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

Optimal. Leaf size=76 \[ \frac {1}{4} \log (x+3)-\frac {3}{8} \log \left (-\frac {1}{2} (1-x)^{2/3}-\sqrt [3]{x+1}\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(1-x)^{2/3}}{\sqrt {3} \sqrt [3]{x+1}}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {753, 123} \begin {gather*} \frac {1}{4} \log (x+3)-\frac {3}{8} \log \left (-\frac {1}{2} (1-x)^{2/3}-\sqrt [3]{x+1}\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(1-x)^{2/3}}{\sqrt {3} \sqrt [3]{x+1}}\right ) \end {gather*}

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)^(2/3)/2
- (1 + x)^(1/3)])/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} \sqrt [3]{1+x} (3+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 (-\frac {1}{2} (1-x)^{2/3}-\sqrt [3]{1+x}\right )\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 68, normalized size = 0.89 \begin {gather*} -\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}} \end {gather*}

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))

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IntegrateAlgebraic [A]  time = 0.13, size = 110, normalized size = 1.45 \begin {gather*} -\frac {1}{4} \log \left (2 \sqrt [3]{1-x^2}-x+1\right )+\frac {1}{8} \log \left (x^2+4 \left (1-x^2\right )^{2/3}+(2 x-2) \sqrt [3]{1-x^2}-2 x+1\right )-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{\sqrt [3]{1-x^2}+x-1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.69, size = 115, normalized size = 1.51 \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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 3\right )}}\,{d x} \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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maple [C]  time = 2.87, size = 618, normalized size = 8.13 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x+3)/(-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)

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

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