Integrand size = 12, antiderivative size = 110 \[ \int \frac {1}{\sqrt [3]{-6-6 x+x^3}} \, dx=\frac {1}{2} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-6-6 x+x^3}}}{\sqrt {3}}\right )-\frac {\arctan \left (\frac {1-\frac {2 (2+x)}{\sqrt [3]{-6-6 x+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {3}{4} \log \left (x-\sqrt [3]{-6-6 x+x^3}\right )+\frac {1}{4} \log \left (2+x+\sqrt [3]{-6-6 x+x^3}\right ) \] Output:
1/2*3^(1/2)*arctan(1/3*(1+2*x/(x^3-6*x-6)^(1/3))*3^(1/2))-1/6*arctan(1/3*( 1-2*(2+x)/(x^3-6*x-6)^(1/3))*3^(1/2))*3^(1/2)-3/4*ln(x-(x^3-6*x-6)^(1/3))+ 1/4*ln(2+x+(x^3-6*x-6)^(1/3))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.15 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.14 \[ \int \frac {1}{\sqrt [3]{-6-6 x+x^3}} \, dx=-\frac {3 \left (\sqrt [3]{2}+2^{2/3}-x\right ) \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{2}+2^{2/3}-x}{\sqrt [3]{2}+2^{2/3}-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,2\right ]},\frac {\sqrt [3]{2}+2^{2/3}-x}{\sqrt [3]{2}+2^{2/3}-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,3\right ]}\right ) \sqrt [3]{\frac {x-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,2\right ]}{\sqrt [3]{2}+2^{2/3}-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,2\right ]}} \sqrt [3]{\frac {x-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,3\right ]}{\sqrt [3]{2}+2^{2/3}-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,3\right ]}}}{2 \sqrt [3]{-6-6 x+x^3}} \] Input:
Integrate[(-6 - 6*x + x^3)^(-1/3),x]
Output:
(-3*(2^(1/3) + 2^(2/3) - x)*AppellF1[2/3, 1/3, 1/3, 5/3, (2^(1/3) + 2^(2/3 ) - x)/(2^(1/3) + 2^(2/3) - Root[-6 - 6*#1 + #1^3 & , 2, 0]), (2^(1/3) + 2 ^(2/3) - x)/(2^(1/3) + 2^(2/3) - Root[-6 - 6*#1 + #1^3 & , 3, 0])]*((x - R oot[-6 - 6*#1 + #1^3 & , 2, 0])/(2^(1/3) + 2^(2/3) - Root[-6 - 6*#1 + #1^3 & , 2, 0]))^(1/3)*((x - Root[-6 - 6*#1 + #1^3 & , 3, 0])/(2^(1/3) + 2^(2/ 3) - Root[-6 - 6*#1 + #1^3 & , 3, 0]))^(1/3))/(2*(-6 - 6*x + x^3)^(1/3))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.53 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.65, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2475, 1179, 150}
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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{x^3-6 x-6}} \, dx\) |
| \(\Big \downarrow \) 2475 |
| \(\displaystyle \frac {\sqrt [3]{x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )} \sqrt [3]{x^2+\sqrt [3]{2} \left (1+\sqrt [3]{2}\right ) x+2^{2/3}+2 \sqrt [3]{2}-2} \int \frac {1}{\sqrt [3]{x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )} \sqrt [3]{x^2+\sqrt [3]{2} \left (1+\sqrt [3]{2}\right ) x+2^{2/3}+2 \sqrt [3]{2}-2}}dx}{\sqrt [3]{x^3-6 x-6}}\) |
| \(\Big \downarrow \) 1179 |
| \(\displaystyle \frac {\sqrt [3]{1-\frac {2 \left (\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )-x\right )}{3 \left (\sqrt [3]{2}+2^{2/3}\right )-i \sqrt {3 \left (-4+2 \sqrt [3]{2}+2^{2/3}\right )}}} \sqrt [3]{1-\frac {2 \left (\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )-x\right )}{3 \left (\sqrt [3]{2}+2^{2/3}\right )+i \sqrt {3 \left (-4+2 \sqrt [3]{2}+2^{2/3}\right )}}} \sqrt [3]{x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )} \int \frac {1}{\sqrt [3]{x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )} \sqrt [3]{\frac {2 \left (x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )\right )}{3 \left (\sqrt [3]{2}+2^{2/3}\right )-i \sqrt {3 \left (-4+2 \sqrt [3]{2}+2^{2/3}\right )}}+1} \sqrt [3]{\frac {2 \left (x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )\right )}{3 \left (\sqrt [3]{2}+2^{2/3}\right )+i \sqrt {3 \left (-4+2 \sqrt [3]{2}+2^{2/3}\right )}}+1}}d\left (x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )\right )}{\sqrt [3]{x^3-6 x-6}}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {3 \sqrt [3]{1-\frac {2 \left (\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )-x\right )}{3 \left (\sqrt [3]{2}+2^{2/3}\right )-i \sqrt {3 \left (-4+2 \sqrt [3]{2}+2^{2/3}\right )}}} \sqrt [3]{1-\frac {2 \left (\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )-x\right )}{3 \left (\sqrt [3]{2}+2^{2/3}\right )+i \sqrt {3 \left (-4+2 \sqrt [3]{2}+2^{2/3}\right )}}} \left (x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )\right ) \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {2 \left (x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )\right )}{3 \left (\sqrt [3]{2}+2^{2/3}\right )-i \sqrt {3 \left (-4+2 \sqrt [3]{2}+2^{2/3}\right )}},-\frac {2 \left (x-\sqrt [3]{2} \left (1+\sqrt [3]{2}\right )\right )}{3 \left (\sqrt [3]{2}+2^{2/3}\right )+i \sqrt {3 \left (-4+2 \sqrt [3]{2}+2^{2/3}\right )}}\right )}{2 \sqrt [3]{x^3-6 x-6}}\) |
Input:
Int[(-6 - 6*x + x^3)^(-1/3),x]
Output:
(3*(1 - (2*(2^(1/3)*(1 + 2^(1/3)) - x))/(3*(2^(1/3) + 2^(2/3)) - I*Sqrt[3* (-4 + 2*2^(1/3) + 2^(2/3))]))^(1/3)*(1 - (2*(2^(1/3)*(1 + 2^(1/3)) - x))/( 3*(2^(1/3) + 2^(2/3)) + I*Sqrt[3*(-4 + 2*2^(1/3) + 2^(2/3))]))^(1/3)*(-(2^ (1/3)*(1 + 2^(1/3))) + x)*AppellF1[2/3, 1/3, 1/3, 5/3, (-2*(-(2^(1/3)*(1 + 2^(1/3))) + x))/(3*(2^(1/3) + 2^(2/3)) - I*Sqrt[3*(-4 + 2*2^(1/3) + 2^(2/ 3))]), (-2*(-(2^(1/3)*(1 + 2^(1/3))) + x))/(3*(2^(1/3) + 2^(2/3)) + I*Sqrt [3*(-4 + 2*2^(1/3) + 2^(2/3))])])/(2*(-6 - 6*x + x^3)^(1/3))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) ^p) Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d - e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m , p}, x]
Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9* a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Simp[(a + b*x + d*x^3)^p /(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3 )*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/1 8^(1/3))*x + d^2*x^2, x]^p) Int[Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d* (2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; Free Q[{a, b, d, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && !IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.06 (sec) , antiderivative size = 2474, normalized size of antiderivative = 22.49
Input:
int(1/(x^3-6*x-6)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/6*RootOf(_Z^2-_Z+1)*ln(-274356776-44490288*x-542385696*(x^3-6*x-6)^(1/3) +185376200*x^3+467148024*x^2-155716008*x^4-44490288*x^5+975977860*RootOf(_ Z^2-_Z+1)+20391382*x^6+6594445*RootOf(_Z^2-_Z+1)^2*x^6+118700010*RootOf(_Z ^2-_Z+1)^2*x^5+1582666800*RootOf(_Z^2-_Z+1)^2*x-738577840*RootOf(_Z^2-_Z+1 )^2*x^3+642913956*RootOf(_Z^2-_Z+1)*x^4+949600080*RootOf(_Z^2-_Z+1)^2*x^2- 451823156*RootOf(_Z^2-_Z+1)*x^3-1928741868*RootOf(_Z^2-_Z+1)*x^2-286636200 *RootOf(_Z^2-_Z+1)*x-74392657*RootOf(_Z^2-_Z+1)*x^6-316533360*RootOf(_Z^2- _Z+1)^2*x^4+124898964*RootOf(_Z^2-_Z+1)*x^5-13796937*RootOf(_Z^2-_Z+1)^2*( x^3-6*x-6)^(2/3)*x^4+54001275*RootOf(_Z^2-_Z+1)^2*(x^3-6*x-6)^(1/3)*x^5+55 187748*RootOf(_Z^2-_Z+1)^2*(x^3-6*x-6)^(2/3)*x^3-108002550*RootOf(_Z^2-_Z+ 1)^2*(x^3-6*x-6)^(1/3)*x^4-54001275*RootOf(_Z^2-_Z+1)*(x^3-6*x-6)^(2/3)*x^ 4-121799487*RootOf(_Z^2-_Z+1)*(x^3-6*x-6)^(1/3)*x^5-324007650*RootOf(_Z^2- _Z+1)^2*(x^3-6*x-6)^(1/3)*x^3+188411226*RootOf(_Z^2-_Z+1)*(x^3-6*x-6)^(2/3 )*x^3+351601524*RootOf(_Z^2-_Z+1)*(x^3-6*x-6)^(1/3)*x^4-110375496*RootOf(_ Z^2-_Z+1)^2*(x^3-6*x-6)^(2/3)*x+324007650*RootOf(_Z^2-_Z+1)^2*(x^3-6*x-6)^ (1/3)*x^2+55187748*RootOf(_Z^2-_Z+1)*(x^3-6*x-6)^(2/3)*x^2+514791822*RootO f(_Z^2-_Z+1)*(x^3-6*x-6)^(1/3)*x^3+648015300*RootOf(_Z^2-_Z+1)^2*(x^3-6*x- 6)^(1/3)*x-376822452*RootOf(_Z^2-_Z+1)*(x^3-6*x-6)^(2/3)*x-1162807122*Root Of(_Z^2-_Z+1)*(x^3-6*x-6)^(1/3)*x^2-1029583644*RootOf(_Z^2-_Z+1)*(x^3-6*x- 6)^(1/3)*x+67798212*(x^3-6*x-6)^(1/3)*x^5-108002550*(x^3-6*x-6)^(2/3)*x...
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (88) = 176\).
Time = 0.68 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.16 \[ \int \frac {1}{\sqrt [3]{-6-6 x+x^3}} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {855898969178282 \, \sqrt {3} {\left (27198383 \, x^{4} - 135596424 \, x^{3} + 53605784 \, x^{2} + 271192848 \, x - 53605784\right )} {\left (x^{3} - 6 \, x - 6\right )}^{\frac {2}{3}} + 1711797938356564 \, \sqrt {3} {\left (6700723 \, x^{5} + 13796937 \, x^{4} - 94601104 \, x^{3} - 68589194 \, x^{2} + 189202208 \, x + 108793532\right )} {\left (x^{3} - 6 \, x - 6\right )}^{\frac {1}{3}} + \sqrt {3} {\left (17374675894957259856889 \, x^{6} - 46219635606812930309550 \, x^{5} - 116056839525861257663568 \, x^{4} + 207819110061048542429744 \, x^{3} + 348170518577583772990704 \, x^{2} - 137643405802780927149264 \, x - 261759515040609011253968\right )}}{17713176234176547783351 \, x^{6} - 209511611757144982062054 \, x^{5} + 206465108704171390723896 \, x^{4} + 1186216965606163701238920 \, x^{3} - 619395326112514172171688 \, x^{2} - 2089023111465502637944224 \, x - 293038351808176151191760}\right ) - \frac {1}{12} \, \log \left (18 \, x^{5} - 36 \, x^{4} - 180 \, x^{3} + 108 \, x^{2} + 3 \, {\left (x^{4} - 6 \, x^{3} - 8 \, x^{2} + 36 \, x + 44\right )} {\left (x^{3} - 6 \, x - 6\right )}^{\frac {2}{3}} - 3 \, {\left (x^{5} - 22 \, x^{3} - 2 \, x^{2} + 92 \, x + 80\right )} {\left (x^{3} - 6 \, x - 6\right )}^{\frac {1}{3}} + 648 \, x + 436\right ) \] Input:
integrate(1/(x^3-6*x-6)^(1/3),x, algorithm="fricas")
Output:
1/6*sqrt(3)*arctan((855898969178282*sqrt(3)*(27198383*x^4 - 135596424*x^3 + 53605784*x^2 + 271192848*x - 53605784)*(x^3 - 6*x - 6)^(2/3) + 171179793 8356564*sqrt(3)*(6700723*x^5 + 13796937*x^4 - 94601104*x^3 - 68589194*x^2 + 189202208*x + 108793532)*(x^3 - 6*x - 6)^(1/3) + sqrt(3)*(17374675894957 259856889*x^6 - 46219635606812930309550*x^5 - 116056839525861257663568*x^4 + 207819110061048542429744*x^3 + 348170518577583772990704*x^2 - 137643405 802780927149264*x - 261759515040609011253968))/(17713176234176547783351*x^ 6 - 209511611757144982062054*x^5 + 206465108704171390723896*x^4 + 11862169 65606163701238920*x^3 - 619395326112514172171688*x^2 - 2089023111465502637 944224*x - 293038351808176151191760)) - 1/12*log(18*x^5 - 36*x^4 - 180*x^3 + 108*x^2 + 3*(x^4 - 6*x^3 - 8*x^2 + 36*x + 44)*(x^3 - 6*x - 6)^(2/3) - 3 *(x^5 - 22*x^3 - 2*x^2 + 92*x + 80)*(x^3 - 6*x - 6)^(1/3) + 648*x + 436)
\[ \int \frac {1}{\sqrt [3]{-6-6 x+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{3} - 6 x - 6}}\, dx \] Input:
integrate(1/(x**3-6*x-6)**(1/3),x)
Output:
Integral((x**3 - 6*x - 6)**(-1/3), x)
\[ \int \frac {1}{\sqrt [3]{-6-6 x+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - 6 \, x - 6\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(x^3-6*x-6)^(1/3),x, algorithm="maxima")
Output:
integrate((x^3 - 6*x - 6)^(-1/3), x)
\[ \int \frac {1}{\sqrt [3]{-6-6 x+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - 6 \, x - 6\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(x^3-6*x-6)^(1/3),x, algorithm="giac")
Output:
integrate((x^3 - 6*x - 6)^(-1/3), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{-6-6 x+x^3}} \, dx=\int \frac {1}{{\left (x^3-6\,x-6\right )}^{1/3}} \,d x \] Input:
int(1/(x^3 - 6*x - 6)^(1/3),x)
Output:
int(1/(x^3 - 6*x - 6)^(1/3), x)
\[ \int \frac {1}{\sqrt [3]{-6-6 x+x^3}} \, dx=\int \frac {1}{\left (x^{3}-6 x -6\right )^{\frac {1}{3}}}d x \] Input:
int(1/(x^3-6*x-6)^(1/3),x)
Output:
int(1/(x**3 - 6*x - 6)**(1/3),x)