3.696 \(\int \frac {1}{1+x-\sqrt {2+x}} \, dx\)

Optimal. Leaf size=61 \[ \frac {1}{5} \left (5-\sqrt {5}\right ) \log \left (-2 \sqrt {x+2}-\sqrt {5}+1\right )+\frac {1}{5} \left (5+\sqrt {5}\right ) \log \left (-2 \sqrt {x+2}+\sqrt {5}+1\right ) \]

[Out]

1/5*ln(1-5^(1/2)-2*(2+x)^(1/2))*(5-5^(1/2))+1/5*ln(1+5^(1/2)-2*(2+x)^(1/2))*(5+5^(1/2))

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Rubi [A]  time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {632, 31} \[ \frac {1}{5} \left (5-\sqrt {5}\right ) \log \left (-2 \sqrt {x+2}-\sqrt {5}+1\right )+\frac {1}{5} \left (5+\sqrt {5}\right ) \log \left (-2 \sqrt {x+2}+\sqrt {5}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x - Sqrt[2 + x])^(-1),x]

[Out]

((5 - Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[2 + x]])/5 + ((5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[2 + x]])/5

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rubi steps

\begin {align*} \int \frac {1}{1+x-\sqrt {2+x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{-1-x+x^2} \, dx,x,\sqrt {2+x}\right )\\ &=\frac {1}{5} \left (5-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {2+x}\right )+\frac {1}{5} \left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {2+x}\right )\\ &=\frac {1}{5} \left (5-\sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {2+x}\right )+\frac {1}{5} \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {2+x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 58, normalized size = 0.95 \[ \frac {1}{5} \left (\left (5+\sqrt {5}\right ) \log \left (-2 \sqrt {x+2}+\sqrt {5}+1\right )-\left (\sqrt {5}-5\right ) \log \left (-2 \sqrt {x+2}-\sqrt {5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((-5 + Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[2 + x]]) + (5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[2 + x]])/5

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fricas [A]  time = 0.44, size = 63, normalized size = 1.03 \[ \frac {1}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (x + 3\right )} - {\left (\sqrt {5} {\left (2 \, x + 1\right )} - 5\right )} \sqrt {x + 2} + 7 \, x + 3}{x^{2} + x - 1}\right ) + \log \left (x - \sqrt {x + 2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(5)*log((2*x^2 - sqrt(5)*(x + 3) - (sqrt(5)*(2*x + 1) - 5)*sqrt(x + 2) + 7*x + 3)/(x^2 + x - 1)) + log
(x - sqrt(x + 2) + 1)

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giac [A]  time = 0.33, size = 50, normalized size = 0.82 \[ \frac {1}{5} \, \sqrt {5} \log \left (\frac {{\left | -\sqrt {5} + 2 \, \sqrt {x + 2} - 1 \right |}}{{\left | \sqrt {5} + 2 \, \sqrt {x + 2} - 1 \right |}}\right ) + \log \left ({\left | x - \sqrt {x + 2} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(5)*log(abs(-sqrt(5) + 2*sqrt(x + 2) - 1)/abs(sqrt(5) + 2*sqrt(x + 2) - 1)) + log(abs(x - sqrt(x + 2)
+ 1))

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maple [A]  time = 0.01, size = 91, normalized size = 1.49 \[ -\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x +1\right ) \sqrt {5}}{5}\right )}{5}-\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {x +2}-1\right ) \sqrt {5}}{5}\right )}{5}-\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {x +2}+1\right ) \sqrt {5}}{5}\right )}{5}+\frac {\ln \left (x +1-\sqrt {x +2}\right )}{2}-\frac {\ln \left (x +1+\sqrt {x +2}\right )}{2}+\frac {\ln \left (x^{2}+x -1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*5^(1/2)*arctanh(1/5*(2*x+1)*5^(1/2))+1/2*ln(x^2+x-1)-1/2*ln(1+x+(x+2)^(1/2))-1/5*5^(1/2)*arctanh(1/5*(2*(
x+2)^(1/2)+1)*5^(1/2))+1/2*ln(1+x-(x+2)^(1/2))-1/5*5^(1/2)*arctanh(1/5*(2*(x+2)^(1/2)-1)*5^(1/2))

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maxima [A]  time = 1.48, size = 46, normalized size = 0.75 \[ \frac {1}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {x + 2} + 1}{\sqrt {5} + 2 \, \sqrt {x + 2} - 1}\right ) + \log \left (x - \sqrt {x + 2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(x + 2) + 1)/(sqrt(5) + 2*sqrt(x + 2) - 1)) + log(x - sqrt(x + 2) + 1)

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mupad [B]  time = 3.24, size = 71, normalized size = 1.16 \[ \ln \left (2\,\sqrt {x+2}-\left (\frac {\sqrt {5}}{5}+1\right )\,\left (2\,\sqrt {x+2}-1\right )\right )\,\left (\frac {\sqrt {5}}{5}+1\right )-\ln \left (2\,\sqrt {x+2}+\left (\frac {\sqrt {5}}{5}-1\right )\,\left (2\,\sqrt {x+2}-1\right )\right )\,\left (\frac {\sqrt {5}}{5}-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - (x + 2)^(1/2) + 1),x)

[Out]

log(2*(x + 2)^(1/2) - (5^(1/2)/5 + 1)*(2*(x + 2)^(1/2) - 1))*(5^(1/2)/5 + 1) - log(2*(x + 2)^(1/2) + (5^(1/2)/
5 - 1)*(2*(x + 2)^(1/2) - 1))*(5^(1/2)/5 - 1)

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sympy [A]  time = 2.55, size = 94, normalized size = 1.54 \[ 4 \left (\begin {cases} - \frac {\sqrt {5} \operatorname {acoth}{\left (\frac {2 \sqrt {5} \left (\sqrt {x + 2} - \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {x + 2} - \frac {1}{2}\right )^{2} > \frac {5}{4} \\- \frac {\sqrt {5} \operatorname {atanh}{\left (\frac {2 \sqrt {5} \left (\sqrt {x + 2} - \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {x + 2} - \frac {1}{2}\right )^{2} < \frac {5}{4} \end {cases}\right ) + \log {\left (x - \sqrt {x + 2} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*Piecewise((-sqrt(5)*acoth(2*sqrt(5)*(sqrt(x + 2) - 1/2)/5)/10, (sqrt(x + 2) - 1/2)**2 > 5/4), (-sqrt(5)*atan
h(2*sqrt(5)*(sqrt(x + 2) - 1/2)/5)/10, (sqrt(x + 2) - 1/2)**2 < 5/4)) + log(x - sqrt(x + 2) + 1)

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