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

Optimal. Leaf size=194 \[ -\frac {1}{8} \tan ^{-1}\left (\frac {-\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}+x+\frac {1}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {1}{8} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}+x-\frac {1}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{1-3 x^2}+1}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {3} x \sqrt [3]{1-3 x^2}-2 \sqrt {3} x}{3 x^2+4 \left (1-3 x^2\right )^{2/3}-2 \sqrt [3]{1-3 x^2}+1}\right )}{8 \sqrt {3}} \]

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Rubi [A]  time = 0.01, antiderivative size = 81, normalized size of antiderivative = 0.42, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {395} \begin {gather*} -\frac {1}{4} \tan ^{-1}\left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )+\frac {\tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[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 395

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)])/(1
2*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*} \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx &=-\frac {1}{4} \tan ^{-1}\left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 126, normalized size = 0.65 \begin {gather*} \frac {9 x F_1\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 (x^2-3\right ) \left (2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};3 x^2,\frac {x^2}{3}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};3 x^2,\frac {x^2}{3}\right )\right )+9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};3 x^2,\frac {x^2}{3}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[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])))

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IntegrateAlgebraic [F]  time = 3.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [B]  time = 2.23, size = 1790, normalized size = 9.23

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*sqrt(6)*sqrt(3)*sqrt(2)*arctan(1/9*(36*sqrt(6)*sqrt(3)*sqrt(2)*(3*x^11 - 1117*x^9 + 3918*x^7 - 1866*x^5
+ 255*x^3 - 9*x) + sqrt(3)*(sqrt(6)*sqrt(3)*sqrt(2)*(x^12 + 2184*x^10 - 211215*x^8 + 94152*x^6 - 13581*x^4 + 4
32*x^2 + 27) + 12*(sqrt(6)*sqrt(3)*sqrt(2)*(x^10 - 107*x^8 - 7262*x^6 + 2322*x^4 - 243*x^2 + 9) - 48*sqrt(3)*(
5*x^9 - 245*x^7 + 183*x^5 - 15*x^3))*(-3*x^2 + 1)^(2/3) - 12*sqrt(3)*(29*x^11 + 293*x^9 - 2670*x^7 + 4986*x^5
- 1215*x^3 + 81*x) - 6*(sqrt(6)*sqrt(3)*sqrt(2)*(49*x^10 - 5043*x^8 + 3658*x^6 + 378*x^4 - 171*x^2 + 9) - 2*sq
rt(3)*(x^11 + 917*x^9 - 40566*x^7 + 15786*x^5 - 2043*x^3 + 81*x))*(-3*x^2 + 1)^(1/3))*sqrt((x^6 - 93*x^4 + 4*s
qrt(6)*sqrt(2)*(x^5 + 13*x^3) - 117*x^2 - 2*(4*sqrt(6)*sqrt(2)*x^3 - 3*x^4 - 18*x^2 + 9)*(-3*x^2 + 1)^(2/3) +
(6*x^4 - sqrt(6)*sqrt(2)*(x^5 - 10*x^3 - 27*x) - 108*x^2 - 18)*(-3*x^2 + 1)^(1/3) + 9)/(x^6 - 9*x^4 + 27*x^2 -
27)) + 12*(2*sqrt(6)*sqrt(3)*sqrt(2)*(35*x^9 - 4860*x^7 + 2106*x^5 - 396*x^3 + 27*x) - 3*sqrt(3)*(x^10 + 589*
x^8 + 3946*x^6 - 774*x^4 - 27*x^2 + 9))*(-3*x^2 + 1)^(2/3) - 3*sqrt(3)*(x^12 + 3150*x^10 + 77991*x^8 + 4260*x^
6 - 14337*x^4 + 2862*x^2 - 135) - 6*(sqrt(6)*sqrt(3)*sqrt(2)*(x^11 - 1591*x^9 + 42426*x^7 - 15102*x^5 + 1269*x
^3 - 27*x) - 6*sqrt(3)*(27*x^10 + 2307*x^8 + 4574*x^6 - 2538*x^4 + 279*x^2 - 9))*(-3*x^2 + 1)^(1/3))/(x^12 - 4
986*x^10 + 327519*x^8 - 159660*x^6 + 25839*x^4 - 2106*x^2 + 81)) - 1/72*sqrt(6)*sqrt(3)*sqrt(2)*arctan(1/9*(36
*sqrt(6)*sqrt(3)*sqrt(2)*(3*x^11 - 1117*x^9 + 3918*x^7 - 1866*x^5 + 255*x^3 - 9*x) + sqrt(3)*(sqrt(6)*sqrt(3)*
sqrt(2)*(x^12 + 2184*x^10 - 211215*x^8 + 94152*x^6 - 13581*x^4 + 432*x^2 + 27) + 12*(sqrt(6)*sqrt(3)*sqrt(2)*(
x^10 - 107*x^8 - 7262*x^6 + 2322*x^4 - 243*x^2 + 9) + 48*sqrt(3)*(5*x^9 - 245*x^7 + 183*x^5 - 15*x^3))*(-3*x^2
+ 1)^(2/3) + 12*sqrt(3)*(29*x^11 + 293*x^9 - 2670*x^7 + 4986*x^5 - 1215*x^3 + 81*x) - 6*(sqrt(6)*sqrt(3)*sqrt
(2)*(49*x^10 - 5043*x^8 + 3658*x^6 + 378*x^4 - 171*x^2 + 9) + 2*sqrt(3)*(x^11 + 917*x^9 - 40566*x^7 + 15786*x^
5 - 2043*x^3 + 81*x))*(-3*x^2 + 1)^(1/3))*sqrt((x^6 - 93*x^4 - 4*sqrt(6)*sqrt(2)*(x^5 + 13*x^3) - 117*x^2 + 2*
(4*sqrt(6)*sqrt(2)*x^3 + 3*x^4 + 18*x^2 - 9)*(-3*x^2 + 1)^(2/3) + (6*x^4 + sqrt(6)*sqrt(2)*(x^5 - 10*x^3 - 27*
x) - 108*x^2 - 18)*(-3*x^2 + 1)^(1/3) + 9)/(x^6 - 9*x^4 + 27*x^2 - 27)) + 12*(2*sqrt(6)*sqrt(3)*sqrt(2)*(35*x^
9 - 4860*x^7 + 2106*x^5 - 396*x^3 + 27*x) + 3*sqrt(3)*(x^10 + 589*x^8 + 3946*x^6 - 774*x^4 - 27*x^2 + 9))*(-3*
x^2 + 1)^(2/3) + 3*sqrt(3)*(x^12 + 3150*x^10 + 77991*x^8 + 4260*x^6 - 14337*x^4 + 2862*x^2 - 135) - 6*(sqrt(6)
*sqrt(3)*sqrt(2)*(x^11 - 1591*x^9 + 42426*x^7 - 15102*x^5 + 1269*x^3 - 27*x) + 6*sqrt(3)*(27*x^10 + 2307*x^8 +
4574*x^6 - 2538*x^4 + 279*x^2 - 9))*(-3*x^2 + 1)^(1/3))/(x^12 - 4986*x^10 + 327519*x^8 - 159660*x^6 + 25839*x
^4 - 2106*x^2 + 81)) + 1/288*sqrt(6)*sqrt(2)*log(12*(x^6 - 93*x^4 + 4*sqrt(6)*sqrt(2)*(x^5 + 13*x^3) - 117*x^2
- 2*(4*sqrt(6)*sqrt(2)*x^3 - 3*x^4 - 18*x^2 + 9)*(-3*x^2 + 1)^(2/3) + (6*x^4 - sqrt(6)*sqrt(2)*(x^5 - 10*x^3
- 27*x) - 108*x^2 - 18)*(-3*x^2 + 1)^(1/3) + 9)/(x^6 - 9*x^4 + 27*x^2 - 27)) - 1/288*sqrt(6)*sqrt(2)*log(12*(x
^6 - 93*x^4 - 4*sqrt(6)*sqrt(2)*(x^5 + 13*x^3) - 117*x^2 + 2*(4*sqrt(6)*sqrt(2)*x^3 + 3*x^4 + 18*x^2 - 9)*(-3*
x^2 + 1)^(2/3) + (6*x^4 + sqrt(6)*sqrt(2)*(x^5 - 10*x^3 - 27*x) - 108*x^2 - 18)*(-3*x^2 + 1)^(1/3) + 9)/(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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 11.37, size = 538, normalized size = 2.77

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-192 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-3\right ) \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )^{2} x -8 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-3\right )^{2} \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +384 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}-3\right ) x +16 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )^{2} x +12 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+24 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+12 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )+96 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +3 x^{2}+3}{x^{2}-3}\right )}{12}-\frac {\ln \left (\frac {-48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}-96 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -4 \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 ) \RootOf \left (\textit {\_Z}^{2}-3\right )}{12}-\ln \left (\frac {-48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}-96 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -4 \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 ) \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )\)	\(538\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*RootOf(_Z^2-3)*ln((-192*(-3*x^2+1)^(1/3)*RootOf(_Z^2-3)*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)^2*x-8*(-3*
x^2+1)^(1/3)*RootOf(_Z^2-3)^2*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*x+384*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1
)^2*RootOf(_Z^2-3)*x+16*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3)^2*x+12*RootOf(4*_Z*RootOf(_Z^2-3)
+48*_Z^2+1)*RootOf(_Z^2-3)*x^2+24*(-3*x^2+1)^(1/3)*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3)+6*(-3*
x^2+1)^(2/3)+12*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*RootOf(_Z^2-3)+96*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)*
x+4*RootOf(_Z^2-3)*x+3*x^2+3)/(x^2-3))-1/12*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*RootOf(_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))*RootOf(_Z^2-3)-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*RootOf(_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))*RootOf(4*_Z*RootOf(_Z^2-3)+48*_Z^2+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x^2-3\right )\,{\left (1-3\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1 - 3 x^{2}} \left (x^{2} - 3\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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