3.1628 \(\int \frac{\sqrt [3]{-1+x}}{\sqrt [3]{1+x}} \, dx\)

Optimal. Leaf size=77 \[ \sqrt [3]{x-1} (x+1)^{2/3}+\frac{1}{3} \log (x-1)+\log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{x-1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0138222, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {50, 59} \[ \sqrt [3]{x-1} (x+1)^{2/3}+\frac{1}{3} \log (x-1)+\log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{x-1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x)^(1/3)/(1 + x)^(1/3),x]

[Out]

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

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 59

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

Rubi steps

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

Mathematica [C]  time = 0.0187196, size = 48, normalized size = 0.62 \[ \frac{3 \left (\frac{x-1}{x+1}\right )^{4/3} (x+1)^{4/3} \, _2F_1\left (\frac{1}{3},\frac{4}{3};\frac{7}{3};\frac{1-x}{2}\right )}{4 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((-1 + x)/(1 + x))^(4/3)*(1 + x)^(4/3)*Hypergeometric2F1[1/3, 4/3, 7/3, (1 - x)/2])/(4*2^(1/3))

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{-1+x}{\frac{1}{\sqrt [3]{1+x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.0401, size = 359, normalized size = 4.66 \begin{align*} -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (x + 1\right )} + 2 \, \sqrt{3}{\left (x + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}^{\frac{1}{3}}}{3 \,{\left (x + 1\right )}}\right ) +{\left (x + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}^{\frac{1}{3}} - \frac{1}{3} \, \log \left (\frac{{\left (x + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}^{\frac{1}{3}} +{\left (x + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )}^{\frac{2}{3}} + x + 1}{x + 1}\right ) + \frac{2}{3} \, \log \left (\frac{{\left (x + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}^{\frac{1}{3}} - x - 1}{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 2.48307, size = 39, normalized size = 0.51 \begin{align*} \frac{2^{\frac{2}{3}} \left (x - 1\right )^{\frac{4}{3}} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{\left (x - 1\right ) e^{i \pi }}{2}} \right )}}{2 \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)**(1/3),x)

[Out]

2**(2/3)*(x - 1)**(4/3)*gamma(4/3)*hyper((1/3, 4/3), (7/3,), (x - 1)*exp_polar(I*pi)/2)/(2*gamma(7/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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