\(\int \frac {1}{\sqrt [3]{1-x^2} (9-x^2)} \, dx\) [372]

Optimal result

Integrand size = 21, antiderivative size = 74 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}+\frac {1}{12} \text {arctanh}\left (\frac {x}{3}\right )-\frac {1}{12} \text {arctanh}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right ) \] Output:

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

Mathematica [C] (verified)

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

Time = 9.49 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=\frac {\sqrt [3]{\frac {-1+x}{-3+x}} \sqrt [3]{\frac {1+x}{-3+x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {4}{-3+x},-\frac {2}{-3+x}\right )-\sqrt [3]{\frac {-1+x}{3+x}} \sqrt [3]{\frac {1+x}{3+x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {2}{3+x},\frac {4}{3+x}\right )}{4 \sqrt [3]{1-x^2}} \] Input:

Integrate[1/((1 - x^2)^(1/3)*(9 - x^2)),x]
 

Output:

(((-1 + x)/(-3 + x))^(1/3)*((1 + x)/(-3 + x))^(1/3)*AppellF1[2/3, 1/3, 1/3 
, 5/3, -4/(-3 + x), -2/(-3 + x)] - ((-1 + x)/(3 + x))^(1/3)*((1 + x)/(3 + 
x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, 2/(3 + x), 4/(3 + x)])/(4*(1 - x^2) 
^(1/3))
 

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 74, 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.

\(\displaystyle \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx\)

\(\Big \downarrow \) 307

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}-\frac {1}{12} \text {arctanh}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )+\frac {1}{12} \text {arctanh}\left (\frac {x}{3}\right )\)

Input:

Int[1/((1 - x^2)^(1/3)*(9 - x^2)),x]
 

Output:

ArcTan[(Sqrt[3]*(1 - (1 - x^2)^(1/3)))/x]/(4*Sqrt[3]) + ArcTanh[x/3]/12 - 
ArcTanh[(1 - (1 - x^2)^(1/3))^2/(3*x)]/12
 

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] (verified)

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

Time = 3.60 (sec) , antiderivative size = 539, normalized size of antiderivative = 7.28

method	result	size
trager	\(-\frac {\ln \left (\frac {288 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x -576 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x -6 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-24 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +3 \left (-x^{2}+1\right )^{\frac {2}{3}}+x \left (-x^{2}+1\right )^{\frac {1}{3}}-18 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-3 \left (-x^{2}+1\right )^{\frac {1}{3}}}{\left (-3+x \right ) \left (3+x \right )}\right )}{12}-\ln \left (\frac {288 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x -576 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x -6 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-24 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +3 \left (-x^{2}+1\right )^{\frac {2}{3}}+x \left (-x^{2}+1\right )^{\frac {1}{3}}-18 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-3 \left (-x^{2}+1\right )^{\frac {1}{3}}}{\left (-3+x \right ) \left (3+x \right )}\right ) \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {576 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x -1152 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +24 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +12 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+72 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-144 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +6 \left (-x^{2}+1\right )^{\frac {2}{3}}+x^{2}+36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-4 x +3}{\left (-3+x \right ) \left (3+x \right )}\right )\)	\(539\)

Input:

int(1/(-x^2+1)^(1/3)/(-x^2+9),x,method=_RETURNVERBOSE)
 

Output:

-1/12*ln((288*(-x^2+1)^(1/3)*RootOf(144*_Z^2+12*_Z+1)^2*x-576*RootOf(144*_ 
Z^2+12*_Z+1)^2*x+36*RootOf(144*_Z^2+12*_Z+1)*(-x^2+1)^(1/3)*x-6*RootOf(144 
*_Z^2+12*_Z+1)*x^2-36*RootOf(144*_Z^2+12*_Z+1)*(-x^2+1)^(1/3)-24*RootOf(14 
4*_Z^2+12*_Z+1)*x+3*(-x^2+1)^(2/3)+x*(-x^2+1)^(1/3)-18*RootOf(144*_Z^2+12* 
_Z+1)-3*(-x^2+1)^(1/3))/(-3+x)/(3+x))-ln((288*(-x^2+1)^(1/3)*RootOf(144*_Z 
^2+12*_Z+1)^2*x-576*RootOf(144*_Z^2+12*_Z+1)^2*x+36*RootOf(144*_Z^2+12*_Z+ 
1)*(-x^2+1)^(1/3)*x-6*RootOf(144*_Z^2+12*_Z+1)*x^2-36*RootOf(144*_Z^2+12*_ 
Z+1)*(-x^2+1)^(1/3)-24*RootOf(144*_Z^2+12*_Z+1)*x+3*(-x^2+1)^(2/3)+x*(-x^2 
+1)^(1/3)-18*RootOf(144*_Z^2+12*_Z+1)-3*(-x^2+1)^(1/3))/(-3+x)/(3+x))*Root 
Of(144*_Z^2+12*_Z+1)+RootOf(144*_Z^2+12*_Z+1)*ln((576*(-x^2+1)^(1/3)*RootO 
f(144*_Z^2+12*_Z+1)^2*x-1152*RootOf(144*_Z^2+12*_Z+1)^2*x+24*RootOf(144*_Z 
^2+12*_Z+1)*(-x^2+1)^(1/3)*x+12*RootOf(144*_Z^2+12*_Z+1)*x^2+72*RootOf(144 
*_Z^2+12*_Z+1)*(-x^2+1)^(1/3)-144*RootOf(144*_Z^2+12*_Z+1)*x+6*(-x^2+1)^(2 
/3)+x^2+36*RootOf(144*_Z^2+12*_Z+1)-4*x+3)/(-3+x)/(3+x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (53) = 106\).

Time = 0.46 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.64 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=-\frac {1}{36} \, \sqrt {3} \arctan \left (\frac {36 \, \sqrt {3} {\left (x^{4} - 32 \, x^{3} - 42 \, x^{2} + 9\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 12 \, \sqrt {3} {\left (x^{5} + 27 \, x^{4} - 210 \, x^{3} - 54 \, x^{2} + 81 \, x + 27\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{6} - 108 \, x^{5} - 567 \, x^{4} + 1080 \, x^{3} + 459 \, x^{2} - 972 \, x - 405\right )}}{3 \, {\left (x^{6} + 108 \, x^{5} - 1647 \, x^{4} - 1080 \, x^{3} + 891 \, x^{2} + 972 \, x + 243\right )}}\right ) - \frac {1}{72} \, \log \left (\frac {x^{3} + 33 \, x^{2} + 18 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )} - 6 \, {\left (x^{2} + 6 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 9 \, x - 9}{x^{3} + 9 \, x^{2} + 27 \, x + 27}\right ) + \frac {1}{36} \, \log \left (-\frac {x^{3} - 33 \, x^{2} + 18 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 6 \, {\left (x^{2} - 6 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 9 \, x + 9}{x^{3} + 9 \, x^{2} + 27 \, x + 27}\right ) \] Input:

integrate(1/(-x^2+1)^(1/3)/(-x^2+9),x, algorithm="fricas")
 

Output:

-1/36*sqrt(3)*arctan(1/3*(36*sqrt(3)*(x^4 - 32*x^3 - 42*x^2 + 9)*(-x^2 + 1 
)^(2/3) + 12*sqrt(3)*(x^5 + 27*x^4 - 210*x^3 - 54*x^2 + 81*x + 27)*(-x^2 + 
 1)^(1/3) + sqrt(3)*(x^6 - 108*x^5 - 567*x^4 + 1080*x^3 + 459*x^2 - 972*x 
- 405))/(x^6 + 108*x^5 - 1647*x^4 - 1080*x^3 + 891*x^2 + 972*x + 243)) - 1 
/72*log((x^3 + 33*x^2 + 18*(-x^2 + 1)^(2/3)*(x + 1) - 6*(x^2 + 6*x - 3)*(- 
x^2 + 1)^(1/3) - 9*x - 9)/(x^3 + 9*x^2 + 27*x + 27)) + 1/36*log(-(x^3 - 33 
*x^2 + 18*(-x^2 + 1)^(2/3)*(x - 1) + 6*(x^2 - 6*x - 3)*(-x^2 + 1)^(1/3) - 
9*x + 9)/(x^3 + 9*x^2 + 27*x + 27))
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=- \int \frac {1}{x^{2} \sqrt [3]{1 - x^{2}} - 9 \sqrt [3]{1 - x^{2}}}\, dx \] Input:

integrate(1/(-x**2+1)**(1/3)/(-x**2+9),x)
 

Output:

-Integral(1/(x**2*(1 - x**2)**(1/3) - 9*(1 - x**2)**(1/3)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=\int { -\frac {1}{{\left (x^{2} - 9\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(-x^2+1)^(1/3)/(-x^2+9),x, algorithm="maxima")
 

Output:

-integrate(1/((x^2 - 9)*(-x^2 + 1)^(1/3)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=\int { -\frac {1}{{\left (x^{2} - 9\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(-x^2+1)^(1/3)/(-x^2+9),x, algorithm="giac")
 

Output:

integrate(-1/((x^2 - 9)*(-x^2 + 1)^(1/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=-\int \frac {1}{{\left (1-x^2\right )}^{1/3}\,\left (x^2-9\right )} \,d x \] Input:

int(-1/((1 - x^2)^(1/3)*(x^2 - 9)),x)
 

Output:

-int(1/((1 - x^2)^(1/3)*(x^2 - 9)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=-\left (\int \frac {1}{\left (-x^{2}+1\right )^{\frac {1}{3}} x^{2}-9 \left (-x^{2}+1\right )^{\frac {1}{3}}}d x \right ) \] Input:

int(1/(-x^2+1)^(1/3)/(-x^2+9),x)
 

Output:

 - int(1/(( - x**2 + 1)**(1/3)*x**2 - 9*( - x**2 + 1)**(1/3)),x)