\(\int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx\) [277]

Optimal result

Integrand size = 17, antiderivative size = 26 \[ \int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx=6 \sqrt [3]{x}-3 x^{2/3}+x-6 \log \left (1+\sqrt [3]{x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {383, 78} \[ \int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx=-3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]

[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 78

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]))))

Rule 383

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]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {(-1+x) x^2}{1+x} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx=6 \sqrt [3]{x}-3 x^{2/3}+x-6 \log \left (1+\sqrt [3]{x}\right ) \]

[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)]

Maple [A] (verified)

Time = 3.92 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(6 x^{\frac {1}{3}}-3 x^{\frac {2}{3}}+x -6 \ln \left (1+x^{\frac {1}{3}}\right )\)	\(21\)
default	\(6 x^{\frac {1}{3}}-3 x^{\frac {2}{3}}+x -6 \ln \left (1+x^{\frac {1}{3}}\right )\)	\(21\)
trager	\(-1+x +6 x^{\frac {1}{3}}-3 x^{\frac {2}{3}}-2 \ln \left (-3 x^{\frac {2}{3}}-3 x^{\frac {1}{3}}-x -1\right )\)	\(32\)
meijerg	\(\frac {x^{\frac {1}{3}} \left (4 x^{\frac {2}{3}}-6 x^{\frac {1}{3}}+12\right )}{4}-6 \ln \left (1+x^{\frac {1}{3}}\right )+\frac {x^{\frac {1}{3}} \left (-3 x^{\frac {1}{3}}+6\right )}{2}\)	\(39\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx=x - 3 \, x^{\frac {2}{3}} + 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx=- 3 x^{\frac {2}{3}} + 6 \sqrt [3]{x} + x - 6 \log {\left (\sqrt [3]{x} + 1 \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx=x - 3 \, x^{\frac {2}{3}} + 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx=x - 3 \, x^{\frac {2}{3}} + 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx=x-6\,\ln \left (x^{1/3}+1\right )+6\,x^{1/3}-3\,x^{2/3} \]

[In]

int((x^(1/3) - 1)/(x^(1/3) + 1),x)

[Out]

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