3.240 \(\int \frac{1}{1-\sqrt{1+x}} \, dx\)

Optimal. Leaf size=24 \[ -2 \sqrt{x+1}-2 \log \left (1-\sqrt{x+1}\right ) \]

[Out]

-2*Sqrt[1 + x] - 2*Log[1 - Sqrt[1 + x]]

________________________________________________________________________________________

Rubi [A]  time = 0.010346, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 190, 43} \[ -2 \sqrt{x+1}-2 \log \left (1-\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sqrt[1 + x])^(-1),x]

[Out]

-2*Sqrt[1 + x] - 2*Log[1 - Sqrt[1 + x]]

Rule 247

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

Rule 190

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]

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}{1-\sqrt{1+x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{x}} \, dx,x,1+x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x}{1-x} \, dx,x,\sqrt{1+x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-1+\frac{1}{1-x}\right ) \, dx,x,\sqrt{1+x}\right )\\ &=-2 \sqrt{1+x}-2 \log \left (1-\sqrt{1+x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0100692, size = 24, normalized size = 1. \[ -2 \sqrt{x+1}-2 \log \left (1-\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sqrt[1 + x])^(-1),x]

[Out]

-2*Sqrt[1 + x] - 2*Log[1 - Sqrt[1 + x]]

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 31, normalized size = 1.3 \begin{align*} -\ln \left ( x \right ) -2\,\sqrt{1+x}-\ln \left ( -1+\sqrt{1+x} \right ) +\ln \left ( 1+\sqrt{1+x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.93336, size = 24, normalized size = 1. \begin{align*} -2 \, \sqrt{x + 1} - 2 \, \log \left (\sqrt{x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(x + 1) - 2*log(sqrt(x + 1) - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.65158, size = 55, normalized size = 2.29 \begin{align*} -2 \, \sqrt{x + 1} - 2 \, \log \left (\sqrt{x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(x + 1) - 2*log(sqrt(x + 1) - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.118422, size = 20, normalized size = 0.83 \begin{align*} - 2 \sqrt{x + 1} - 2 \log{\left (\sqrt{x + 1} - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(x + 1) - 2*log(sqrt(x + 1) - 1)

________________________________________________________________________________________

Giac [A]  time = 1.12016, size = 26, normalized size = 1.08 \begin{align*} -2 \, \sqrt{x + 1} - 2 \, \log \left ({\left | \sqrt{x + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(x + 1) - 2*log(abs(sqrt(x + 1) - 1))