3.1455 \(\int \frac{\sqrt [3]{1-x}}{1+x} \, dx\)

Optimal. Leaf size=84 \[ 3 \sqrt [3]{1-x}+\frac{3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac{\log (x+1)}{2^{2/3}}-\sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x}+1}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0377397, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {50, 57, 617, 204, 31} \[ 3 \sqrt [3]{1-x}+\frac{3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac{\log (x+1)}{2^{2/3}}-\sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x}+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(1/3)/(1 + x),x]

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.047317, size = 104, normalized size = 1.24 \[ 3 \sqrt [3]{1-x}+\sqrt [3]{2} \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )-\frac{\log \left ((1-x)^{2/3}+\sqrt [3]{2-2 x}+2^{2/3}\right )}{2^{2/3}}-\sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x}+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(1/3)/(1 + x),x]

[Out]

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

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Maple [A]  time = 0.005, size = 84, normalized size = 1. \begin{align*} 3\,\sqrt [3]{1-x}+\sqrt [3]{2}\ln \left ( \sqrt [3]{1-x}-\sqrt [3]{2} \right ) -{\frac{\sqrt [3]{2}}{2}\ln \left ( \left ( 1-x \right ) ^{{\frac{2}{3}}}+\sqrt [3]{2}\sqrt [3]{1-x}+{2}^{{\frac{2}{3}}} \right ) }-\sqrt [3]{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+{2}^{{\frac{2}{3}}}\sqrt [3]{1-x} \right ) } \right ) \sqrt{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46232, size = 116, normalized size = 1.38 \begin{align*} -\sqrt{3} 2^{\frac{1}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{2} \cdot 2^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}} +{\left (-x + 1\right )}^{\frac{2}{3}}\right ) + 2^{\frac{1}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x + 1\right )}^{\frac{1}{3}}\right ) + 3 \,{\left (-x + 1\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x + 1)^(1/3))) - 1/2*2^(1/3)*log(2^(2/3) + 2^(1/3)*
(-x + 1)^(1/3) + (-x + 1)^(2/3)) + 2^(1/3)*log(-2^(1/3) + (-x + 1)^(1/3)) + 3*(-x + 1)^(1/3)

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Fricas [A]  time = 2.19034, size = 275, normalized size = 3.27 \begin{align*} -\sqrt{3} 2^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3} 2^{\frac{2}{3}}{\left (-x + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - \frac{1}{2} \cdot 2^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}} +{\left (-x + 1\right )}^{\frac{2}{3}}\right ) + 2^{\frac{1}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x + 1\right )}^{\frac{1}{3}}\right ) + 3 \,{\left (-x + 1\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*2^(1/3)*arctan(1/3*sqrt(3)*2^(2/3)*(-x + 1)^(1/3) + 1/3*sqrt(3)) - 1/2*2^(1/3)*log(2^(2/3) + 2^(1/3)*
(-x + 1)^(1/3) + (-x + 1)^(2/3)) + 2^(1/3)*log(-2^(1/3) + (-x + 1)^(1/3)) + 3*(-x + 1)^(1/3)

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Sympy [C]  time = 1.92376, size = 170, normalized size = 2.02 \begin{align*} \frac{4 \sqrt [3]{-1} \sqrt [3]{x - 1} \Gamma \left (\frac{4}{3}\right )}{\Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{-2} e^{- \frac{i \pi }{3}} \log{\left (- \frac{2^{\frac{2}{3}} \sqrt [3]{x - 1} e^{\frac{i \pi }{3}}}{2} + 1 \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} - \frac{4 \sqrt [3]{-2} \log{\left (- \frac{2^{\frac{2}{3}} \sqrt [3]{x - 1} e^{i \pi }}{2} + 1 \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{-2} e^{\frac{i \pi }{3}} \log{\left (- \frac{2^{\frac{2}{3}} \sqrt [3]{x - 1} e^{\frac{5 i \pi }{3}}}{2} + 1 \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(-1)**(1/3)*(x - 1)**(1/3)*gamma(4/3)/gamma(7/3) + 4*(-2)**(1/3)*exp(-I*pi/3)*log(-2**(2/3)*(x - 1)**(1/3)*e
xp_polar(I*pi/3)/2 + 1)*gamma(4/3)/(3*gamma(7/3)) - 4*(-2)**(1/3)*log(-2**(2/3)*(x - 1)**(1/3)*exp_polar(I*pi)
/2 + 1)*gamma(4/3)/(3*gamma(7/3)) + 4*(-2)**(1/3)*exp(I*pi/3)*log(-2**(2/3)*(x - 1)**(1/3)*exp_polar(5*I*pi/3)
/2 + 1)*gamma(4/3)/(3*gamma(7/3))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError