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

Optimal. Leaf size=33 \[ \frac{3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0101715, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 190, 43} \[ \frac{3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + (1 + x)^(1/3))^(-1),x]

[Out]

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

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, 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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0120075, size = 33, normalized size = 1. \[ \frac{3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + (1 + x)^(1/3))^(-1),x]

[Out]

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

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Maple [A]  time = 0.008, size = 47, normalized size = 1.4 \begin{align*} \ln \left ( 2+x \right ) +{\frac{3}{2} \left ( 1+x \right ) ^{{\frac{2}{3}}}}+2\,\ln \left ( 1+\sqrt [3]{1+x} \right ) -\ln \left ( \left ( 1+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{1+x}+1 \right ) -3\,\sqrt [3]{1+x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.955634, size = 34, normalized size = 1.03 \begin{align*} \frac{3}{2} \,{\left (x + 1\right )}^{\frac{2}{3}} - 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\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]

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

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Fricas [A]  time = 1.87864, size = 86, normalized size = 2.61 \begin{align*} \frac{3}{2} \,{\left (x + 1\right )}^{\frac{2}{3}} - 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\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]

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

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Sympy [A]  time = 0.115146, size = 29, normalized size = 0.88 \begin{align*} \frac{3 \left (x + 1\right )^{\frac{2}{3}}}{2} - 3 \sqrt [3]{x + 1} + 3 \log{\left (\sqrt [3]{x + 1} + 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 + 1)**(2/3)/2 - 3*(x + 1)**(1/3) + 3*log((x + 1)**(1/3) + 1)

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Giac [A]  time = 1.04661, size = 34, normalized size = 1.03 \begin{align*} \frac{3}{2} \,{\left (x + 1\right )}^{\frac{2}{3}} - 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\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]

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