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

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 = {383, 78} \begin {gather*} -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \end {gather*}

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 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*} \int \frac {-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx &=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*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} 6 \sqrt [3]{x}-3 x^{2/3}+x-6 \log \left (1+\sqrt [3]{x}\right ) \end {gather*}

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 = 0.24, size = 21, normalized size = 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	\(x -1+6 x^{\frac {1}{3}}-3 x^{\frac {2}{3}}-2 \ln \left (-3 x^{\frac {2}{3}}-3 x^{\frac {1}{3}}-x -1\right )\)	\(32\)
meijerg	\(\frac {x^{\frac {1}{3}} \left (4 x^{\frac {2}{3}}-6 x^{\frac {1}{3}}+12\right )}{4}-6 \ln \left (1+x^{\frac {1}{3}}\right )+\frac {x^{\frac {1}{3}} \left (-3 x^{\frac {1}{3}}+6\right )}{2}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Maxima [A]
time = 0.26, size = 20, normalized size = 0.77 \begin {gather*} x - 3 \, x^{\frac {2}{3}} + 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} + 1\right ) \end {gather*}

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 = 2.09, size = 20, normalized size = 0.77 \begin {gather*} x - 3 \, x^{\frac {2}{3}} + 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} + 1\right ) \end {gather*}

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 = 0.14, size = 24, normalized size = 0.92 \begin {gather*} - 3 x^{\frac {2}{3}} + 6 \sqrt [3]{x} + x - 6 \log {\left (\sqrt [3]{x} + 1 \right )} \end {gather*}

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 = 1.43, size = 20, normalized size = 0.77 \begin {gather*} x - 3 \, x^{\frac {2}{3}} + 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} + 1\right ) \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 20, normalized size = 0.77 \begin {gather*} x-6\,\ln \left (x^{1/3}+1\right )+6\,x^{1/3}-3\,x^{2/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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