[next] [prev] [prev-tail] [tail] [up]

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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.02, 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 = {753, 123} \[ -\frac {1}{4} \log (3-x)+\frac {3}{8} \log \left (-\frac {1}{2} (x+1)^{2/3}-\sqrt [3]{1-x}\right )-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(x+1)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((3 - x)*(1 - x^2)^(1/3)),x]

[Out]

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

Rule 123

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 753

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*}

________________________________________________________________________________________

Mathematica [C]  time = 0.06, size = 68, normalized size = 0.87 \[ \frac {3 \sqrt [3]{\frac {x-1}{x-3}} \sqrt [3]{\frac {x+1}{x-3}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};-\frac {4}{x-3},-\frac {2}{x-3}\right )}{2 \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*((-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)])/
(2*(1 - x^2)^(1/3))

________________________________________________________________________________________

fricas [A]  time = 1.88, size = 113, normalized size = 1.45 \[ -\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {18031 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + \sqrt {3} {\left (5054 \, x^{2} - 8497 \, x + 23659\right )} + 57889 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{6859 \, x^{2} + 240699 \, x - 220122}\right ) + \frac {1}{8} \, \log \left (\frac {x^{2} + 6 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} - 6 \, x + 12 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 9}{x^{2} - 6 \, x + 9}\right ) \]

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))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [C]  time = 2.80, size = 618, normalized size = 7.92 \[ \frac {\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {48 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-91 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+144 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-49 x^{2}+216 \left (-x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-102 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+237 \left (-x^{2}+1\right )^{\frac {1}{3}} x -546 x -432 \left (-x^{2}+1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+216 \left (-x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-171 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-474 \left (-x^{2}+1\right )^{\frac {2}{3}}+237 \left (-x^{2}+1\right )^{\frac {1}{3}}+399}{\left (x -3\right )^{2}}\right )}{2}-\frac {\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {96 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+278 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+288 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+17 x^{2}-432 \left (-x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+492 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+258 \left (-x^{2}+1\right )^{\frac {1}{3}} x -918 x +864 \left (-x^{2}+1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-432 \left (-x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+342 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-516 \left (-x^{2}+1\right )^{\frac {2}{3}}+258 \left (-x^{2}+1\right )^{\frac {1}{3}}+969}{\left (x -3\right )^{2}}\right )}{2}-\frac {\ln \left (-\frac {96 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+278 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+288 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+17 x^{2}-432 \left (-x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+492 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+258 \left (-x^{2}+1\right )^{\frac {1}{3}} x -918 x +864 \left (-x^{2}+1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-432 \left (-x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+342 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-516 \left (-x^{2}+1\right )^{\frac {2}{3}}+258 \left (-x^{2}+1\right )^{\frac {1}{3}}+969}{\left (x -3\right )^{2}}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*RootO
f(4*_Z^2+2*_Z+1)+237*(-x^2+1)^(1/3)-546*x+399)/(x-3)^2)-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)+25
8*(-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+17*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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{x \sqrt [3]{1 - x^{2}} - 3 \sqrt [3]{1 - x^{2}}}\, dx \]

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)

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]