Integrand size = 19, antiderivative size = 194 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=-\frac {1}{8} \arctan \left (\frac {\frac {1}{\sqrt {3}}+x-\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {1}{8} \arctan \left (\frac {-\frac {1}{\sqrt {3}}+x+\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{1+2 \sqrt [3]{1-3 x^2}}\right )}{4 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {-2 \sqrt {3} x+2 \sqrt {3} x \sqrt [3]{1-3 x^2}}{1+3 x^2-2 \sqrt [3]{1-3 x^2}+4 \left (1-3 x^2\right )^{2/3}}\right )}{8 \sqrt {3}} \]
-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 = 4.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )\right )\right )} \]
(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 |
| \(\displaystyle -\frac {1}{4} \arctan \left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )+\frac {\text {arctanh}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}\) |
-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])
3.25.13.3.1 Defintions of rubi rules used
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 = 10.04 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.79
| 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}-\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}-\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 )\) | \(541\) |
-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*RootOf( 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))-1/1 2*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*Root Of(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))*R ootOf(_Z^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*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(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 1208, normalized size of antiderivative = 6.23 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\text {Too large to display} \]
-1/144*sqrt(6)*sqrt(-I*sqrt(3) - 1)*log(-(42*x^5 + 276*x^3 + sqrt(6)*(x^6 + 105*x^4 + 63*x^2 - 9)*sqrt(-I*sqrt(3) - 1) - 3*(40*x^3 + sqrt(6)*(x^4 + 12*x^2 - sqrt(3)*(-I*x^4 - 12*I*x^2 - 3*I) + 3)*sqrt(-I*sqrt(3) - 1) + 72* x)*(-3*x^2 + 1)^(2/3) + 6*sqrt(3)*(-7*I*x^5 - 46*I*x^3 + 9*I*x) - 3*(2*x^5 + 52*x^3 - sqrt(6)*(5*x^4 + 18*x^2 + sqrt(3)*(-5*I*x^4 - 18*I*x^2 + 3*I) - 3)*sqrt(-I*sqrt(3) - 1) - 2*sqrt(3)*(-I*x^5 - 26*I*x^3 - 9*I*x) + 18*x)* (-3*x^2 + 1)^(1/3) - 54*x)/(x^6 - 9*x^4 + 27*x^2 - 27)) + 1/144*sqrt(6)*sq rt(-I*sqrt(3) - 1)*log(-(42*x^5 + 276*x^3 - sqrt(6)*(x^6 + 105*x^4 + 63*x^ 2 - 9)*sqrt(-I*sqrt(3) - 1) - 3*(40*x^3 - sqrt(6)*(x^4 + 12*x^2 + sqrt(3)* (I*x^4 + 12*I*x^2 + 3*I) + 3)*sqrt(-I*sqrt(3) - 1) + 72*x)*(-3*x^2 + 1)^(2 /3) + 6*sqrt(3)*(-7*I*x^5 - 46*I*x^3 + 9*I*x) - 3*(2*x^5 + 52*x^3 + sqrt(6 )*(5*x^4 + 18*x^2 - sqrt(3)*(5*I*x^4 + 18*I*x^2 - 3*I) - 3)*sqrt(-I*sqrt(3 ) - 1) - 2*sqrt(3)*(-I*x^5 - 26*I*x^3 - 9*I*x) + 18*x)*(-3*x^2 + 1)^(1/3) - 54*x)/(x^6 - 9*x^4 + 27*x^2 - 27)) - 1/144*sqrt(6)*sqrt(I*sqrt(3) - 1)*l og(-(42*x^5 + 276*x^3 - 24*(5*x^3 + 9*x)*(-3*x^2 + 1)^(2/3) + 6*sqrt(3)*(7 *I*x^5 + 46*I*x^3 - 9*I*x) - (3*sqrt(6)*(x^4 + 12*x^2 - sqrt(3)*(I*x^4 + 1 2*I*x^2 + 3*I) + 3)*(-3*x^2 + 1)^(2/3) - 3*sqrt(6)*(5*x^4 + 18*x^2 + sqrt( 3)*(5*I*x^4 + 18*I*x^2 - 3*I) - 3)*(-3*x^2 + 1)^(1/3) - sqrt(6)*(x^6 + 105 *x^4 + 63*x^2 - 9))*sqrt(I*sqrt(3) - 1) - 6*(x^5 + 26*x^3 - sqrt(3)*(I*x^5 + 26*I*x^3 + 9*I*x) + 9*x)*(-3*x^2 + 1)^(1/3) - 54*x)/(x^6 - 9*x^4 + 2...
\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\int \frac {1}{\sqrt [3]{1 - 3 x^{2}} \left (x^{2} - 3\right )}\, dx \]
\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\int \frac {1}{\left (x^2-3\right )\,{\left (1-3\,x^2\right )}^{1/3}} \,d x \]