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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {253, 196, 45} \begin {gather*} -2 \sqrt {x+1}-2 \log \left (1-\sqrt {x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{1-\sqrt {1+x}} \, dx &=\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]
time = 0.01, size = 22, normalized size = 0.92 \begin {gather*} -2 \sqrt {1+x}-2 \log \left (-1+\sqrt {1+x}\right ) \end {gather*}

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 = 0.04, size = 31, normalized size = 1.29

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	\(-\ln \left (x \right )-2 \sqrt {1+x}-\ln \left (-1+\sqrt {1+x}\right )+\ln \left (1+\sqrt {1+x}\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 2.46, size = 18, normalized size = 0.75 \begin {gather*} -2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} - 1\right ) \end {gather*}

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 = 0.59, size = 18, normalized size = 0.75 \begin {gather*} -2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} - 1\right ) \end {gather*}

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 = 0.04, size = 20, normalized size = 0.83 \begin {gather*} - 2 \sqrt {x + 1} - 2 \log {\left (\sqrt {x + 1} - 1 \right )} \end {gather*}

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 = 0.43, size = 19, normalized size = 0.79 \begin {gather*} -2 \, \sqrt {x + 1} - 2 \, \log \left ({\left | \sqrt {x + 1} - 1 \right |}\right ) \end {gather*}

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))

________________________________________________________________________________________

Mupad [B]
time = 0.11, size = 18, normalized size = 0.75 \begin {gather*} -2\,\ln \left (\sqrt {x+1}-1\right )-2\,\sqrt {x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((x + 1)^(1/2) - 1),x)

[Out]

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

________________________________________________________________________________________