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

Optimal. Leaf size=81 \[ \frac {1}{4} \tan ^{-1}\left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {404} \begin {gather*} \frac {1}{4} \text {ArcTan}\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]

ArcTan[(1 - (1 - 3*x^2)^(1/3))/x]/4 + ArcTanh[x/Sqrt[3]]/(4*Sqrt[3]) - ArcTanh[(1 - (1 - 3*x^2)^(1/3))^2/(3*Sq
rt[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*} \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] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 4.95, size = 126, normalized size = 1.56 \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 (-3+x^2\right ) \left (9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};3 x^2,\frac {x^2}{3}\right )+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 )\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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.62, size = 651, normalized size = 8.04

method	result	size
trager	\(-48 \ln \left (-\frac {18432 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} \left (-3 x^{2}+1\right )^{\frac {1}{3}} x -36864 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x +768 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} \left (-3 x^{2}+1\right )^{\frac {1}{3}} x -2304 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x -48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} x^{2}-96 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} \left (-3 x^{2}+1\right )^{\frac {1}{3}}-48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}-32 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x -2 \left (-3 x^{2}+1\right )^{\frac {2}{3}}-x^{2}-1}{x^{2}-3}\right ) \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3}+\RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {9216 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} \left (-3 x^{2}+1\right )^{\frac {1}{3}} x -18432 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x +576 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} \left (-3 x^{2}+1\right )^{\frac {1}{3}} x -768 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x +24 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} x^{2}+48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} \left (-3 x^{2}+1\right )^{\frac {1}{3}}+8 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) \left (-3 x^{2}+1\right )^{\frac {1}{3}} x +24 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}-\left (-3 x^{2}+1\right )^{\frac {2}{3}}+\left (-3 x^{2}+1\right )^{\frac {1}{3}}}{x^{2}-3}\right )-\ln \left (-\frac {18432 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} \left (-3 x^{2}+1\right )^{\frac {1}{3}} x -36864 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x +768 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} \left (-3 x^{2}+1\right )^{\frac {1}{3}} x -2304 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x -48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} x^{2}-96 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2} \left (-3 x^{2}+1\right )^{\frac {1}{3}}-48 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}-32 \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x -2 \left (-3 x^{2}+1\right )^{\frac {2}{3}}-x^{2}-1}{x^{2}-3}\right ) \RootOf \left (2304 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )\)	\(651\)

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]

-48*ln(-(18432*RootOf(2304*_Z^4+48*_Z^2+1)^5*(-3*x^2+1)^(1/3)*x-36864*RootOf(2304*_Z^4+48*_Z^2+1)^5*x+768*Root
Of(2304*_Z^4+48*_Z^2+1)^3*(-3*x^2+1)^(1/3)*x-2304*RootOf(2304*_Z^4+48*_Z^2+1)^3*x-48*RootOf(2304*_Z^4+48*_Z^2+
1)^2*x^2-96*RootOf(2304*_Z^4+48*_Z^2+1)^2*(-3*x^2+1)^(1/3)-48*RootOf(2304*_Z^4+48*_Z^2+1)^2-32*RootOf(2304*_Z^
4+48*_Z^2+1)*x-2*(-3*x^2+1)^(2/3)-x^2-1)/(x^2-3))*RootOf(2304*_Z^4+48*_Z^2+1)^3+RootOf(2304*_Z^4+48*_Z^2+1)*ln
(-(9216*RootOf(2304*_Z^4+48*_Z^2+1)^5*(-3*x^2+1)^(1/3)*x-18432*RootOf(2304*_Z^4+48*_Z^2+1)^5*x+576*RootOf(2304
*_Z^4+48*_Z^2+1)^3*(-3*x^2+1)^(1/3)*x-768*RootOf(2304*_Z^4+48*_Z^2+1)^3*x+24*RootOf(2304*_Z^4+48*_Z^2+1)^2*x^2
+48*RootOf(2304*_Z^4+48*_Z^2+1)^2*(-3*x^2+1)^(1/3)+8*RootOf(2304*_Z^4+48*_Z^2+1)*(-3*x^2+1)^(1/3)*x+24*RootOf(
2304*_Z^4+48*_Z^2+1)^2-(-3*x^2+1)^(2/3)+(-3*x^2+1)^(1/3))/(x^2-3))-ln(-(18432*RootOf(2304*_Z^4+48*_Z^2+1)^5*(-
3*x^2+1)^(1/3)*x-36864*RootOf(2304*_Z^4+48*_Z^2+1)^5*x+768*RootOf(2304*_Z^4+48*_Z^2+1)^3*(-3*x^2+1)^(1/3)*x-23
04*RootOf(2304*_Z^4+48*_Z^2+1)^3*x-48*RootOf(2304*_Z^4+48*_Z^2+1)^2*x^2-96*RootOf(2304*_Z^4+48*_Z^2+1)^2*(-3*x
^2+1)^(1/3)-48*RootOf(2304*_Z^4+48*_Z^2+1)^2-32*RootOf(2304*_Z^4+48*_Z^2+1)*x-2*(-3*x^2+1)^(2/3)-x^2-1)/(x^2-3
))*RootOf(2304*_Z^4+48*_Z^2+1)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1792 vs. \(2 (59) = 118\).
time = 2.21, size = 1792, normalized size = 22.12 \begin {gather*} \text {Too large to display} \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="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 + 43
2*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*sqr
t(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*sq
rt(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 - 49
86*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)*s
qrt(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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{x^{2} \sqrt [3]{1 - 3 x^{2}} - 3 \sqrt [3]{1 - 3 x^{2}}}\, 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/(x**2*(1 - 3*x**2)**(1/3) - 3*(1 - 3*x**2)**(1/3)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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