3.1.12 \(\int \frac {x}{x+\sqrt {1-\sqrt {1+x}}} \, dx\) [12]

Optimal. Leaf size=73 \[ 2 \sqrt {1+x}-4 \sqrt {1-\sqrt {1+x}}+\left (1-\sqrt {1+x}\right )^2+\frac {8 \tanh ^{-1}\left (\frac {1+2 \sqrt {1-\sqrt {1+x}}}{\sqrt {5}}\right )}{\sqrt {5}} \]

[Out]

8/5*arctanh(1/5*(1+2*(1-(1+x)^(1/2))^(1/2))*5^(1/2))*5^(1/2)+(1-(1+x)^(1/2))^2+2*(1+x)^(1/2)-4*(1-(1+x)^(1/2))
^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1642, 632, 212} \begin {gather*} \left (1-\sqrt {x+1}\right )^2-4 \sqrt {1-\sqrt {x+1}}+2 \sqrt {x+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {1-\sqrt {x+1}}+1}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(x + Sqrt[1 - Sqrt[1 + x]]),x]

[Out]

2*Sqrt[1 + x] - 4*Sqrt[1 - Sqrt[1 + x]] + (1 - Sqrt[1 + x])^2 + (8*ArcTanh[(1 + 2*Sqrt[1 - Sqrt[1 + x]])/Sqrt[
5]])/Sqrt[5]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {x}{x+\sqrt {1-\sqrt {1+x}}} \, dx &=2 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )}{-1+\sqrt {1-x}+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=4 \text {Subst}\left (\int \frac {x^2 (1+x) \left (-2+x^2\right )}{-1+x+x^2} \, dx,x,\sqrt {1-\sqrt {1+x}}\right )\\ &=4 \text {Subst}\left (\int \left (-1-x+x^3-\frac {1}{-1+x+x^2}\right ) \, dx,x,\sqrt {1-\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-4 \sqrt {1-\sqrt {1+x}}+\left (1-\sqrt {1+x}\right )^2-4 \text {Subst}\left (\int \frac {1}{-1+x+x^2} \, dx,x,\sqrt {1-\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-4 \sqrt {1-\sqrt {1+x}}+\left (1-\sqrt {1+x}\right )^2+8 \text {Subst}\left (\int \frac {1}{5-x^2} \, dx,x,1+2 \sqrt {1-\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-4 \sqrt {1-\sqrt {1+x}}+\left (1-\sqrt {1+x}\right )^2+\frac {8 \tanh ^{-1}\left (\frac {1+2 \sqrt {1-\sqrt {1+x}}}{\sqrt {5}}\right )}{\sqrt {5}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 52, normalized size = 0.71 \begin {gather*} x-4 \sqrt {1-\sqrt {1+x}}+\frac {8 \tanh ^{-1}\left (\frac {1+2 \sqrt {1-\sqrt {1+x}}}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - 4*Sqrt[1 - Sqrt[1 + x]] + (8*ArcTanh[(1 + 2*Sqrt[1 - Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded in comparison} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x/(x + Sqrt[1 - Sqrt[1 + x]]),x]')

[Out]

cought exception: maximum recursion depth exceeded in comparison

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Maple [A]
time = 0.02, size = 60, normalized size = 0.82

method	result	size
derivativedivides	\(\left (1-\sqrt {1+x}\right )^{2}-2+2 \sqrt {1+x}-4 \sqrt {1-\sqrt {1+x}}+\frac {8 \arctanh \left (\frac {\left (1+2 \sqrt {1-\sqrt {1+x}}\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{5}\)	\(60\)
default	\(\left (1-\sqrt {1+x}\right )^{2}-2+2 \sqrt {1+x}-4 \sqrt {1-\sqrt {1+x}}+\frac {8 \arctanh \left (\frac {\left (1+2 \sqrt {1-\sqrt {1+x}}\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{5}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1-(1+x)^(1/2))^2-2+2*(1+x)^(1/2)-4*(1-(1+x)^(1/2))^(1/2)+8/5*arctanh(1/5*(1+2*(1-(1+x)^(1/2))^(1/2))*5^(1/2))
*5^(1/2)

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Maxima [A]
time = 0.37, size = 77, normalized size = 1.05 \begin {gather*} {\left (\sqrt {x + 1} - 1\right )}^{2} - \frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {-\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {-\sqrt {x + 1} + 1} + 1}\right ) + 2 \, \sqrt {x + 1} - 4 \, \sqrt {-\sqrt {x + 1} + 1} - 2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x + 1) - 1)^2 - 4/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(-sqrt(x + 1) + 1) - 1)/(sqrt(5) + 2*sqrt(-sqrt(x + 1)
+ 1) + 1)) + 2*sqrt(x + 1) - 4*sqrt(-sqrt(x + 1) + 1) - 2

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Fricas [A]
time = 0.42, size = 110, normalized size = 1.51 \begin {gather*} \frac {4}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (3 \, x + 1\right )} + {\left (\sqrt {5} {\left (x + 2\right )} - 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} - {\left (\sqrt {5} {\left (2 \, x - 1\right )} - 5\right )} \sqrt {x + 1} - 5 \, x\right )} \sqrt {-\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + x - 4 \, \sqrt {-\sqrt {x + 1} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*sqrt(5)*log((2*x^2 - sqrt(5)*(3*x + 1) + (sqrt(5)*(x + 2) - 5*x)*sqrt(x + 1) + (sqrt(5)*(x + 2) - (sqrt(5)
*(2*x - 1) - 5)*sqrt(x + 1) - 5*x)*sqrt(-sqrt(x + 1) + 1) + 3*x + 3)/(x^2 - x - 1)) + x - 4*sqrt(-sqrt(x + 1)
+ 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{x + \sqrt {1 - \sqrt {x + 1}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(x + sqrt(1 - sqrt(x + 1))), x)

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Giac [A]
time = 0.00, size = 106, normalized size = 1.45 \begin {gather*} 4 \left (\frac {\left (-\sqrt {x+1}+1\right )^{2}}{4}-\frac {-\sqrt {x+1}+1}{2}-\sqrt {-\sqrt {x+1}+1}-\frac {\ln \left (\frac {\left |2 \sqrt {-\sqrt {x+1}+1}+1-\sqrt {5}\right |}{2 \sqrt {-\sqrt {x+1}+1}+1+\sqrt {5}}\right )}{\sqrt {5}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x + 1) - 1)^2 - 4/5*sqrt(5)*log(abs(-sqrt(5) + 2*sqrt(-sqrt(x + 1) + 1) + 1)/(sqrt(5) + 2*sqrt(-sqrt(x +
1) + 1) + 1)) + 2*sqrt(x + 1) - 4*sqrt(-sqrt(x + 1) + 1) - 2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{x+\sqrt {1-\sqrt {x+1}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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