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

Optimal. Leaf size=33 \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]

[Out]

6*(1 + x)^(1/6) + 3*(1 + x)^(1/3) + 6*Log[1 - (1 + x)^(1/6)]

________________________________________________________________________________________

Rubi [A]  time = 0.0190299, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2012, 1593, 266, 43} \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

6*(1 + x)^(1/6) + 3*(1 + x)^(1/3) + 6*Log[1 - (1 + x)^(1/6)]

Rule 2012

Int[((a_.)*(u_)^(j_.) + (b_.)*(u_)^(n_.))^(p_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a*x^j + b
*x^n)^p, x], x, u], x] /; FreeQ[{a, b, j, n, p}, x] && LinearQ[u, x] && NeQ[u, x]

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

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])

Rubi steps

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

Mathematica [A]  time = 0.0214325, size = 33, normalized size = 1. \[ 3 \left (\sqrt [3]{x+1}+2 \sqrt [6]{x+1}+2 \log \left (1-\sqrt [6]{x+1}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*(2*(1 + x)^(1/6) + (1 + x)^(1/3) + 2*Log[1 - (1 + x)^(1/6)])

________________________________________________________________________________________

Maple [B]  time = 0.031, size = 111, normalized size = 3.4 \begin{align*} 6\,\sqrt [6]{1+x}+3\,\sqrt [3]{1+x}+\ln \left ( x \right ) +2\,\ln \left ( -1+\sqrt [6]{1+x} \right ) -\ln \left ( \sqrt [3]{1+x}+\sqrt [6]{1+x}+1 \right ) -2\,\ln \left ( 1+\sqrt [6]{1+x} \right ) +\ln \left ( \sqrt [3]{1+x}-\sqrt [6]{1+x}+1 \right ) -\ln \left ( 1+\sqrt{1+x} \right ) +\ln \left ( -1+\sqrt{1+x} \right ) +2\,\ln \left ( -1+\sqrt [3]{1+x} \right ) -\ln \left ( \left ( 1+x \right ) ^{{\frac{2}{3}}}+\sqrt [3]{1+x}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*(1+x)^(1/6)+3*(1+x)^(1/3)+ln(x)+2*ln(-1+(1+x)^(1/6))-ln((1+x)^(1/3)+(1+x)^(1/6)+1)-2*ln(1+(1+x)^(1/6))+ln((1
+x)^(1/3)-(1+x)^(1/6)+1)-ln(1+(1+x)^(1/2))+ln(-1+(1+x)^(1/2))+2*ln(-1+(1+x)^(1/3))-ln((1+x)^(2/3)+(1+x)^(1/3)+
1)

________________________________________________________________________________________

Maxima [A]  time = 0.927302, size = 34, normalized size = 1.03 \begin{align*} 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 6 \,{\left (x + 1\right )}^{\frac{1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac{1}{6}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.72372, size = 84, normalized size = 2.55 \begin{align*} 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 6 \,{\left (x + 1\right )}^{\frac{1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac{1}{6}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x + 1\right )^{\frac{2}{3}} - \sqrt{x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((x + 1)**(2/3) - sqrt(x + 1)), x)

________________________________________________________________________________________

Giac [A]  time = 1.0838, size = 35, normalized size = 1.06 \begin{align*} 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 6 \,{\left (x + 1\right )}^{\frac{1}{6}} + 6 \, \log \left ({\left |{\left (x + 1\right )}^{\frac{1}{6}} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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