Integrand size = 17, antiderivative size = 81 \[ \int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx=\frac {1}{2} \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 (-1+x)}{\sqrt {3} \sqrt [3]{-5+7 x-3 x^2+x^3}}\right )+\frac {1}{4} \log (1-x)-\frac {3}{4} \log \left (1-x+\sqrt [3]{-5+7 x-3 x^2+x^3}\right ) \]
1/4*ln(1-x)-3/4*ln(1-x+(x^3-3*x^2+7*x-5)^(1/3))+1/2*arctan(1/3*3^(1/2)+2/3 *(-1+x)/(x^3-3*x^2+7*x-5)^(1/3)*3^(1/2))*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx=\frac {3 \sqrt [3]{(2-i)+i x} \sqrt [3]{i (-1+x)} ((-1+2 i)+x) \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {1}{4} i ((-1+2 i)+x),-\frac {1}{2} i ((-1+2 i)+x)\right )}{4 \sqrt [3]{-5+7 x-3 x^2+x^3}} \]
(3*((2 - I) + I*x)^(1/3)*(I*(-1 + x))^(1/3)*((-1 + 2*I) + x)*AppellF1[2/3, 1/3, 1/3, 5/3, (-1/4*I)*((-1 + 2*I) + x), (-1/2*I)*((-1 + 2*I) + x)])/(4* (-5 + 7*x - 3*x^2 + x^3)^(1/3))
Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2481, 1917, 266, 807, 769}
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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{x^3-3 x^2+7 x-5}} \, dx\) |
| \(\Big \downarrow \) 2481 |
| \(\displaystyle \int \frac {1}{\sqrt [3]{(x-1)^3+4 (x-1)}}d(x-1)\) |
| \(\Big \downarrow \) 1917 |
| \(\displaystyle \frac {\sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \int \frac {1}{\sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1}}d(x-1)}{\sqrt [3]{(x-1)^3+4 (x-1)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \int \frac {\sqrt [3]{x-1}}{\sqrt [3]{(x-1)^2+4}}d\sqrt [3]{x-1}}{\sqrt [3]{(x-1)^3+4 (x-1)}}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {3 \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \int \frac {1}{\sqrt [3]{x+3}}d(x-1)^{2/3}}{2 \sqrt [3]{(x-1)^3+4 (x-1)}}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle \frac {3 \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \left (\frac {\arctan \left (\frac {\frac {2 (x-1)^{2/3}}{\sqrt [3]{x+3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{x+3}+1\right )\right )}{2 \sqrt [3]{(x-1)^3+4 (x-1)}}\) |
(3*(4 + (-1 + x)^2)^(1/3)*(-1 + x)^(1/3)*(ArcTan[(1 + (2*(-1 + x)^(2/3))/( 3 + x)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[1 - x + (3 + x)^(1/3)]/2))/(2*(4*(-1 + x) + (-1 + x)^3)^(1/3))
3.1.40.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]) Int[ x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !Integ erQ[p] && NeQ[n, j] && PosQ[n - j]
Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] , c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Subst[Int[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, c/(3 *d) + x]] /; FreeQ[p, x] && PolyQ[Px, x, 3]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.95 (sec) , antiderivative size = 653, normalized size of antiderivative = 8.06
| method | result | size |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-304 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+624 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}+624 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x +608 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +928 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+51 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}-624 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}+51 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x -1856 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -253 x^{2}-51 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}+2356 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+506 x -713\right )}{2}-\frac {\ln \left (-304 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-624 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}-624 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x +608 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -320 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}+624 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}+675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x +640 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +371 x^{2}-675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}-2356 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-742 x +1643\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{2}+\frac {\ln \left (-304 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-624 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}-624 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x +608 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -320 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}+624 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}+675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x +640 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +371 x^{2}-675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}-2356 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-742 x +1643\right )}{2}\) | \(653\) |
1/2*RootOf(_Z^2-_Z+1)*ln(-304*RootOf(_Z^2-_Z+1)^2*x^2+624*RootOf(_Z^2-_Z+1 )*(x^3-3*x^2+7*x-5)^(2/3)+624*RootOf(_Z^2-_Z+1)*(x^3-3*x^2+7*x-5)^(1/3)*x+ 608*RootOf(_Z^2-_Z+1)^2*x+928*RootOf(_Z^2-_Z+1)*x^2+51*(x^3-3*x^2+7*x-5)^( 2/3)-624*RootOf(_Z^2-_Z+1)*(x^3-3*x^2+7*x-5)^(1/3)+51*(x^3-3*x^2+7*x-5)^(1 /3)*x-1856*RootOf(_Z^2-_Z+1)*x-253*x^2-51*(x^3-3*x^2+7*x-5)^(1/3)+2356*Roo tOf(_Z^2-_Z+1)+506*x-713)-1/2*ln(-304*RootOf(_Z^2-_Z+1)^2*x^2-624*RootOf(_ Z^2-_Z+1)*(x^3-3*x^2+7*x-5)^(2/3)-624*RootOf(_Z^2-_Z+1)*(x^3-3*x^2+7*x-5)^ (1/3)*x+608*RootOf(_Z^2-_Z+1)^2*x-320*RootOf(_Z^2-_Z+1)*x^2+675*(x^3-3*x^2 +7*x-5)^(2/3)+624*RootOf(_Z^2-_Z+1)*(x^3-3*x^2+7*x-5)^(1/3)+675*(x^3-3*x^2 +7*x-5)^(1/3)*x+640*RootOf(_Z^2-_Z+1)*x+371*x^2-675*(x^3-3*x^2+7*x-5)^(1/3 )-2356*RootOf(_Z^2-_Z+1)-742*x+1643)*RootOf(_Z^2-_Z+1)+1/2*ln(-304*RootOf( _Z^2-_Z+1)^2*x^2-624*RootOf(_Z^2-_Z+1)*(x^3-3*x^2+7*x-5)^(2/3)-624*RootOf( _Z^2-_Z+1)*(x^3-3*x^2+7*x-5)^(1/3)*x+608*RootOf(_Z^2-_Z+1)^2*x-320*RootOf( _Z^2-_Z+1)*x^2+675*(x^3-3*x^2+7*x-5)^(2/3)+624*RootOf(_Z^2-_Z+1)*(x^3-3*x^ 2+7*x-5)^(1/3)+675*(x^3-3*x^2+7*x-5)^(1/3)*x+640*RootOf(_Z^2-_Z+1)*x+371*x ^2-675*(x^3-3*x^2+7*x-5)^(1/3)-2356*RootOf(_Z^2-_Z+1)-742*x+1643)
Time = 0.40 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx=-\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {22791076 \, \sqrt {3} {\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (20389537 \, x^{2} - 40779074 \, x + 53222437\right )} + 17987998 \, \sqrt {3} {\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {2}{3}}}{7204617 \, x^{2} - 14409234 \, x - 20666867}\right ) - \frac {1}{4} \, \log \left (3 \, {\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 3 \, {\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {2}{3}} + 4\right ) \]
-1/2*sqrt(3)*arctan((22791076*sqrt(3)*(x^3 - 3*x^2 + 7*x - 5)^(1/3)*(x - 1 ) + sqrt(3)*(20389537*x^2 - 40779074*x + 53222437) + 17987998*sqrt(3)*(x^3 - 3*x^2 + 7*x - 5)^(2/3))/(7204617*x^2 - 14409234*x - 20666867)) - 1/4*lo g(3*(x^3 - 3*x^2 + 7*x - 5)^(1/3)*(x - 1) - 3*(x^3 - 3*x^2 + 7*x - 5)^(2/3 ) + 4)
\[ \int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{3} - 3 x^{2} + 7 x - 5}}\, dx \]
\[ \int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx=\int \frac {1}{{\left (x^3-3\,x^2+7\,x-5\right )}^{1/3}} \,d x \]