Integrand size = 19, antiderivative size = 194 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=-\frac {1}{8} \arctan \left (\frac {\frac {1}{\sqrt {3}}+x-\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {1}{8} \arctan \left (\frac {-\frac {1}{\sqrt {3}}+x+\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{1+2 \sqrt [3]{1-3 x^2}}\right )}{4 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {-2 \sqrt {3} x+2 \sqrt {3} x \sqrt [3]{1-3 x^2}}{1+3 x^2-2 \sqrt [3]{1-3 x^2}+4 \left (1-3 x^2\right )^{2/3}}\right )}{8 \sqrt {3}} \] Output:
-1/8*arctan((1/3*3^(1/2)+x-1/3*(-3*x^2+1)^(1/3)*3^(1/2))/(-3*x^2+1)^(1/3)) -1/8*arctan((-1/3*3^(1/2)+x+1/3*(-3*x^2+1)^(1/3)*3^(1/2))/(-3*x^2+1)^(1/3) )-1/12*arctanh(3^(1/2)*x/(1+2*(-3*x^2+1)^(1/3)))*3^(1/2)-1/24*arctanh((-2* x*3^(1/2)+2*(-3*x^2+1)^(1/3)*3^(1/2)*x)/(1+3*x^2-2*(-3*x^2+1)^(1/3)+4*(-3* x^2+1)^(2/3)))*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 4.53 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )\right )\right )} \] Input:
Integrate[1/((1 - 3*x^2)^(1/3)*(-3 + x^2)),x]
Output:
(9*x*AppellF1[1/2, 1/3, 1, 3/2, 3*x^2, x^2/3])/((1 - 3*x^2)^(1/3)*(-3 + x^ 2)*(9*AppellF1[1/2, 1/3, 1, 3/2, 3*x^2, x^2/3] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, 3*x^2, x^2/3] + 3*AppellF1[3/2, 4/3, 1, 5/2, 3*x^2, x^2/3])))
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.42, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {307}
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.
\(\displaystyle \int \frac {1}{\sqrt [3]{1-3 x^2} \left (x^2-3\right )} \, dx\) |
\(\Big \downarrow \) 307 |
\(\displaystyle -\frac {1}{4} \arctan \left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )+\frac {\text {arctanh}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}\) |
Input:
Int[1/((1 - 3*x^2)^(1/3)*(-3 + x^2)),x]
Output:
-1/4*ArcTan[(1 - (1 - 3*x^2)^(1/3))/x] - ArcTanh[x/Sqrt[3]]/(4*Sqrt[3]) + ArcTanh[(1 - (1 - 3*x^2)^(1/3))^2/(3*Sqrt[3]*x)]/(4*Sqrt[3])
Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[(-q)*(ArcTanh[q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Si mp[q*(ArcTanh[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)]/(12*Rt[a , 3]*d)), x] - Simp[q*(ArcTan[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[ a, 3]*q*x)]/(4*Sqrt[3]*Rt[a, 3]*d)), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[ b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && NegQ[b/a]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.47 (sec) , antiderivative size = 538, normalized size of antiderivative = 2.77
method | result | size |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {8 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )^{2} \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -192 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) {\operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )}^{2} x -16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )^{2} \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +384 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) {\operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )}^{2} x -12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x^{2}-24 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )-12 \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}-96 \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +3 x^{2}+3}{x^{2}-3}\right )}{12}-\ln \left (\frac {2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}-96 \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -3 x^{2}+6 \left (-3 x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}-3}\right ) \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )+\frac {\ln \left (\frac {2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}-96 \operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -3 x^{2}+6 \left (-3 x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}-3}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{12}\) | \(538\) |
Input:
int(1/(-3*x^2+1)^(1/3)/(x^2-3),x,method=_RETURNVERBOSE)
Output:
-1/12*RootOf(_Z^2-3)*ln((8*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)^2*RootOf(-4*_Z* RootOf(_Z^2-3)+48*_Z^2+1)*x-192*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)*RootOf(-4* _Z*RootOf(_Z^2-3)+48*_Z^2+1)^2*x-16*RootOf(_Z^2-3)^2*RootOf(-4*_Z*RootOf(_ Z^2-3)+48*_Z^2+1)*x+384*RootOf(_Z^2-3)*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2 +1)^2*x-12*RootOf(_Z^2-3)*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x^2-24*(- 3*x^2+1)^(1/3)*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3)-12*Ro otOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3)+4*RootOf(_Z^2-3)*x+6*( -3*x^2+1)^(2/3)-96*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x+3*x^2+3)/(x^2- 3))-ln((2*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)*x-48*(-3*x^2+1)^(1/3)*RootOf(-4* _Z*RootOf(_Z^2-3)+48*_Z^2+1)*x+4*RootOf(_Z^2-3)*x+6*(-3*x^2+1)^(2/3)-96*Ro otOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x-3*x^2+6*(-3*x^2+1)^(1/3)-3)/(x^2-3) )*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)+1/12*ln((2*(-3*x^2+1)^(1/3)*RootO f(_Z^2-3)*x-48*(-3*x^2+1)^(1/3)*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x+4 *RootOf(_Z^2-3)*x+6*(-3*x^2+1)^(2/3)-96*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^ 2+1)*x-3*x^2+6*(-3*x^2+1)^(1/3)-3)/(x^2-3))*RootOf(_Z^2-3)
Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (158) = 316\).
Time = 1.13 (sec) , antiderivative size = 954, normalized size of antiderivative = 4.92 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/(-3*x^2+1)^(1/3)/(x^2-3),x, algorithm="fricas")
Output:
1/72*sqrt(3)*log(-(x^12 + 2598*x^10 + 55143*x^8 + 114228*x^6 - 22113*x^4 - 7290*x^2 + 8*(3*x^10 + 576*x^8 + 5598*x^6 + 5832*x^4 - 729*x^2 + sqrt(3)* (41*x^9 + 1368*x^7 + 4482*x^5 + 864*x^3 - 243*x))*(-3*x^2 + 1)^(2/3) + 4*s qrt(3)*(25*x^11 + 2359*x^9 + 15426*x^7 + 6966*x^5 - 4347*x^3 + 243*x) - 4* (84*x^10 + 4536*x^8 + 20880*x^6 + 5832*x^4 - 2916*x^2 + sqrt(3)*(x^11 + 52 1*x^9 + 7362*x^7 + 10746*x^5 - 1971*x^3 - 243*x))*(-3*x^2 + 1)^(1/3) + 729 )/(x^12 - 18*x^10 + 135*x^8 - 540*x^6 + 1215*x^4 - 1458*x^2 + 729)) - 1/14 4*sqrt(3)*log((x^6 - 93*x^4 - 117*x^2 + 2*(3*x^4 + 8*sqrt(3)*x^3 + 18*x^2 - 9)*(-3*x^2 + 1)^(2/3) - 8*sqrt(3)*(x^5 + 13*x^3) + 2*(3*x^4 - 54*x^2 + s qrt(3)*(x^5 - 10*x^3 - 27*x) - 9)*(-3*x^2 + 1)^(1/3) + 9)/(x^6 - 9*x^4 + 2 7*x^2 - 27)) + 1/144*sqrt(3)*log((x^6 - 93*x^4 - 117*x^2 + 2*(3*x^4 - 8*sq rt(3)*x^3 + 18*x^2 - 9)*(-3*x^2 + 1)^(2/3) + 8*sqrt(3)*(x^5 + 13*x^3) + 2* (3*x^4 - 54*x^2 - sqrt(3)*(x^5 - 10*x^3 - 27*x) - 9)*(-3*x^2 + 1)^(1/3) + 9)/(x^6 - 9*x^4 + 27*x^2 - 27)) - 1/24*arctan(1/3*(216*x^11 - 80424*x^9 + 282096*x^7 - 134352*x^5 + 18360*x^3 + 12*(140*x^9 - 19440*x^7 + 8424*x^5 - 1584*x^3 + sqrt(3)*(x^10 + 589*x^8 + 3946*x^6 - 774*x^4 - 27*x^2 + 9) + 1 08*x)*(-3*x^2 + 1)^(2/3) + sqrt(3)*(x^12 + 3150*x^10 + 77991*x^8 + 4260*x^ 6 - 14337*x^4 + 2862*x^2 - 135) - 12*(x^11 - 1591*x^9 + 42426*x^7 - 15102* x^5 + 1269*x^3 + sqrt(3)*(27*x^10 + 2307*x^8 + 4574*x^6 - 2538*x^4 + 279*x ^2 - 9) - 27*x)*(-3*x^2 + 1)^(1/3) - 648*x)/(x^12 - 4986*x^10 + 327519*...
\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\int \frac {1}{\sqrt [3]{1 - 3 x^{2}} \left (x^{2} - 3\right )}\, dx \] Input:
integrate(1/(-3*x**2+1)**(1/3)/(x**2-3),x)
Output:
Integral(1/((1 - 3*x**2)**(1/3)*(x**2 - 3)), x)
\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(-3*x^2+1)^(1/3)/(x^2-3),x, algorithm="maxima")
Output:
integrate(1/((x^2 - 3)*(-3*x^2 + 1)^(1/3)), x)
\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(-3*x^2+1)^(1/3)/(x^2-3),x, algorithm="giac")
Output:
integrate(1/((x^2 - 3)*(-3*x^2 + 1)^(1/3)), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\int \frac {1}{\left (x^2-3\right )\,{\left (1-3\,x^2\right )}^{1/3}} \,d x \] Input:
int(1/((x^2 - 3)*(1 - 3*x^2)^(1/3)),x)
Output:
int(1/((x^2 - 3)*(1 - 3*x^2)^(1/3)), x)
\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\int \frac {1}{\left (-3 x^{2}+1\right )^{\frac {1}{3}} x^{2}-3 \left (-3 x^{2}+1\right )^{\frac {1}{3}}}d x \] Input:
int(1/(-3*x^2+1)^(1/3)/(x^2-3),x)
Output:
int(1/(( - 3*x**2 + 1)**(1/3)*x**2 - 3*( - 3*x**2 + 1)**(1/3)),x)