Integrand size = 21, antiderivative size = 233 \[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-q+3 q x+(-1-2 q) x^2+x^3}}{-2 \sqrt [3]{q}+2 \sqrt [3]{q} x+\sqrt [3]{-q+3 q x+(-1-2 q) x^2+x^3}}\right )}{2 \sqrt [3]{q}}-\frac {\log \left (\sqrt [3]{q}-\sqrt [3]{q} x+\sqrt [3]{-q+3 q x+(-1-2 q) x^2+x^3}\right )}{2 \sqrt [3]{q}}+\frac {\log \left (q^{2/3}-2 q^{2/3} x+q^{2/3} x^2+\left (-\sqrt [3]{q}+\sqrt [3]{q} x\right ) \sqrt [3]{-q+3 q x+(-1-2 q) x^2+x^3}+\left (-q+3 q x+(-1-2 q) x^2+x^3\right )^{2/3}\right )}{4 \sqrt [3]{q}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*(-q+3*q*x+(-1-2*q)*x^2+x^3)^(1/3)/(-2*q^(1/3)+ 2*q^(1/3)*x+(-q+3*q*x+(-1-2*q)*x^2+x^3)^(1/3)))/q^(1/3)-1/2*ln(q^(1/3)-q^( 1/3)*x+(-q+3*q*x+(-1-2*q)*x^2+x^3)^(1/3))/q^(1/3)+1/4*ln(q^(2/3)-2*q^(2/3) *x+q^(2/3)*x^2+(-q^(1/3)+q^(1/3)*x)*(-q+3*q*x+(-1-2*q)*x^2+x^3)^(1/3)+(-q+ 3*q*x+(-1-2*q)*x^2+x^3)^(2/3))/q^(1/3)
Time = 2.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\frac {\sqrt [3]{-1+x} \sqrt [3]{q-2 q x+x^2} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{q} (-1+x)^{2/3}}{\sqrt [3]{q} (-1+x)^{2/3}+2 \sqrt [3]{q-2 q x+x^2}}\right )-2 \log \left (-\sqrt [3]{q} (-1+x)^{2/3}+\sqrt [3]{q-2 q x+x^2}\right )+\log \left (q^{2/3} (-1+x)^{4/3}+\sqrt [3]{q} (-1+x)^{2/3} \sqrt [3]{q-2 q x+x^2}+\left (q-2 q x+x^2\right )^{2/3}\right )\right )}{4 \sqrt [3]{q} \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \]
((-1 + x)^(1/3)*(q - 2*q*x + x^2)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*q^(1/3) *(-1 + x)^(2/3))/(q^(1/3)*(-1 + x)^(2/3) + 2*(q - 2*q*x + x^2)^(1/3))] - 2 *Log[-(q^(1/3)*(-1 + x)^(2/3)) + (q - 2*q*x + x^2)^(1/3)] + Log[q^(2/3)*(- 1 + x)^(4/3) + q^(1/3)*(-1 + x)^(2/3)*(q - 2*q*x + x^2)^(1/3) + (q - 2*q*x + x^2)^(2/3)]))/(4*q^(1/3)*((-1 + x)*(q - 2*q*x + x^2))^(1/3))
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.
\(\displaystyle \int \frac {1}{x \sqrt [3]{(x-1) \left (-2 q x+q+x^2\right )}} \, dx\) |
\(\Big \downarrow \) 2490 |
\(\displaystyle \int \frac {1}{\left (\frac {1}{3} (-2 q-1)+\frac {1}{3} (2 q+1)+x\right ) \sqrt [3]{\left (\frac {1}{3} (-2 q-1)+x\right )^3-\frac {1}{3} (1-4 q) (1-q) \left (\frac {1}{3} (-2 q-1)+x\right )-\frac {2}{27} (1-q)^2 (8 q+1)}}d\left (\frac {1}{3} (-2 q-1)+x\right )\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {1}{\left (\frac {1}{3} (-2 q-1)+\frac {1}{3} (2 q+1)+x\right ) \sqrt [3]{\left (\frac {1}{3} (-2 q-1)+x\right )^3-\frac {1}{3} (1-4 q) (1-q) \left (\frac {1}{3} (-2 q-1)+x\right )-\frac {2}{27} (1-q)^2 (8 q+1)}}d\left (\frac {1}{3} (-2 q-1)+x\right )\) |
3.27.35.3.1 Defintions of rubi rules used
Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 , x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 *d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c , 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
\[\int \frac {1}{x {\left (\left (-1+x \right ) \left (-2 q x +x^{2}+q \right )\right )}^{\frac {1}{3}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (194) = 388\).
Time = 14.40 (sec) , antiderivative size = 1496, normalized size of antiderivative = 6.42 \[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\text {Too large to display} \]
[1/12*(sqrt(3)*q*sqrt((-q)^(1/3)/q)*log(-((q^3 - 30*q^2 - 51*q - 1)*x^6 + 54*(q^3 + 6*q^2 + 2*q)*x^5 - 27*(17*q^3 + 26*q^2 + 2*q)*x^4 + 486*q^3*x + 540*(2*q^3 + q^2)*x^3 - 81*q^3 - 135*(8*q^3 + q^2)*x^2 + 9*((2*q^2 - q - 1 )*x^4 - 6*(q^2 - q)*x^3 + 3*(q^2 - q)*x^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(1/3) + 9*((q^2 + 7*q + 1)*x^5 - (19*q^2 + 25*q + 1)*x^4 + 9*(7*q^2 + 3*q)*x^3 + 45*q^2*x - 9*(9*q^2 + q)*x^2 - 9*q^2)*(-(2*q + 1)*x^ 2 + x^3 + 3*q*x - q)^(1/3)*(-q)^(2/3) + sqrt(3)*(3*((4*q^2 + 13*q + 1)*x^4 - 6*(7*q^2 + 5*q)*x^3 - 72*q^2*x + 3*(31*q^2 + 5*q)*x^2 + 18*q^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(2/3) + 3*((q^3 - 5*q^2 - 5*q)*x^5 + 5*(q^3 + 7*q^2 + q)*x^4 - 45*q^3*x - 45*(q^3 + q^2)*x^3 + 9*q^3 + 15*(5* q^3 + q^2)*x^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(1/3) + ((q^3 + 24*q^2 + 3*q - 1)*x^6 - 54*(q^3 + 2*q^2)*x^5 + 81*(3*q^3 + 2*q^2)*x^4 - 162*q^3*x - 108*(4*q^3 + q^2)*x^3 + 27*q^3 + 27*(14*q^3 + q^2)*x^2)*(-q)^(1/3))*sqr t((-q)^(1/3)/q))/x^6) - 2*(-q)^(2/3)*log(((-q)^(2/3)*(q - 1)*x^2 + 3*(-(2* q + 1)*x^2 + x^3 + 3*q*x - q)^(1/3)*(q*x - q)*(-q)^(1/3) + 3*(-(2*q + 1)*x ^2 + x^3 + 3*q*x - q)^(2/3)*q)/x^2) + (-q)^(2/3)*log((3*((2*q + 1)*x^2 - 6 *q*x + 3*q)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(2/3) + 3*((q^2 + 2*q)*x^3 + 9*q^2*x - (7*q^2 + 2*q)*x^2 - 3*q^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(1/3) - ((q^2 + 7*q + 1)*x^4 - 18*(q^2 + q)*x^3 - 36*q^2*x + 9* (5*q^2 + q)*x^2 + 9*q^2)*(-q)^(1/3))/x^4))/q, 1/12*(2*sqrt(3)*q*sqrt(-(...
\[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\int \frac {1}{x \sqrt [3]{\left (x - 1\right ) \left (- 2 q x + q + x^{2}\right )}}\, dx \]
\[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\int { \frac {1}{\left (-{\left (2 \, q x - x^{2} - q\right )} {\left (x - 1\right )}\right )^{\frac {1}{3}} x} \,d x } \]
\[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\int { \frac {1}{\left (-{\left (2 \, q x - x^{2} - q\right )} {\left (x - 1\right )}\right )^{\frac {1}{3}} x} \,d x } \]
Timed out. \[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\int \frac {1}{x\,{\left (\left (x-1\right )\,\left (x^2-2\,q\,x+q\right )\right )}^{1/3}} \,d x \]