\(\int \frac {1}{1+\sqrt [3]{1+x}} \, dx\) [47]

Optimal result

Integrand size = 11, antiderivative size = 33 \[ \int \frac {1}{1+\sqrt [3]{1+x}} \, dx=-3 \sqrt [3]{1+x}+\frac {3}{2} (1+x)^{2/3}+3 \log \left (1+\sqrt [3]{1+x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {253, 196, 45} \[ \int \frac {1}{1+\sqrt [3]{1+x}} \, dx=\frac {3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]

[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 45

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 196

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 253

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

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+\sqrt [3]{1+x}} \, dx=\frac {3}{2} \sqrt [3]{1+x} \left (-2+\sqrt [3]{1+x}\right )+3 \log \left (1+\sqrt [3]{1+x}\right ) \]

[In]

Integrate[(1 + (1 + x)^(1/3))^(-1),x]

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 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	\(\ln \left (2+x \right )+\frac {3 \left (1+x \right )^{\frac {2}{3}}}{2}+2 \ln \left (1+\left (1+x \right )^{\frac {1}{3}}\right )-\ln \left (\left (1+x \right )^{\frac {2}{3}}-\left (1+x \right )^{\frac {1}{3}}+1\right )-3 \left (1+x \right )^{\frac {1}{3}}\)	\(47\)

[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))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1}{1+\sqrt [3]{1+x}} \, dx=\frac {3}{2} \, {\left (x + 1\right )}^{\frac {2}{3}} - 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac {1}{3}} + 1\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {1}{1+\sqrt [3]{1+x}} \, dx=\frac {3 \left (x + 1\right )^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x + 1} + 3 \log {\left (\sqrt [3]{x + 1} + 1 \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1}{1+\sqrt [3]{1+x}} \, dx=\frac {3}{2} \, {\left (x + 1\right )}^{\frac {2}{3}} - 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac {1}{3}} + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1}{1+\sqrt [3]{1+x}} \, dx=\frac {3}{2} \, {\left (x + 1\right )}^{\frac {2}{3}} - 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac {1}{3}} + 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1}{1+\sqrt [3]{1+x}} \, dx=3\,\ln \left ({\left (x+1\right )}^{1/3}+1\right )-3\,{\left (x+1\right )}^{1/3}+\frac {3\,{\left (x+1\right )}^{2/3}}{2} \]

[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