3.8.15 \(\int \frac {1}{(3+x) \sqrt [3]{1-x^2}} \, dx\) [715]

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

[Out]

1/4*ln(3+x)-3/8*ln(-1/2*(1-x)^(2/3)-(1+x)^(1/3))-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 = 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 = {767, 124} \begin {gather*} \frac {1}{4} \sqrt {3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {(1-x)^{2/3}}{\sqrt {3} \sqrt [3]{x+1}}\right )+\frac {1}{4} \log (x+3)-\frac {3}{8} \log \left (-\frac {1}{2} (1-x)^{2/3}-\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[-1/2*(1 - x)^(2/
3) - (1 + x)^(1/3)])/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} \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 [A]
time = 0.09, size = 105, normalized size = 1.38 \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)] +
Log[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 = 2.00, size = 616, normalized size = 8.11

method	result	size
trager	\(\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}+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 -144 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x +91 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}-474 \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}}-237 \left (-x^{2}+1\right )^{\frac {1}{3}} x -102 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -49 x^{2}+237 \left (-x^{2}+1\right )^{\frac {1}{3}}+171 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+546 x +399}{\left (3+x \right )^{2}}\right )}{2}+\frac {\ln \left (\frac {-96 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}+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 +288 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x +278 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}+516 \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}}+258 \left (-x^{2}+1\right )^{\frac {1}{3}} x -492 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -17 x^{2}-258 \left (-x^{2}+1\right )^{\frac {1}{3}}+342 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-918 x -969}{\left (3+x \right )^{2}}\right )}{4}-\frac {\ln \left (\frac {-96 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}+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 +288 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x +278 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}+516 \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}}+258 \left (-x^{2}+1\right )^{\frac {1}{3}} x -492 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -17 x^{2}-258 \left (-x^{2}+1\right )^{\frac {1}{3}}+342 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-918 x -969}{\left (3+x \right )^{2}}\right ) \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{2}\)	\(616\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(4*_Z^2-2*_Z+1)*ln(-(48*RootOf(4*_Z^2-2*_Z+1)^2*x^2+432*RootOf(4*_Z^2-2*_Z+1)*(-x^2+1)^(2/3)+216*Roo
tOf(4*_Z^2-2*_Z+1)*(-x^2+1)^(1/3)*x-144*RootOf(4*_Z^2-2*_Z+1)^2*x+91*RootOf(4*_Z^2-2*_Z+1)*x^2-474*(-x^2+1)^(2
/3)-216*RootOf(4*_Z^2-2*_Z+1)*(-x^2+1)^(1/3)-237*(-x^2+1)^(1/3)*x-102*RootOf(4*_Z^2-2*_Z+1)*x-49*x^2+237*(-x^2
+1)^(1/3)+171*RootOf(4*_Z^2-2*_Z+1)+546*x+399)/(3+x)^2)+1/4*ln((-96*RootOf(4*_Z^2-2*_Z+1)^2*x^2+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+288*RootOf(4*_Z^2-2*_Z+1)^2*x+278*RootOf
(4*_Z^2-2*_Z+1)*x^2+516*(-x^2+1)^(2/3)-432*RootOf(4*_Z^2-2*_Z+1)*(-x^2+1)^(1/3)+258*(-x^2+1)^(1/3)*x-492*RootO
f(4*_Z^2-2*_Z+1)*x-17*x^2-258*(-x^2+1)^(1/3)+342*RootOf(4*_Z^2-2*_Z+1)-918*x-969)/(3+x)^2)-1/2*ln((-96*RootOf(
4*_Z^2-2*_Z+1)^2*x^2+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+288*R
ootOf(4*_Z^2-2*_Z+1)^2*x+278*RootOf(4*_Z^2-2*_Z+1)*x^2+516*(-x^2+1)^(2/3)-432*RootOf(4*_Z^2-2*_Z+1)*(-x^2+1)^(
1/3)+258*(-x^2+1)^(1/3)*x-492*RootOf(4*_Z^2-2*_Z+1)*x-17*x^2-258*(-x^2+1)^(1/3)+342*RootOf(4*_Z^2-2*_Z+1)-918*
x-969)/(3+x)^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*} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (56) = 112\).
time = 3.76, 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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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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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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