3.2.62 \(\int \frac {1}{\sqrt [3]{1-x^2} (9-x^2)} \, dx\) [162]

Optimal. Leaf size=74 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}+\frac {1}{12} \tanh ^{-1}\left (\frac {x}{3}\right )-\frac {1}{12} \tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 74, 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 {\text {ArcTan}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}-\frac {1}{12} \tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )+\frac {1}{12} \tanh ^{-1}\left (\frac {x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x^2)^(1/3)*(9 - x^2)),x]

[Out]

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

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-x^2} \left (9-x^2\right )} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}+\frac {1}{12} \tanh ^{-1}\left (\frac {x}{3}\right )-\frac {1}{12} \tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 9.75, size = 125, normalized size = 1.69 \begin {gather*} \frac {\sqrt [3]{\frac {-1+x}{-3+x}} \sqrt [3]{\frac {1+x}{-3+x}} F_1\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}} F_1\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-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))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.61, size = 365, normalized size = 4.93

method	result	size
trager	\(-\frac {\ln \left (\frac {576 \left (-x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +24 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x -1152 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +12 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+6 \left (-x^{2}+1\right )^{\frac {2}{3}}+72 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-144 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +x^{2}+36 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-4 x +3}{\left (3+x \right ) \left (x -3\right )}\right )}{12}-\frac {\ln \left (\frac {48 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +6 \left (-x^{2}+1\right )^{\frac {2}{3}}+2 \left (-x^{2}+1\right )^{\frac {1}{3}} x +96 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -x^{2}+6 \left (-x^{2}+1\right )^{\frac {1}{3}}+4 x -3}{\left (3+x \right ) \left (x -3\right )}\right )}{12}-\ln \left (\frac {48 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +6 \left (-x^{2}+1\right )^{\frac {2}{3}}+2 \left (-x^{2}+1\right )^{\frac {1}{3}} x +96 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -x^{2}+6 \left (-x^{2}+1\right )^{\frac {1}{3}}+4 x -3}{\left (3+x \right ) \left (x -3\right )}\right ) \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )\)	\(365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*ln((576*(-x^2+1)^(1/3)*RootOf(144*_Z^2+12*_Z+1)^2*x+24*RootOf(144*_Z^2+12*_Z+1)*(-x^2+1)^(1/3)*x-1152*Ro
otOf(144*_Z^2+12*_Z+1)^2*x+12*RootOf(144*_Z^2+12*_Z+1)*x^2+6*(-x^2+1)^(2/3)+72*RootOf(144*_Z^2+12*_Z+1)*(-x^2+
1)^(1/3)-144*RootOf(144*_Z^2+12*_Z+1)*x+x^2+36*RootOf(144*_Z^2+12*_Z+1)-4*x+3)/(3+x)/(x-3))-1/12*ln((48*RootOf
(144*_Z^2+12*_Z+1)*(-x^2+1)^(1/3)*x+6*(-x^2+1)^(2/3)+2*(-x^2+1)^(1/3)*x+96*RootOf(144*_Z^2+12*_Z+1)*x-x^2+6*(-
x^2+1)^(1/3)+4*x-3)/(3+x)/(x-3))-ln((48*RootOf(144*_Z^2+12*_Z+1)*(-x^2+1)^(1/3)*x+6*(-x^2+1)^(2/3)+2*(-x^2+1)^
(1/3)*x+96*RootOf(144*_Z^2+12*_Z+1)*x-x^2+6*(-x^2+1)^(1/3)+4*x-3)/(3+x)/(x-3))*RootOf(144*_Z^2+12*_Z+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/(-x^2+1)^(1/3)/(-x^2+9),x, algorithm="maxima")

[Out]

-integrate(1/((x^2 - 9)*(-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. 269 vs. \(2 (53) = 106\).
time = 1.49, size = 269, normalized size = 3.64 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 + 2
7))

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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 - x^{2}} - 9 \sqrt [3]{1 - x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(-1/((x^2 - 9)*(-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 (1-x^2\right )}^{1/3}\,\left (x^2-9\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((1 - x^2)^(1/3)*(x^2 - 9)),x)

[Out]

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

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