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

Optimal. Leaf size=26 \[ -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]

[Out]

6*x^(1/3) - 3*x^(2/3) + x - 6*Log[1 + x^(1/3)]

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Rubi [A]  time = 0.0159524, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {376, 77} \[ -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^(1/3))/(1 + x^(1/3)),x]

[Out]

6*x^(1/3) - 3*x^(2/3) + x - 6*Log[1 + x^(1/3)]

Rule 376

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{(-1+x) x^2}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (2-2 x+x^2-\frac{2}{1+x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=6 \sqrt [3]{x}-3 x^{2/3}+x-6 \log \left (1+\sqrt [3]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0103988, size = 26, normalized size = 1. \[ -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

6*x^(1/3) - 3*x^(2/3) + x - 6*Log[1 + x^(1/3)]

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Maple [A]  time = 0.003, size = 21, normalized size = 0.8 \begin{align*} 6\,\sqrt [3]{x}-3\,{x}^{2/3}+x-6\,\ln \left ( \sqrt [3]{x}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*x^(1/3)-3*x^(2/3)+x-6*ln(x^(1/3)+1)

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Maxima [A]  time = 0.940811, size = 27, normalized size = 1.04 \begin{align*} x - 3 \, x^{\frac{2}{3}} + 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} + 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="maxima")

[Out]

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

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Fricas [A]  time = 1.3296, size = 65, normalized size = 2.5 \begin{align*} x - 3 \, x^{\frac{2}{3}} + 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} + 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]

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

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Sympy [A]  time = 0.166887, size = 24, normalized size = 0.92 \begin{align*} - 3 x^{\frac{2}{3}} + 6 \sqrt [3]{x} + x - 6 \log{\left (\sqrt [3]{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)

[Out]

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

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Giac [A]  time = 1.10426, size = 27, normalized size = 1.04 \begin{align*} x - 3 \, x^{\frac{2}{3}} + 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} + 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="giac")

[Out]

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