### 3.293 $$\int \frac{1}{\frac{1}{\sqrt {x}}+x} \, dx$$

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

[Out]

(3*Log[1 + x^(4/3)])/4

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Rubi [A]  time = 0.0034391, antiderivative size = 12, normalized size of antiderivative = 1., 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 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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0017443, size = 12, normalized size = 1. $\frac{3}{4} \log \left (x^{4/3}+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^(-1/3) + x)^(-1),x]

[Out]

(3*Log[1 + x^(4/3)])/4

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Maple [A]  time = 0.003, size = 9, normalized size = 0.8 \begin{align*}{\frac{3}{4}\ln \left ( 1+{x}^{{\frac{4}{3}}} \right ) } \end{align*}

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 = 1.41533, size = 11, normalized size = 0.92 \begin{align*} \frac{3}{4} \, \log \left (x^{\frac{4}{3}} + 1\right ) \end{align*}

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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Fricas [A]  time = 2.1519, size = 30, normalized size = 2.5 \begin{align*} \frac{3}{4} \, \log \left (x^{\frac{4}{3}} + 1\right ) \end{align*}

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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Sympy [A]  time = 0.217919, size = 10, normalized size = 0.83 \begin{align*} \frac{3 \log{\left (x^{\frac{4}{3}} + 1 \right )}}{4} \end{align*}

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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Giac [B]  time = 1.05467, size = 43, normalized size = 3.58 \begin{align*} \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 ) \end{align*}

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)