\(\int \frac {1}{1-\sqrt {1+x}} \, dx\) [240]

Optimal result

Integrand size = 13, antiderivative size = 24 \[ \int \frac {1}{1-\sqrt {1+x}} \, dx=-2 \sqrt {1+x}-2 \log \left (1-\sqrt {1+x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 196, 45} \[ \int \frac {1}{1-\sqrt {1+x}} \, dx=-2 \sqrt {x+1}-2 \log \left (1-\sqrt {x+1}\right ) \]

[In]

Int[(1 - Sqrt[1 + x])^(-1),x]

[Out]

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

Rule 45

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])

Rule 196

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 253

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1-\sqrt {x}} \, dx,x,1+x\right ) \\ & = 2 \text {Subst}\left (\int \frac {x}{1-x} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{1-\sqrt {1+x}} \, dx=-2 \sqrt {1+x}-2 \log \left (-1+\sqrt {1+x}\right ) \]

[In]

Integrate[(1 - Sqrt[1 + x])^(-1),x]

[Out]

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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(-2 \sqrt {1+x}-2 \ln \left (-1+\sqrt {1+x}\right )\)	\(19\)
trager	\(-2 \sqrt {1+x}-\ln \left (2 \sqrt {1+x}-2-x \right )\)	\(24\)
default	\(-\ln \left (x \right )-2 \sqrt {1+x}-\ln \left (-1+\sqrt {1+x}\right )+\ln \left (1+\sqrt {1+x}\right )\)	\(31\)

[In]

int(1/(1-(1+x)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {1}{1-\sqrt {1+x}} \, dx=-2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} - 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {1}{1-\sqrt {1+x}} \, dx=- 2 \sqrt {x + 1} - 2 \log {\left (\sqrt {x + 1} - 1 \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {1}{1-\sqrt {1+x}} \, dx=-2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} - 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {1}{1-\sqrt {1+x}} \, dx=-2 \, \sqrt {x + 1} - 2 \, \log \left ({\left | \sqrt {x + 1} - 1 \right |}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {1}{1-\sqrt {1+x}} \, dx=-2\,\ln \left (\sqrt {x+1}-1\right )-2\,\sqrt {x+1} \]

[In]

int(-1/((x + 1)^(1/2) - 1),x)

[Out]

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