2-17 x+7\right ) \log (6)-37 x+17\right )}{1369 \left (x^2-x+1\right )}+\frac {(-3 x-4) \left (15 x^2+\left (-6 x^2+17 x-7\right ) \log (3 x-7)+\left (6 x^2-17 x+7\right ) \log (6)-37 x+17\right )}{37 \left (x^2-x+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {11}{37} \sqrt {3} \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )+\frac {17 (10-11 x)}{111 \left (x^2-x+1\right )}-\frac {11-x}{3 \left (x^2-x+1\right )}+\frac {5 (11-x) x}{37 \left (x^2-x+1\right )}+\frac {81 x^2 \log (3 x-7)}{1369}+\frac {9 (74+79 \log (6)-21 \log (36)) \log \left (x^2-x+1\right )}{1369}-\frac {9}{37} \log (6) \log \left (x^2-x+1\right )-\frac {45}{74} \log \left (x^2-x+1\right )-\frac {\log (6)}{x^2-x+1}-\frac {81 (1-2 x)^2 \log (3 x-7)}{5476}+\frac {2 \left (-\sqrt {3}+i\right ) \log (7-3 x)}{3 \sqrt {3}+11 i}+\frac {2 \left (\sqrt {3}+i\right ) \log (7-3 x)}{-3 \sqrt {3}+11 i}-\frac {2 \log (7-3 x)}{11+3 i \sqrt {3}}-\frac {2 \log (7-3 x)}{11-3 i \sqrt {3}}+\frac {657 \log (7-3 x)}{5476}-\frac {2 \left (\sqrt {3}+i\right ) \log \left (-2 x-i \sqrt {3}+1\right )}{-3 \sqrt {3}+11 i}+\frac {2 \log \left (-2 x-i \sqrt {3}+1\right )}{11+3 i \sqrt {3}}-\frac {2 \left (-\sqrt {3}+i\right ) \log \left (-2 x+i \sqrt {3}+1\right )}{3 \sqrt {3}+11 i}+\frac {2 \log \left (-2 x+i \sqrt {3}+1\right )}{11-3 i \sqrt {3}}+\frac {27 (7-3 x) \log (3 x-7)}{1369}+\frac {2 \left (1-i \sqrt {3}\right ) \log (3 x-7)}{3 \left (-2 x-i \sqrt {3}+1\right )}-\frac {2 \log (3 x-7)}{3 \left (-2 x-i \sqrt {3}+1\right )}+\frac {2 \left (1+i \sqrt {3}\right ) \log (3 x-7)}{3 \left (-2 x+i \sqrt {3}+1\right )}-\frac {2 \log (3 x-7)}{3 \left (-2 x+i \sqrt {3}+1\right )}\) |
Input:
Int[(17 - 37*x + 15*x^2 + (7 - 17*x + 6*x^2)*Log[6] + (-7 + 17*x - 6*x^2)* Log[-7 + 3*x])/(-7 + 17*x - 27*x^2 + 23*x^3 - 13*x^4 + 3*x^5),x]
Output:
(17*(10 - 11*x))/(111*(1 - x + x^2)) - (11 - x)/(3*(1 - x + x^2)) + (5*(11 - x)*x)/(37*(1 - x + x^2)) + (11*Sqrt[3]*ArcTan[(1 - 2*x)/Sqrt[3]])/37 - Log[6]/(1 - x + x^2) + (657*Log[7 - 3*x])/5476 - (2*Log[7 - 3*x])/(11 - (3 *I)*Sqrt[3]) - (2*Log[7 - 3*x])/(11 + (3*I)*Sqrt[3]) + (2*(I + Sqrt[3])*Lo g[7 - 3*x])/(11*I - 3*Sqrt[3]) + (2*(I - Sqrt[3])*Log[7 - 3*x])/(11*I + 3* Sqrt[3]) + (2*Log[1 - I*Sqrt[3] - 2*x])/(11 + (3*I)*Sqrt[3]) - (2*(I + Sqr t[3])*Log[1 - I*Sqrt[3] - 2*x])/(11*I - 3*Sqrt[3]) + (2*Log[1 + I*Sqrt[3] - 2*x])/(11 - (3*I)*Sqrt[3]) - (2*(I - Sqrt[3])*Log[1 + I*Sqrt[3] - 2*x])/ (11*I + 3*Sqrt[3]) + (27*(7 - 3*x)*Log[-7 + 3*x])/1369 - (81*(1 - 2*x)^2*L og[-7 + 3*x])/5476 - (2*Log[-7 + 3*x])/(3*(1 - I*Sqrt[3] - 2*x)) + (2*(1 - I*Sqrt[3])*Log[-7 + 3*x])/(3*(1 - I*Sqrt[3] - 2*x)) - (2*Log[-7 + 3*x])/( 3*(1 + I*Sqrt[3] - 2*x)) + (2*(1 + I*Sqrt[3])*Log[-7 + 3*x])/(3*(1 + I*Sqr t[3] - 2*x)) + (81*x^2*Log[-7 + 3*x])/1369 - (45*Log[1 - x + x^2])/74 - (9 *Log[6]*Log[1 - x + x^2])/37 + (9*(74 + 79*Log[6] - 21*Log[36])*Log[1 - x + x^2])/1369
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {\ln \left (3 x -7\right )-\ln \left (6\right )-2}{x^{2}-x +1}\) | \(24\) |
risch | \(\frac {\ln \left (3 x -7\right )}{x^{2}-x +1}-\frac {\ln \left (3\right )}{x^{2}-x +1}-\frac {\ln \left (2\right )}{x^{2}-x +1}-\frac {2}{x^{2}-x +1}\) | \(59\) |
derivativedivides | \(-\frac {9 \ln \left (6\right )}{\left (3 x -7\right )^{2}+33 x -40}-\frac {9 \ln \left (3 x -7\right ) \left (3 x -7\right ) \left (4+3 x \right )}{37 \left (\left (3 x -7\right )^{2}+33 x -40\right )}-\frac {18}{\left (3 x -7\right )^{2}+33 x -40}+\frac {9 \ln \left (3 x -7\right )}{37}\) | \(76\) |
default | \(-\frac {9 \ln \left (6\right )}{\left (3 x -7\right )^{2}+33 x -40}-\frac {9 \ln \left (3 x -7\right ) \left (3 x -7\right ) \left (4+3 x \right )}{37 \left (\left (3 x -7\right )^{2}+33 x -40\right )}-\frac {18}{\left (3 x -7\right )^{2}+33 x -40}+\frac {9 \ln \left (3 x -7\right )}{37}\) | \(76\) |
parallelrisch | \(-\frac {-49 \ln \left (x -\frac {7}{3}\right ) x^{2}-49 x^{2} \ln \left (6\right )+49 x^{2} \ln \left (3 x -7\right )+49 \ln \left (x -\frac {7}{3}\right ) x +49 x \ln \left (6\right )-98 x^{2}-49 x \ln \left (3 x -7\right )-49 \ln \left (x -\frac {7}{3}\right )+98 x}{49 \left (x^{2}-x +1\right )}\) | \(76\) |
parts | \(\frac {-37 \ln \left (6\right )-74}{37 x^{2}-37 x +37}-\frac {9 \ln \left (x^{2}-x +1\right )}{74}-\frac {11 \sqrt {3}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {3}}{3}\right )}{37}+\frac {9 \ln \left (3 x -7\right )}{37}+\frac {9 \ln \left (\left (3 x -7\right )^{2}+33 x -40\right )}{74}+\frac {11 \sqrt {3}\, \arctan \left (\frac {\left (-3+6 x \right ) \sqrt {3}}{9}\right )}{37}-\frac {9 \ln \left (3 x -7\right ) \left (3 x -7\right ) \left (4+3 x \right )}{37 \left (\left (3 x -7\right )^{2}+33 x -40\right )}\) | \(118\) |
orering | \(-\frac {\left (15 x^{2}-37 x +17\right ) \left (\left (-6 x^{2}+17 x -7\right ) \ln \left (3 x -7\right )+\left (6 x^{2}-17 x +7\right ) \ln \left (6\right )+15 x^{2}-37 x +17\right )}{\left (12 x -17\right ) \left (3 x^{5}-13 x^{4}+23 x^{3}-27 x^{2}+17 x -7\right )}-\frac {\left (x^{2}-x +1\right ) \left (3 x -7\right ) \left (\frac {\left (-12 x +17\right ) \ln \left (3 x -7\right )+\frac {-18 x^{2}+51 x -21}{3 x -7}+\left (12 x -17\right ) \ln \left (6\right )+30 x -37}{3 x^{5}-13 x^{4}+23 x^{3}-27 x^{2}+17 x -7}-\frac {\left (\left (-6 x^{2}+17 x -7\right ) \ln \left (3 x -7\right )+\left (6 x^{2}-17 x +7\right ) \ln \left (6\right )+15 x^{2}-37 x +17\right ) \left (15 x^{4}-52 x^{3}+69 x^{2}-54 x +17\right )}{\left (3 x^{5}-13 x^{4}+23 x^{3}-27 x^{2}+17 x -7\right )^{2}}\right )}{12 x -17}\) | \(272\) |
Input:
int(((-6*x^2+17*x-7)*ln(3*x-7)+(6*x^2-17*x+7)*ln(6)+15*x^2-37*x+17)/(3*x^5 -13*x^4+23*x^3-27*x^2+17*x-7),x,method=_RETURNVERBOSE)
Output:
(ln(3*x-7)-ln(6)-2)/(x^2-x+1)
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {17-37 x+15 x^2+\left (7-17 x+6 x^2\right ) \log (6)+\left (-7+17 x-6 x^2\right ) \log (-7+3 x)}{-7+17 x-27 x^2+23 x^3-13 x^4+3 x^5} \, dx=-\frac {\log \left (6\right ) - \log \left (3 \, x - 7\right ) + 2}{x^{2} - x + 1} \] Input:
integrate(((-6*x^2+17*x-7)*log(3*x-7)+(6*x^2-17*x+7)*log(6)+15*x^2-37*x+17 )/(3*x^5-13*x^4+23*x^3-27*x^2+17*x-7),x, algorithm="fricas")
Output:
-(log(6) - log(3*x - 7) + 2)/(x^2 - x + 1)
Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {17-37 x+15 x^2+\left (7-17 x+6 x^2\right ) \log (6)+\left (-7+17 x-6 x^2\right ) \log (-7+3 x)}{-7+17 x-27 x^2+23 x^3-13 x^4+3 x^5} \, dx=\frac {\log {\left (3 x - 7 \right )}}{x^{2} - x + 1} + \frac {-2 - \log {\left (6 \right )}}{x^{2} - x + 1} \] Input:
integrate(((-6*x**2+17*x-7)*ln(3*x-7)+(6*x**2-17*x+7)*ln(6)+15*x**2-37*x+1 7)/(3*x**5-13*x**4+23*x**3-27*x**2+17*x-7),x)
Output:
log(3*x - 7)/(x**2 - x + 1) + (-2 - log(6))/(x**2 - x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 255, normalized size of antiderivative = 10.20 \[ \int \frac {17-37 x+15 x^2+\left (7-17 x+6 x^2\right ) \log (6)+\left (-7+17 x-6 x^2\right ) \log (-7+3 x)}{-7+17 x-27 x^2+23 x^3-13 x^4+3 x^5} \, dx=-\frac {7}{24642} \, {\left (2222 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {222 \, {\left (11 \, x - 10\right )}}{x^{2} - x + 1} + 243 \, \log \left (x^{2} - x + 1\right ) - 486 \, \log \left (3 \, x - 7\right )\right )} \log \left (6\right ) - \frac {1}{4107} \, {\left (1754 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {222 \, {\left (10 \, x + 1\right )}}{x^{2} - x + 1} + 1323 \, \log \left (x^{2} - x + 1\right ) - 2646 \, \log \left (3 \, x - 7\right )\right )} \log \left (6\right ) + \frac {17}{24642} \, {\left (1534 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {222 \, {\left (x - 11\right )}}{x^{2} - x + 1} + 567 \, \log \left (x^{2} - x + 1\right ) - 1134 \, \log \left (3 \, x - 7\right )\right )} \log \left (6\right ) - \frac {{\left (9 \, x^{2} - 9 \, x - 28\right )} \log \left (3 \, x - 7\right )}{37 \, {\left (x^{2} - x + 1\right )}} - \frac {17 \, {\left (11 \, x - 10\right )}}{111 \, {\left (x^{2} - x + 1\right )}} + \frac {5 \, {\left (10 \, x + 1\right )}}{37 \, {\left (x^{2} - x + 1\right )}} + \frac {x - 11}{3 \, {\left (x^{2} - x + 1\right )}} + \frac {9}{37} \, \log \left (3 \, x - 7\right ) \] Input:
integrate(((-6*x^2+17*x-7)*log(3*x-7)+(6*x^2-17*x+7)*log(6)+15*x^2-37*x+17 )/(3*x^5-13*x^4+23*x^3-27*x^2+17*x-7),x, algorithm="maxima")
Output:
-7/24642*(2222*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 222*(11*x - 10)/(x^ 2 - x + 1) + 243*log(x^2 - x + 1) - 486*log(3*x - 7))*log(6) - 1/4107*(175 4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 222*(10*x + 1)/(x^2 - x + 1) + 1 323*log(x^2 - x + 1) - 2646*log(3*x - 7))*log(6) + 17/24642*(1534*sqrt(3)* arctan(1/3*sqrt(3)*(2*x - 1)) + 222*(x - 11)/(x^2 - x + 1) + 567*log(x^2 - x + 1) - 1134*log(3*x - 7))*log(6) - 1/37*(9*x^2 - 9*x - 28)*log(3*x - 7) /(x^2 - x + 1) - 17/111*(11*x - 10)/(x^2 - x + 1) + 5/37*(10*x + 1)/(x^2 - x + 1) + 1/3*(x - 11)/(x^2 - x + 1) + 9/37*log(3*x - 7)
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {17-37 x+15 x^2+\left (7-17 x+6 x^2\right ) \log (6)+\left (-7+17 x-6 x^2\right ) \log (-7+3 x)}{-7+17 x-27 x^2+23 x^3-13 x^4+3 x^5} \, dx=-\frac {\log \left (6\right ) + 2}{x^{2} - x + 1} + \frac {\log \left (3 \, x - 7\right )}{x^{2} - x + 1} \] Input:
integrate(((-6*x^2+17*x-7)*log(3*x-7)+(6*x^2-17*x+7)*log(6)+15*x^2-37*x+17 )/(3*x^5-13*x^4+23*x^3-27*x^2+17*x-7),x, algorithm="giac")
Output:
-(log(6) + 2)/(x^2 - x + 1) + log(3*x - 7)/(x^2 - x + 1)
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {17-37 x+15 x^2+\left (7-17 x+6 x^2\right ) \log (6)+\left (-7+17 x-6 x^2\right ) \log (-7+3 x)}{-7+17 x-27 x^2+23 x^3-13 x^4+3 x^5} \, dx=\frac {\ln \left (\frac {x}{2}-\frac {7}{6}\right )-2}{x^2-x+1} \] Input:
int((log(6)*(6*x^2 - 17*x + 7) - 37*x - log(3*x - 7)*(6*x^2 - 17*x + 7) + 15*x^2 + 17)/(17*x - 27*x^2 + 23*x^3 - 13*x^4 + 3*x^5 - 7),x)
Output:
(log(x/2 - 7/6) - 2)/(x^2 - x + 1)
Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {17-37 x+15 x^2+\left (7-17 x+6 x^2\right ) \log (6)+\left (-7+17 x-6 x^2\right ) \log (-7+3 x)}{-7+17 x-27 x^2+23 x^3-13 x^4+3 x^5} \, dx=\frac {\mathrm {log}\left (3 x -7\right )-\mathrm {log}\left (6\right )-2}{x^{2}-x +1} \] Input:
int(((-6*x^2+17*x-7)*log(3*x-7)+(6*x^2-17*x+7)*log(6)+15*x^2-37*x+17)/(3*x ^5-13*x^4+23*x^3-27*x^2+17*x-7),x)
Output:
(log(3*x - 7) - log(6) - 2)/(x**2 - x + 1)