\(\int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx\) [223]

Optimal result

Integrand size = 9, antiderivative size = 32 \[ \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx=3 \sqrt [3]{x}-\frac {3 x^{2/3}}{2}+x-3 \log \left (1+\frac {1}{\sqrt [3]{x}}\right )-\log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {196, 46} \[ \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx=-\frac {3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\frac {1}{\sqrt [3]{x}}+1\right )-\log (x) \]

[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 46

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] && Lt
Q[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]

Rubi steps \begin{align*} \text {integral}& = -\left (3 \text {Subst}\left (\int \frac {1}{x^4 (1+x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -\left (3 \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx=3 \sqrt [3]{x}-\frac {3 x^{2/3}}{2}+x-3 \log \left (1+\sqrt [3]{x}\right ) \]

[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)]

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(x -\frac {3 x^{\frac {2}{3}}}{2}+3 x^{\frac {1}{3}}-3 \ln \left (x^{\frac {1}{3}}+1\right )\)	\(21\)
default	\(x -\frac {3 x^{\frac {2}{3}}}{2}+3 x^{\frac {1}{3}}-3 \ln \left (x^{\frac {1}{3}}+1\right )\)	\(21\)
meijerg	\(\frac {x^{\frac {1}{3}} \left (4 x^{\frac {2}{3}}-6 x^{\frac {1}{3}}+12\right )}{4}-3 \ln \left (x^{\frac {1}{3}}+1\right )\)	\(27\)
trager	\(-1+x +3 x^{\frac {1}{3}}-\frac {3 x^{\frac {2}{3}}}{2}-\ln \left (-3 x^{\frac {2}{3}}-3 x^{\frac {1}{3}}-x -1\right )\)	\(32\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx=x - \frac {3}{2} \, x^{\frac {2}{3}} + 3 \, x^{\frac {1}{3}} - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx=- \frac {3 x^{\frac {2}{3}}}{2} + 3 \sqrt [3]{x} + x - 3 \log {\left (\sqrt [3]{x} + 1 \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx=-\frac {1}{2} \, x {\left (\frac {3}{x^{\frac {1}{3}}} - \frac {6}{x^{\frac {2}{3}}} - 2\right )} - \log \left (x\right ) - 3 \, \log \left (\frac {1}{x^{\frac {1}{3}}} + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx=x - \frac {3}{2} \, x^{\frac {2}{3}} + 3 \, x^{\frac {1}{3}} - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx=x-3\,\ln \left (x^{1/3}+1\right )+3\,x^{1/3}-\frac {3\,x^{2/3}}{2} \]

[In]

int(1/(1/x^(1/3) + 1),x)

[Out]

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