\(\int \frac {1}{\sqrt [3]{1-3 x^2} (3-x^2)} \, dx\) [77]

Optimal result

Integrand size = 21, antiderivative size = 81 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=\frac {1}{4} \arctan \left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )+\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}} \] Output:

1/4*arctan((1-(-3*x^2+1)^(1/3))/x)+1/12*arctanh(1/3*x*3^(1/2))*3^(1/2)-1/1 
2*arctanh(1/9*(1-(-3*x^2+1)^(1/3))^2/x*3^(1/2))*3^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 4.16 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=-\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )\right )\right )} \] 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])))
 

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, 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.

\(\displaystyle \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx\)

\(\Big \downarrow \) 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:

ArcTan[(1 - (1 - 3*x^2)^(1/3))/x]/4 + ArcTanh[x/Sqrt[3]]/(4*Sqrt[3]) - Arc 
Tanh[(1 - (1 - 3*x^2)^(1/3))^2/(3*Sqrt[3]*x)]/(4*Sqrt[3])
 

Defintions of rubi rules used

rule 307

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 8.73 (sec) , antiderivative size = 438, normalized size of antiderivative = 5.41

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-96 \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 )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +4 \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 )^{2} x +192 {\operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -8 \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 )^{2} x +6 \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 ) x^{2}+12 \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 )+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 ) x -\left (-3 x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +3 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+6 \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 )-3 \left (-3 x^{2}+1\right )^{\frac {1}{3}}}{x^{2}-3}\right )}{12}+\operatorname {RootOf}\left (-4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-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 +2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \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 +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -3 x^{2}+6 \left (-3 x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}-3}\right )\)	\(438\)

Input:

int(1/(-3*x^2+1)^(1/3)/(-x^2+3),x,method=_RETURNVERBOSE)
 

Output:

1/12*RootOf(_Z^2-3)*ln((-96*(-3*x^2+1)^(1/3)*RootOf(-4*_Z*RootOf(_Z^2-3)+4 
8*_Z^2+1)^2*RootOf(_Z^2-3)*x+4*(-3*x^2+1)^(1/3)*RootOf(-4*_Z*RootOf(_Z^2-3 
)+48*_Z^2+1)*RootOf(_Z^2-3)^2*x+192*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1) 
^2*RootOf(_Z^2-3)*x-8*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3 
)^2*x+6*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3)*x^2+12*(-3*x 
^2+1)^(1/3)*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3)+24*(-3*x 
^2+1)^(1/3)*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x-(-3*x^2+1)^(1/3)*Root 
Of(_Z^2-3)*x+3*(-3*x^2+1)^(2/3)+6*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*R 
ootOf(_Z^2-3)-3*(-3*x^2+1)^(1/3))/(x^2-3))+RootOf(-4*_Z*RootOf(_Z^2-3)+48* 
_Z^2+1)*ln((-48*(-3*x^2+1)^(1/3)*RootOf(-4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x+ 
2*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)*x+6*(-3*x^2+1)^(2/3)-96*RootOf(-4*_Z*Roo 
tOf(_Z^2-3)+48*_Z^2+1)*x+4*RootOf(_Z^2-3)*x-3*x^2+6*(-3*x^2+1)^(1/3)-3)/(x 
^2-3))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 956 vs. \(2 (59) = 118\).

Time = 1.10 (sec) , antiderivative size = 956, normalized size of antiderivative = 11.80 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=\text {Too large to display} \] 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*...
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=- \int \frac {1}{x^{2} \sqrt [3]{1 - 3 x^{2}} - 3 \sqrt [3]{1 - 3 x^{2}}}\, dx \] Input:

integrate(1/(-3*x**2+1)**(1/3)/(-x**2+3),x)
 

Output:

-Integral(1/(x**2*(1 - 3*x**2)**(1/3) - 3*(1 - 3*x**2)**(1/3)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=\int { -\frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] 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)
 

Giac [F]

\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=\int { -\frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] 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)
 

Mupad [F(-1)]

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)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx=-\left (\int \frac {1}{\left (-3 x^{2}+1\right )^{\frac {1}{3}} x^{2}-3 \left (-3 x^{2}+1\right )^{\frac {1}{3}}}d x \right ) \] 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)