Integrand size = 37, antiderivative size = 459 \[ \int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=-\frac {\arctan \left (\frac {\left (\sqrt {3}+\sqrt {3} x\right ) \left (1-x+x^2\right )}{1+x^3+\sqrt [3]{2} \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{3\ 2^{2/3}}+\frac {\log \left (1+2 x^3+x^6\right )}{6\ 2^{2/3}}+\frac {\log \left (-2-2 x^3+\sqrt [3]{2} \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}\right )}{3\ 2^{2/3}}-\frac {\log \left (4+8 x^3+4 x^6+\left (2 \sqrt [3]{2}+2 \sqrt [3]{2} x^3\right ) \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}+2^{2/3} \left (1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}\right )^{2/3}\right )}{6\ 2^{2/3}}+\frac {1}{3} \text {RootSum}\left [7+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {5 \log \left (1+x^3\right )-5 \log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right )+\log \left (1+x^3\right ) \text {$\#$1}^3-\log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^3}{\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
Time = 0.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \left (\sqrt [3]{2} \left (2 \sqrt {3} \arctan \left (\frac {1-\sqrt [3]{-2+6 x}}{\sqrt {3}}\right )+2 \log \left (2+\sqrt [3]{-2+6 x}\right )-\log \left (4-2 \sqrt [3]{-2+6 x}+(-2+6 x)^{2/3}\right )\right )+4 \text {RootSum}\left [7-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-5 \log \left (\sqrt [3]{-1+3 x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{-1+3 x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ]\right )}{12 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}} \]
-1/12*((-1 + 3*x)^(1/3)*(1 + x^3)*(2^(1/3)*(2*Sqrt[3]*ArcTan[(1 - (-2 + 6* x)^(1/3))/Sqrt[3]] + 2*Log[2 + (-2 + 6*x)^(1/3)] - Log[4 - 2*(-2 + 6*x)^(1 /3) + (-2 + 6*x)^(2/3)]) + 4*RootSum[7 - #1^3 + #1^6 & , (-5*Log[(-1 + 3*x )^(1/3) - #1] + Log[(-1 + 3*x)^(1/3) - #1]*#1^3)/(-#1 + 2*#1^4) & ]))/(-(( -1 + 3*x)*(1 + x^3)^3))^(1/3)
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 613, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {7239, 7270, 7267, 27, 2462, 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.
\(\displaystyle \int \frac {1}{\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {1}{\sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}}dx\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle \frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \int \frac {1}{\sqrt [3]{3 x-1} \left (x^3+1\right )}dx}{\sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle \frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \int \frac {27 \sqrt [3]{3 x-1}}{27 x^3+27}d\sqrt [3]{3 x-1}}{\sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {27 \sqrt [3]{3 x-1} \left (x^3+1\right ) \int \frac {\sqrt [3]{3 x-1}}{27 x^3+27}d\sqrt [3]{3 x-1}}{\sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \frac {27 \sqrt [3]{3 x-1} \left (x^3+1\right ) \int \left (\frac {\sqrt [3]{3 x-1}}{27 (3 x+3)}-\frac {(3 x-6) \sqrt [3]{3 x-1}}{27 \left ((3 x-1)^2-3 x+8\right )}\right )d\sqrt [3]{3 x-1}}{\sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {27 \sqrt [3]{3 x-1} \left (x^3+1\right ) \left (-\frac {\arctan \left (\frac {1-\sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt {3}}\right )}{27\ 2^{2/3} \sqrt {3}}+\frac {\left (-\sqrt {3}+3 i\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1-3 i \sqrt {3}}}}{\sqrt {3}}\right )}{81\ 2^{2/3} \sqrt [3]{1-3 i \sqrt {3}}}-\frac {\left (\sqrt {3}+3 i\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1+3 i \sqrt {3}}}}{\sqrt {3}}\right )}{81\ 2^{2/3} \sqrt [3]{1+3 i \sqrt {3}}}-\frac {\log \left (\sqrt [3]{3 x-1}+2^{2/3}\right )}{81\ 2^{2/3}}-\frac {\left (1-i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{3 x-1}+\sqrt [3]{1-3 i \sqrt {3}}\right )}{81\ 2^{2/3} \sqrt [3]{1-3 i \sqrt {3}}}-\frac {\left (1+i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{3 x-1}+\sqrt [3]{1+3 i \sqrt {3}}\right )}{81\ 2^{2/3} \sqrt [3]{1+3 i \sqrt {3}}}+\frac {\log \left ((3 x-1)^{2/3}-2^{2/3} \sqrt [3]{3 x-1}+2 \sqrt [3]{2}\right )}{162\ 2^{2/3}}+\frac {\left (1-i \sqrt {3}\right ) \log \left (2^{2/3} (3 x-1)^{2/3}+\sqrt [3]{2 \left (1-3 i \sqrt {3}\right )} \sqrt [3]{3 x-1}+\left (1-3 i \sqrt {3}\right )^{2/3}\right )}{162\ 2^{2/3} \sqrt [3]{1-3 i \sqrt {3}}}+\frac {\left (1+i \sqrt {3}\right ) \log \left (2^{2/3} (3 x-1)^{2/3}+\sqrt [3]{2 \left (1+3 i \sqrt {3}\right )} \sqrt [3]{3 x-1}+\left (1+3 i \sqrt {3}\right )^{2/3}\right )}{162\ 2^{2/3} \sqrt [3]{1+3 i \sqrt {3}}}\right )}{\sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}\) |
(27*(-1 + 3*x)^(1/3)*(1 + x^3)*(-1/27*ArcTan[(1 - 2^(1/3)*(-1 + 3*x)^(1/3) )/Sqrt[3]]/(2^(2/3)*Sqrt[3]) + ((3*I - Sqrt[3])*ArcTan[(1 + (2*2^(1/3)*(-1 + 3*x)^(1/3))/(1 - (3*I)*Sqrt[3])^(1/3))/Sqrt[3]])/(81*2^(2/3)*(1 - (3*I) *Sqrt[3])^(1/3)) - ((3*I + Sqrt[3])*ArcTan[(1 + (2*2^(1/3)*(-1 + 3*x)^(1/3 ))/(1 + (3*I)*Sqrt[3])^(1/3))/Sqrt[3]])/(81*2^(2/3)*(1 + (3*I)*Sqrt[3])^(1 /3)) - Log[2^(2/3) + (-1 + 3*x)^(1/3)]/(81*2^(2/3)) - ((1 - I*Sqrt[3])*Log [(1 - (3*I)*Sqrt[3])^(1/3) - 2^(1/3)*(-1 + 3*x)^(1/3)])/(81*2^(2/3)*(1 - ( 3*I)*Sqrt[3])^(1/3)) - ((1 + I*Sqrt[3])*Log[(1 + (3*I)*Sqrt[3])^(1/3) - 2^ (1/3)*(-1 + 3*x)^(1/3)])/(81*2^(2/3)*(1 + (3*I)*Sqrt[3])^(1/3)) + Log[2*2^ (1/3) - 2^(2/3)*(-1 + 3*x)^(1/3) + (-1 + 3*x)^(2/3)]/(162*2^(2/3)) + ((1 - I*Sqrt[3])*Log[(1 - (3*I)*Sqrt[3])^(2/3) + (2*(1 - (3*I)*Sqrt[3]))^(1/3)* (-1 + 3*x)^(1/3) + 2^(2/3)*(-1 + 3*x)^(2/3)])/(162*2^(2/3)*(1 - (3*I)*Sqrt [3])^(1/3)) + ((1 + I*Sqrt[3])*Log[(1 + (3*I)*Sqrt[3])^(2/3) + (2*(1 + (3* I)*Sqrt[3]))^(1/3)*(-1 + 3*x)^(1/3) + 2^(2/3)*(-1 + 3*x)^(2/3)])/(162*2^(2 /3)*(1 + (3*I)*Sqrt[3])^(1/3))))/((1 - 3*x)*(1 + x^3)^3)^(1/3)
3.31.48.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Time = 15.05 (sec) , antiderivative size = 25543, normalized size of antiderivative = 55.65
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\text {Too large to display} \]
1/84*14^(2/3)*(3*I*sqrt(3) - 1)^(1/3)*(sqrt(-3) - 1)*log(-(14^(1/3)*(2*x^3 + 2*sqrt(-3)*(x^3 + 1) - sqrt(3)*(I*x^3 + I*sqrt(-3)*(x^3 + 1) + I) + 2)* (3*I*sqrt(3) - 1)^(2/3) - 28*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^ 3 - 3*x + 1)^(1/3))/(x^3 + 1)) - 1/84*14^(2/3)*(3*I*sqrt(3) - 1)^(1/3)*(sq rt(-3) + 1)*log(-(14^(1/3)*(2*x^3 - 2*sqrt(-3)*(x^3 + 1) - sqrt(3)*(I*x^3 - I*sqrt(-3)*(x^3 + 1) + I) + 2)*(3*I*sqrt(3) - 1)^(2/3) - 28*(-3*x^10 + x ^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3))/(x^3 + 1)) + 1/84*14^ (2/3)*(-3*I*sqrt(3) - 1)^(1/3)*(sqrt(-3) - 1)*log(-(14^(1/3)*(2*x^3 + 2*sq rt(-3)*(x^3 + 1) - sqrt(3)*(-I*x^3 - I*sqrt(-3)*(x^3 + 1) - I) + 2)*(-3*I* sqrt(3) - 1)^(2/3) - 28*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3 *x + 1)^(1/3))/(x^3 + 1)) - 1/84*14^(2/3)*(-3*I*sqrt(3) - 1)^(1/3)*(sqrt(- 3) + 1)*log(-(14^(1/3)*(2*x^3 - 2*sqrt(-3)*(x^3 + 1) - sqrt(3)*(-I*x^3 + I *sqrt(-3)*(x^3 + 1) - I) + 2)*(-3*I*sqrt(3) - 1)^(2/3) - 28*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3))/(x^3 + 1)) + 1/42*14^(2 /3)*(3*I*sqrt(3) - 1)^(1/3)*log((14^(1/3)*(2*x^3 + sqrt(3)*(-I*x^3 - I) + 2)*(3*I*sqrt(3) - 1)^(2/3) + 14*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3 *x^3 - 3*x + 1)^(1/3))/(x^3 + 1)) + 1/42*14^(2/3)*(-3*I*sqrt(3) - 1)^(1/3) *log((14^(1/3)*(2*x^3 + sqrt(3)*(I*x^3 + I) + 2)*(-3*I*sqrt(3) - 1)^(2/3) + 14*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3))/(x^3 + 1)) + 1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*(4^(1/3)*sqrt(3)*(x^3 +...
Not integrable
Time = 0.68 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.08 \[ \int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\int \frac {1}{\sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}}\, dx \]
Not integrable
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.08 \[ \int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\int { \frac {1}{{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.08 \[ \int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\int { \frac {1}{{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 7.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.08 \[ \int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\int \frac {1}{{\left (-3\,x^{10}+x^9-9\,x^7+3\,x^6-9\,x^4+3\,x^3-3\,x+1\right )}^{1/3}} \,d x \]