### 3.223 $$\int \frac{1}{1+\frac{1}{\sqrt [3]{x}}} \, dx$$

Optimal. Leaf size=32 $-\frac{3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\frac{1}{\sqrt [3]{x}}+1\right )-\log (x)$

[Out]

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

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Rubi [A]  time = 0.0132232, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {190, 44} $-\frac{3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\frac{1}{\sqrt [3]{x}}+1\right )-\log (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + x^(-1/3))^(-1),x]

[Out]

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

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 44

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

Rubi steps

\begin{align*} \int \frac{1}{1+\frac{1}{\sqrt [3]{x}}} \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{1}{x^4 (1+x)} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \left (\frac{1}{x^4}-\frac{1}{x^3}+\frac{1}{x^2}-\frac{1}{x}+\frac{1}{1+x}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=3 \sqrt [3]{x}-\frac{3 x^{2/3}}{2}+x-3 \log \left (1+\frac{1}{\sqrt [3]{x}}\right )-\log (x)\\ \end{align*}

Mathematica [A]  time = 0.010988, size = 28, normalized size = 0.88 $-\frac{3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 + x^(-1/3))^(-1),x]

[Out]

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

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Maple [A]  time = 0.002, size = 21, normalized size = 0.7 \begin{align*} x-{\frac{3}{2}{x}^{{\frac{2}{3}}}}+3\,\sqrt [3]{x}-3\,\ln \left ( \sqrt [3]{x}+1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.930783, size = 38, normalized size = 1.19 \begin{align*} -\frac{1}{2} \, x{\left (\frac{3}{x^{\frac{1}{3}}} - \frac{6}{x^{\frac{2}{3}}} - 2\right )} - \log \left (x\right ) - 3 \, \log \left (\frac{1}{x^{\frac{1}{3}}} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.90929, size = 68, normalized size = 2.12 \begin{align*} x - \frac{3}{2} \, x^{\frac{2}{3}} + 3 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.113406, size = 26, normalized size = 0.81 \begin{align*} - \frac{3 x^{\frac{2}{3}}}{2} + 3 \sqrt [3]{x} + x - 3 \log{\left (\sqrt [3]{x} + 1 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04935, size = 27, normalized size = 0.84 \begin{align*} x - \frac{3}{2} \, x^{\frac{2}{3}} + 3 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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