3.278 \(\int \frac{1+x^{2/3}}{-1+x^{2/3}} \, dx\)

Optimal. Leaf size=17 \[ x+6 \sqrt [3]{x}-6 \tanh ^{-1}\left (\sqrt [3]{x}\right ) \]

[Out]

6*x^(1/3) + x - 6*ArcTanh[x^(1/3)]

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Rubi [A]  time = 0.0146946, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {376, 459, 321, 207} \[ x+6 \sqrt [3]{x}-6 \tanh ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^(2/3))/(-1 + x^(2/3)),x]

[Out]

6*x^(1/3) + x - 6*ArcTanh[x^(1/3)]

Rule 376

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1+x^{2/3}}{-1+x^{2/3}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2 \left (1+x^2\right )}{-1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=x+6 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=6 \sqrt [3]{x}+x+6 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=6 \sqrt [3]{x}+x-6 \tanh ^{-1}\left (\sqrt [3]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0036745, size = 17, normalized size = 1. \[ x+6 \sqrt [3]{x}-6 \tanh ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^(2/3))/(-1 + x^(2/3)),x]

[Out]

6*x^(1/3) + x - 6*ArcTanh[x^(1/3)]

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Maple [A]  time = 0.004, size = 24, normalized size = 1.4 \begin{align*} x+6\,\sqrt [3]{x}+3\,\ln \left ( -1+\sqrt [3]{x} \right ) -3\,\ln \left ( \sqrt [3]{x}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x^(2/3))/(-1+x^(2/3)),x)

[Out]

x+6*x^(1/3)+3*ln(-1+x^(1/3))-3*ln(x^(1/3)+1)

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Maxima [A]  time = 0.973211, size = 31, normalized size = 1.82 \begin{align*} x + 6 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) + 3 \, \log \left (x^{\frac{1}{3}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(2/3))/(-1+x^(2/3)),x, algorithm="maxima")

[Out]

x + 6*x^(1/3) - 3*log(x^(1/3) + 1) + 3*log(x^(1/3) - 1)

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Fricas [A]  time = 1.42153, size = 77, normalized size = 4.53 \begin{align*} x + 6 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) + 3 \, \log \left (x^{\frac{1}{3}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(2/3))/(-1+x^(2/3)),x, algorithm="fricas")

[Out]

x + 6*x^(1/3) - 3*log(x^(1/3) + 1) + 3*log(x^(1/3) - 1)

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Sympy [A]  time = 0.223739, size = 27, normalized size = 1.59 \begin{align*} 6 \sqrt [3]{x} + x + 3 \log{\left (\sqrt [3]{x} - 1 \right )} - 3 \log{\left (\sqrt [3]{x} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x**(2/3))/(-1+x**(2/3)),x)

[Out]

6*x**(1/3) + x + 3*log(x**(1/3) - 1) - 3*log(x**(1/3) + 1)

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Giac [A]  time = 1.36138, size = 32, normalized size = 1.88 \begin{align*} x + 6 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) + 3 \, \log \left ({\left | x^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(2/3))/(-1+x^(2/3)),x, algorithm="giac")

[Out]

x + 6*x^(1/3) - 3*log(x^(1/3) + 1) + 3*log(abs(x^(1/3) - 1))