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

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

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \[ \frac {3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]

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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IntegrateAlgebraic [A]  time = 0.02, size = 33, normalized size = 1.00 \[ \frac {3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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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fricas [A]  time = 0.76, size = 25, normalized size = 0.76 \[ \frac {3}{2} \, {\left (x + 1\right )}^{\frac {2}{3}} - 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac {1}{3}} + 1\right ) \]

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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giac [A]  time = 1.05, size = 25, normalized size = 0.76 \[ \frac {3}{2} \, {\left (x + 1\right )}^{\frac {2}{3}} - 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac {1}{3}} + 1\right ) \]

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)

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maple [A]  time = 0.14, size = 26, normalized size = 0.79

method	result	size
derivativedivides	\(-3 \left (1+x \right )^{\frac {1}{3}}+\frac {3 \left (1+x \right )^{\frac {2}{3}}}{2}+3 \ln \left (1+\left (1+x \right )^{\frac {1}{3}}\right )\)	\(26\)
trager	\(-3 \left (1+x \right )^{\frac {1}{3}}+\frac {3 \left (1+x \right )^{\frac {2}{3}}}{2}+\ln \left (-3 \left (1+x \right )^{\frac {2}{3}}-3 \left (1+x \right )^{\frac {1}{3}}-x -2\right )\)	\(36\)
default	\(\ln \left (2+x \right )+\frac {3 \left (1+x \right )^{\frac {2}{3}}}{2}+2 \ln \left (1+\left (1+x \right )^{\frac {1}{3}}\right )-\ln \left (\left (1+x \right )^{\frac {2}{3}}-\left (1+x \right )^{\frac {1}{3}}+1\right )-3 \left (1+x \right )^{\frac {1}{3}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 25, normalized size = 0.76 \[ \frac {3}{2} \, {\left (x + 1\right )}^{\frac {2}{3}} - 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac {1}{3}} + 1\right ) \]

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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mupad [B]  time = 0.20, size = 25, normalized size = 0.76 \[ 3\,\ln \left ({\left (x+1\right )}^{1/3}+1\right )-3\,{\left (x+1\right )}^{1/3}+\frac {3\,{\left (x+1\right )}^{2/3}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x + 1)^(1/3) + 1),x)

[Out]

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

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sympy [A]  time = 0.14, size = 29, normalized size = 0.88 \[ \frac {3 \left (x + 1\right )^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x + 1} + 3 \log {\left (\sqrt [3]{x + 1} + 1 \right )} \]

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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