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

Optimal. Leaf size=12 \[ \frac {3}{4} \log \left (x^{4/3}+1\right ) \]

[Out]

3/4*ln(1+x^(4/3))

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1593, 260} \[ \frac {3}{4} \log \left (x^{4/3}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x^(-1/3) + x)^(-1),x]

[Out]

(3*Log[1 + x^(4/3)])/4

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ \frac {3}{4} \log \left (x^{4/3}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(-1/3) + x)^(-1),x]

[Out]

(3*Log[1 + x^(4/3)])/4

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fricas [A]  time = 0.40, size = 8, normalized size = 0.67 \[ \frac {3}{4} \, \log \left (x^{\frac {4}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*log(x^(4/3) + 1)

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giac [B]  time = 0.90, size = 32, normalized size = 2.67 \[ \frac {3}{4} \, \log \left (\sqrt {2} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {3}{4} \, \log \left (-\sqrt {2} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*log(sqrt(2)*x^(1/3) + x^(2/3) + 1) + 3/4*log(-sqrt(2)*x^(1/3) + x^(2/3) + 1)

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maple [A]  time = 0.00, size = 9, normalized size = 0.75 \[ \frac {3 \ln \left (x^{\frac {4}{3}}+1\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*ln(1+x^(4/3))

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maxima [A]  time = 1.49, size = 8, normalized size = 0.67 \[ \frac {3}{4} \, \log \left (x^{\frac {4}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*log(x^(4/3) + 1)

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mupad [B]  time = 0.22, size = 8, normalized size = 0.67 \[ \frac {3\,\ln \left (x^{4/3}+1\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + 1/x^(1/3)),x)

[Out]

(3*log(x^(4/3) + 1))/4

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sympy [A]  time = 0.24, size = 10, normalized size = 0.83 \[ \frac {3 \log {\left (x^{\frac {4}{3}} + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x**(4/3) + 1)/4

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