Integrand size = 32, antiderivative size = 203 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-b+a x^6\right )}{-d+c x^6} \, dx=\frac {a x \sqrt [3]{-x+x^3}}{2 c}+\frac {a \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{2 \sqrt {3} c}+\frac {a \log \left (-x+\sqrt [3]{-x+x^3}\right )}{6 c}-\frac {a \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right )}{12 c}+\frac {(b c-a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]}{6 c d} \]
Time = 0.00 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-b+a x^6\right )}{-d+c x^6} \, dx=\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \left (a d \left (6 x^{4/3} \sqrt [3]{-1+x^2}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-\log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right )+2 (b c-a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]\right )}{12 c d \left (x \left (-1+x^2\right )\right )^{2/3}} \]
(x^(2/3)*(-1 + x^2)^(2/3)*(a*d*(6*x^(4/3)*(-1 + x^2)^(1/3) + 2*Sqrt[3]*Arc Tan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(-1 + x^2)^(1/3))] + 2*Log[-x^(2/3) + ( -1 + x^2)^(1/3)] - Log[x^(4/3) + x^(2/3)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/ 3)]) + 2*(b*c - a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-2* Log[x^(1/3)]*#1 + Log[(-1 + x^2)^(1/3) - x^(2/3)*#1]*#1)/(-1 + #1^3) & ])) /(12*c*d*(x*(-1 + x^2))^(2/3))
Leaf count is larger than twice the leaf count of optimal. \(1176\) vs. \(2(203)=406\).
Time = 3.77 (sec) , antiderivative size = 1176, normalized size of antiderivative = 5.79, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2467, 2035, 7276, 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.
| \(\displaystyle \int \frac {\sqrt [3]{x^3-x} \left (a x^6-b\right )}{c x^6-d} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x} \int \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \left (b-a x^6\right )}{d-c x^6}dx}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x^3-x} \int \frac {x \sqrt [3]{x^2-1} \left (b-a x^6\right )}{d-c x^6}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x^3-x} \int \left (\frac {a \sqrt [3]{x^2-1} x}{c}+\frac {(b c-a d) \sqrt [3]{x^2-1} x}{c \left (d-c x^6\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x^3-x} \left (\frac {a \sqrt [3]{x^2-1} x^{4/3}}{6 c}+\frac {(-1)^{2/3} (b c-a d) \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} c^{4/3} d^{2/3}}-\frac {\sqrt [3]{-1} (b c-a d) \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} c^{4/3} d^{2/3}}+\frac {(b c-a d) \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} c^{4/3} d^{2/3}}+\frac {a \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} c}+\frac {\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{18 \sqrt {3} c^{4/3} d^{7/9}}-\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} c^{4/3} d^{7/9}}+\frac {\sqrt [3]{-1} \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} c^{4/3} d^{7/9}}-\frac {\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} (b c-a d) \log \left (\sqrt [3]{d}-\sqrt [3]{c} x^2\right )}{108 c^{4/3} d^{7/9}}-\frac {\sqrt [3]{-1} \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} (b c-a d) \log \left (\sqrt [3]{c} x^2+\sqrt [3]{-1} \sqrt [3]{d}\right )}{108 c^{4/3} d^{7/9}}+\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} (b c-a d) \log \left (\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right )}{108 c^{4/3} d^{7/9}}+\frac {(-1)^{2/3} (b c-a d) \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{36 c^{4/3} d^{2/3}}-\frac {\sqrt [3]{-1} (b c-a d) \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{36 c^{4/3} d^{2/3}}+\frac {(b c-a d) \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{36 c^{4/3} d^{2/3}}+\frac {a \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{12 c}-\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} x^{2/3}-\sqrt [9]{d} \sqrt [3]{x^2-1}\right )}{36 c^{4/3} d^{7/9}}+\frac {\sqrt [3]{-1} \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} x^{2/3}-\sqrt [9]{d} \sqrt [3]{x^2-1}\right )}{36 c^{4/3} d^{7/9}}+\frac {\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}+\sqrt [9]{d} \sqrt [3]{x^2-1}\right )}{36 c^{4/3} d^{7/9}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\) |
(3*(-x + x^3)^(1/3)*((a*x^(4/3)*(-1 + x^2)^(1/3))/(6*c) + (a*ArcTan[(1 + ( 2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c) + ((b*c - a*d)*ArcTan [(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(18*Sqrt[3]*c^(4/3)*d^(2/3)) - ((-1)^(1/3)*(b*c - a*d)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[ 3]])/(18*Sqrt[3]*c^(4/3)*d^(2/3)) + ((-1)^(2/3)*(b*c - a*d)*ArcTan[(1 + (2 *x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(18*Sqrt[3]*c^(4/3)*d^(2/3)) + ((c^( 1/3) - d^(1/3))^(1/3)*(b*c - a*d)*ArcTan[(1 - (2*(c^(1/3) - d^(1/3))^(1/3) *x^(2/3))/(d^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(18*Sqrt[3]*c^(4/3)*d^(7/9 )) - ((-1)^(2/3)*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*(b*c - a*d)*ArcTan[( 1 + (2*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(-1 + x^2)^( 1/3)))/Sqrt[3]])/(18*Sqrt[3]*c^(4/3)*d^(7/9)) + ((-1)^(1/3)*(-((-1)^(2/3)* c^(1/3)) + d^(1/3))^(1/3)*(b*c - a*d)*ArcTan[(1 + (2*(-((-1)^(2/3)*c^(1/3) ) + d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(18*Sqrt [3]*c^(4/3)*d^(7/9)) - ((c^(1/3) - d^(1/3))^(1/3)*(b*c - a*d)*Log[d^(1/3) - c^(1/3)*x^2])/(108*c^(4/3)*d^(7/9)) - ((-1)^(1/3)*(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/3)*(b*c - a*d)*Log[(-1)^(1/3)*d^(1/3) + c^(1/3)*x^2])/(108* c^(4/3)*d^(7/9)) + ((-1)^(2/3)*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*(b*c - a*d)*Log[-((-1)^(2/3)*d^(1/3)) + c^(1/3)*x^2])/(108*c^(4/3)*d^(7/9)) + (a *Log[x^(2/3) - (-1 + x^2)^(1/3)])/(12*c) + ((b*c - a*d)*Log[x^(2/3) - (-1 + x^2)^(1/3)])/(36*c^(4/3)*d^(2/3)) - ((-1)^(1/3)*(b*c - a*d)*Log[x^(2/...
3.25.75.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.00 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(\frac {x \left (\left (-a d +b c \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 d \,\textit {\_Z}^{6}+3 d \,\textit {\_Z}^{3}+c -d \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right ) \textit {\_R}}{\textit {\_R}^{3}-1}\right )+a d \left (-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )+3 x \left (x^{3}-x \right )^{\frac {1}{3}}+\ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )\right )}{6 d \left (\left (x^{3}-x \right )^{\frac {2}{3}}+x \left (x +\left (x^{3}-x \right )^{\frac {1}{3}}\right )\right ) \left (-\left (x^{3}-x \right )^{\frac {1}{3}}+x \right ) c}\) | \(206\) |
1/6*x*((-a*d+b*c)*sum(ln((-_R*x+(x^3-x)^(1/3))/x)*_R/(_R^3-1),_R=RootOf(_Z ^9*d-3*_Z^6*d+3*_Z^3*d+c-d))+a*d*(-3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^3- x)^(1/3)))+3*x*(x^3-x)^(1/3)+ln((-x+(x^3-x)^(1/3))/x)-1/2*ln((x^2+x*(x^3-x )^(1/3)+(x^3-x)^(2/3))/x^2)))/d/((x^3-x)^(2/3)+x*(x+(x^3-x)^(1/3)))/(-(x^3 -x)^(1/3)+x)/c
Exception generated. \[ \int \frac {\sqrt [3]{-x+x^3} \left (-b+a x^6\right )}{-d+c x^6} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
Timed out. \[ \int \frac {\sqrt [3]{-x+x^3} \left (-b+a x^6\right )}{-d+c x^6} \, dx=\text {Timed out} \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-b+a x^6\right )}{-d+c x^6} \, dx=\int { \frac {{\left (a x^{6} - b\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{c x^{6} - d} \,d x } \]
Not integrable
Time = 3.20 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.01 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-b+a x^6\right )}{-d+c x^6} \, dx=\int { \frac {{\left (a x^{6} - b\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{c x^{6} - d} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.15 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-b+a x^6\right )}{-d+c x^6} \, dx=\int \frac {{\left (x^3-x\right )}^{1/3}\,\left (b-a\,x^6\right )}{d-c\,x^6} \,d x \]