3.8.14 \(\int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx\) [714]

Optimal. Leaf size=78 \[ -\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(1+x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )-\frac {1}{4} \log (3-x)+\frac {3}{8} \log \left (-\sqrt [3]{1-x}-\frac {1}{2} (1+x)^{2/3}\right ) \]

[Out]

-1/4*ln(3-x)+3/8*ln(-(1-x)^(1/3)-1/2*(1+x)^(2/3))+1/4*arctan(-1/3*3^(1/2)+1/3*(1+x)^(2/3)/(1-x)^(1/3)*3^(1/2))
*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {767, 124} \begin {gather*} -\frac {1}{4} \sqrt {3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {(x+1)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )-\frac {1}{4} \log (3-x)+\frac {3}{8} \log \left (-\frac {1}{2} (x+1)^{2/3}-\sqrt [3]{1-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((3 - x)*(1 - x^2)^(1/3)),x]

[Out]

-1/4*(Sqrt[3]*ArcTan[1/Sqrt[3] - (1 + x)^(2/3)/(Sqrt[3]*(1 - x)^(1/3))]) - Log[3 - x]/4 + (3*Log[-(1 - x)^(1/3
) - (1 + x)^(2/3)/2])/8

Rule 124

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[
b*((b*e - a*f)/(b*c - a*d)^2), 3]}, Simp[-Log[a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[Sqrt[3]*(ArcTan[1/Sqrt[3
] + 2*q*((c + d*x)^(2/3)/(Sqrt[3]*(e + f*x)^(1/3)))]/(2*q*(b*c - a*d))), x] + Simp[3*(Log[q*(c + d*x)^(2/3) -
(e + f*x)^(1/3)]/(4*q*(b*c - a*d))), x])] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]

Rule 767

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> Dist[a^(1/3), Int[1/((d + e*x)*(1 - 3*e*
(x/d))^(1/3)*(1 + 3*e*(x/d))^(1/3)), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] && GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx &=\int \frac {1}{\sqrt [3]{1-x} (3-x) \sqrt [3]{1+x}} \, dx\\ &=-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(1+x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )-\frac {1}{4} \log (3-x)+\frac {3}{8} \log \left (-\sqrt [3]{1-x}-\frac {1}{2} (1+x)^{2/3}\right )\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 107, normalized size = 1.37 \begin {gather*} \frac {1}{8} \left (-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{1+x-\sqrt [3]{1-x^2}}\right )+2 \log \left (1+x+2 \sqrt [3]{1-x^2}\right )-\log \left (1+2 x+x^2-2 (1+x) \sqrt [3]{1-x^2}+4 \left (1-x^2\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^2)^(1/3))/(1 + x - (1 - x^2)^(1/3))] + 2*Log[1 + x + 2*(1 - x^2)^(1/3)] - L
og[1 + 2*x + x^2 - 2*(1 + x)*(1 - x^2)^(1/3) + 4*(1 - x^2)^(2/3)])/8

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.03, size = 618, normalized size = 7.92

method	result	size
trager	\(-\frac {\ln \left (-\frac {96 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+288 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +864 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}-432 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +278 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-432 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+492 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x -516 \left (-x^{2}+1\right )^{\frac {2}{3}}+258 \left (-x^{2}+1\right )^{\frac {1}{3}} x +17 x^{2}+342 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+258 \left (-x^{2}+1\right )^{\frac {1}{3}}-918 x +969}{\left (x -3\right )^{2}}\right )}{4}-\frac {\ln \left (-\frac {96 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+288 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +864 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}-432 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +278 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-432 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+492 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x -516 \left (-x^{2}+1\right )^{\frac {2}{3}}+258 \left (-x^{2}+1\right )^{\frac {1}{3}} x +17 x^{2}+342 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+258 \left (-x^{2}+1\right )^{\frac {1}{3}}-918 x +969}{\left (x -3\right )^{2}}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{2}+\frac {\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {48 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+144 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x -432 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}+216 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x -91 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+216 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-102 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x -474 \left (-x^{2}+1\right )^{\frac {2}{3}}+237 \left (-x^{2}+1\right )^{\frac {1}{3}} x -49 x^{2}-171 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+237 \left (-x^{2}+1\right )^{\frac {1}{3}}-546 x +399}{\left (x -3\right )^{2}}\right )}{2}\)	\(618\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*ln(-(96*RootOf(4*_Z^2+2*_Z+1)^2*x^2+288*RootOf(4*_Z^2+2*_Z+1)^2*x+864*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(2/3
)-432*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(1/3)*x+278*RootOf(4*_Z^2+2*_Z+1)*x^2-432*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^
(1/3)+492*RootOf(4*_Z^2+2*_Z+1)*x-516*(-x^2+1)^(2/3)+258*(-x^2+1)^(1/3)*x+17*x^2+342*RootOf(4*_Z^2+2*_Z+1)+258
*(-x^2+1)^(1/3)-918*x+969)/(x-3)^2)-1/2*ln(-(96*RootOf(4*_Z^2+2*_Z+1)^2*x^2+288*RootOf(4*_Z^2+2*_Z+1)^2*x+864*
RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(2/3)-432*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(1/3)*x+278*RootOf(4*_Z^2+2*_Z+1)*x^2-
432*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(1/3)+492*RootOf(4*_Z^2+2*_Z+1)*x-516*(-x^2+1)^(2/3)+258*(-x^2+1)^(1/3)*x+1
7*x^2+342*RootOf(4*_Z^2+2*_Z+1)+258*(-x^2+1)^(1/3)-918*x+969)/(x-3)^2)*RootOf(4*_Z^2+2*_Z+1)+1/2*RootOf(4*_Z^2
+2*_Z+1)*ln(-(48*RootOf(4*_Z^2+2*_Z+1)^2*x^2+144*RootOf(4*_Z^2+2*_Z+1)^2*x-432*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^
(2/3)+216*RootOf(4*_Z^2+2*_Z+1)*(-x^2+1)^(1/3)*x-91*RootOf(4*_Z^2+2*_Z+1)*x^2+216*RootOf(4*_Z^2+2*_Z+1)*(-x^2+
1)^(1/3)-102*RootOf(4*_Z^2+2*_Z+1)*x-474*(-x^2+1)^(2/3)+237*(-x^2+1)^(1/3)*x-49*x^2-171*RootOf(4*_Z^2+2*_Z+1)+
237*(-x^2+1)^(1/3)-546*x+399)/(x-3)^2)

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

[Out]

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

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Fricas [A]
time = 2.28, size = 113, normalized size = 1.45 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {18031 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + \sqrt {3} {\left (5054 \, x^{2} - 8497 \, x + 23659\right )} + 57889 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{6859 \, x^{2} + 240699 \, x - 220122}\right ) + \frac {1}{8} \, \log \left (\frac {x^{2} + 6 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} - 6 \, x + 12 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 9}{x^{2} - 6 \, x + 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(3)*arctan((18031*sqrt(3)*(-x^2 + 1)^(1/3)*(x + 1) + sqrt(3)*(5054*x^2 - 8497*x + 23659) + 57889*sqrt
(3)*(-x^2 + 1)^(2/3))/(6859*x^2 + 240699*x - 220122)) + 1/8*log((x^2 + 6*(-x^2 + 1)^(1/3)*(x + 1) - 6*x + 12*(
-x^2 + 1)^(2/3) + 9)/(x^2 - 6*x + 9))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((1 - x^2)^(1/3)*(x - 3)),x)

[Out]

-int(1/((1 - x^2)^(1/3)*(x - 3)), x)

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