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

Optimal. Leaf size=172 \[ \frac {\log \left (2 \sqrt [3]{1-x^3}+\sqrt [3]{2} x-\sqrt [3]{2}\right )}{2 \sqrt [3]{2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{1-x^3}-\sqrt [3]{2} x+\sqrt [3]{2}}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (4 \left (1-x^3\right )^{2/3}+\left (2 \sqrt [3]{2}-2 \sqrt [3]{2} x\right ) \sqrt [3]{1-x^3}+2^{2/3} x^2-2\ 2^{2/3} x+2^{2/3}\right )}{4 \sqrt [3]{2}} \]

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Rubi [A]  time = 0.05, antiderivative size = 97, normalized size of antiderivative = 0.56, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2148} \begin {gather*} \frac {3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}-\frac {\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 + x)*(1 - x^3)^(1/3)),x]

[Out]

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

Rule 2148

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*ArcTan[(1 - (2^(1/3)*Rt[b,
3]*(c - d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(2^(4/3)*Rt[b, 3]*c), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rubi steps

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

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Mathematica [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.99, size = 172, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+\left (2 \sqrt [3]{2}-2 \sqrt [3]{2} x\right ) \sqrt [3]{1-x^3}+4 \left (1-x^3\right )^{2/3}\right )}{4 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 3.31, size = 301, normalized size = 1.75 \begin {gather*} \frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (13 \, x^{6} + 2 \, x^{5} + 19 \, x^{4} - 4 \, x^{3} + 19 \, x^{2} + 2 \, x + 13\right )} - 4 \, \sqrt {2} {\left (5 \, x^{5} - 5 \, x^{4} + 6 \, x^{3} - 6 \, x^{2} + 5 \, x - 5\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 16 \cdot 2^{\frac {1}{6}} {\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (3 \, x^{6} - 18 \, x^{5} - 3 \, x^{4} - 28 \, x^{3} - 3 \, x^{2} - 18 \, x + 3\right )}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {4 \cdot 2^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 1\right )} + 2^{\frac {1}{3}} {\left (5 \, x^{4} + 6 \, x^{2} + 5\right )} - 2 \, {\left (3 \, x^{3} - x^{2} + x - 3\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{2} + 2 \, x + 1\right )} - 2 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 4 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + 2 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(13*x^6 + 2*x^5 + 19*x^4 - 4*x^3 + 19*x^2 + 2*x + 13)
- 4*sqrt(2)*(5*x^5 - 5*x^4 + 6*x^3 - 6*x^2 + 5*x - 5)*(-x^3 + 1)^(1/3) + 16*2^(1/6)*(x^4 + 2*x^3 + 2*x^2 + 2*
x + 1)*(-x^3 + 1)^(2/3))/(3*x^6 - 18*x^5 - 3*x^4 - 28*x^3 - 3*x^2 - 18*x + 3)) - 1/24*2^(2/3)*log((4*2^(2/3)*(
-x^3 + 1)^(2/3)*(x^2 + 1) + 2^(1/3)*(5*x^4 + 6*x^2 + 5) - 2*(3*x^3 - x^2 + x - 3)*(-x^3 + 1)^(1/3))/(x^4 + 4*x
^3 + 6*x^2 + 4*x + 1)) + 1/12*2^(2/3)*log((2^(2/3)*(x^2 + 2*x + 1) - 2*2^(1/3)*(-x^3 + 1)^(1/3)*(x - 1) - 4*(-
x^3 + 1)^(2/3))/(x^2 + 2*x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 8.28, size = 769, normalized size = 4.47

method	result	size
trager	\(\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +10 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -4 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-2 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +9 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x +2 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-9 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )+7 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}+35 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{3}-4\right ) x +30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x -26 \left (-x^{3}+1\right )^{\frac {2}{3}}+7 \RootOf \left (\textit {\_Z}^{3}-4\right )+35 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{\left (1+x \right )^{2}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (-\frac {10 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -8 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-4 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -26 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x +4 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+26 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )-35 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-28 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}-4\right ) x -8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x +36 \left (-x^{3}+1\right )^{\frac {2}{3}}-35 \RootOf \left (\textit {\_Z}^{3}-4\right )-28 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{\left (1+x \right )^{2}}\right )}{4}\)	\(769\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*ln((2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z
^2)*RootOf(_Z^3-4)^3*x+10*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x-4*(-x^3+1)^
(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2-2*(-x^3+1)^(1/3)*RootOf(_Z^3-4)^2*x
+9*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x+2*(-x^3+1)^(1/3)*RootOf
(_Z^3-4)^2-9*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)+7*RootOf(_Z^3-4
)*x^2+35*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2+6*RootOf(_Z^3-4)*x+30*RootOf(RootOf(_Z^3-4)^2
+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x-26*(-x^3+1)^(2/3)+7*RootOf(_Z^3-4)+35*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-
4)+4*_Z^2))/(1+x)^2)+1/4*RootOf(_Z^3-4)*ln(-(10*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^
3-4)^3*x+8*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x-8*(-x^3+1)^(2/3)*RootOf(Ro
otOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2-4*(-x^3+1)^(1/3)*RootOf(_Z^3-4)^2*x-26*(-x^3+1)^(1
/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x+4*(-x^3+1)^(1/3)*RootOf(_Z^3-4)^2+26*
(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)-35*RootOf(_Z^3-4)*x^2-28*Roo
tOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2-10*RootOf(_Z^3-4)*x-8*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf
(_Z^3-4)+4*_Z^2)*x+36*(-x^3+1)^(2/3)-35*RootOf(_Z^3-4)-28*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))
/(1+x)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x^3)^(1/3)*(x + 1)),x)

[Out]

int(1/((1 - x^3)^(1/3)*(x + 1)), 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^{2} + x + 1\right )} \left (x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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