\(\int \frac {1}{\sqrt [3]{1-3 x^2} (-3+x^2)} \, dx\) [2413]

Optimal result

Integrand size = 19, antiderivative size = 194 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=-\frac {1}{8} \arctan \left (\frac {\frac {1}{\sqrt {3}}+x-\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {1}{8} \arctan \left (\frac {-\frac {1}{\sqrt {3}}+x+\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{1+2 \sqrt [3]{1-3 x^2}}\right )}{4 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {-2 \sqrt {3} x+2 \sqrt {3} x \sqrt [3]{1-3 x^2}}{1+3 x^2-2 \sqrt [3]{1-3 x^2}+4 \left (1-3 x^2\right )^{2/3}}\right )}{8 \sqrt {3}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.01 (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 = {404} \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=-\frac {1}{4} \arctan \left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )+\frac {\text {arctanh}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}} \]

[In]

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

[Out]

-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])

Rule 404

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[(-q)*(Arc
Tanh[q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Simp[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]

Rubi steps \begin{align*} \text {integral}& = -\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}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},3 x^2,\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},3 x^2,\frac {x^2}{3}\right )\right )\right )} \]

[In]

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

[Out]

(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])))

Maple [C] (warning: unable to verify)

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 )\)	\(541\)

[In]

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

[Out]

-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*Roo
tOf(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))-1/12*ln(-(2*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)*x+48*(-3*x^2+1)^(1/3)*RootO
f(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)-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)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.35 (sec) , antiderivative size = 1208, normalized size of antiderivative = 6.23 \[ \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-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) + 7
2*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)*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 + sq
rt(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)*log(-(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 + 12*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 + 27*x^2 - 27)) + 1/144*sqrt(6)*sqrt(I*sqrt(3) - 1)*log(-(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 - 12*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 + 27*x^2 - 27
)) + 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*sqrt(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 + 521*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))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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