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

Optimal. Leaf size=196 \[ -4 \text {RootSum}\left [\text {$\#$1}^4-8 \text {$\#$1}^3+6 \text {$\#$1}^2-40 \text {$\#$1}+25\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )+2 \text {$\#$1} \log \left (-\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )+5 \log \left (-\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )}{\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-10}\& \right ]+2 \sqrt {x+\sqrt {x+1}}-3 \log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]

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Rubi [A]  time = 0.23, antiderivative size = 189, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1019, 1076, 621, 206, 1032, 724, 204} \begin {gather*} 2 \sqrt {x+\sqrt {x+1}}+2 \sqrt {\frac {2}{5} \left (\sqrt {5}-2\right )} \tan ^{-1}\left (\frac {-2 \left (2-\sqrt {5}\right ) \sqrt {x+1}+\sqrt {5}+3}{4 \sqrt {\sqrt {5}-1} \sqrt {x+\sqrt {x+1}}}\right )+3 \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )+2 \sqrt {\frac {2}{5} \left (2+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {-2 \left (2+\sqrt {5}\right ) \sqrt {x+1}-\sqrt {5}+3}{4 \sqrt {1+\sqrt {5}} \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

2*Sqrt[x + Sqrt[1 + x]] + 2*Sqrt[(2*(-2 + Sqrt[5]))/5]*ArcTan[(3 + Sqrt[5] - 2*(2 - Sqrt[5])*Sqrt[1 + x])/(4*S
qrt[-1 + Sqrt[5]]*Sqrt[x + Sqrt[1 + x]])] + 3*ArcTanh[(1 + 2*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])] + 2*Sqrt[
(2*(2 + Sqrt[5]))/5]*ArcTanh[(3 - Sqrt[5] - 2*(2 + Sqrt[5])*Sqrt[1 + x])/(4*Sqrt[1 + Sqrt[5]]*Sqrt[x + Sqrt[1
+ x]])]

Rule 204

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

Rule 206

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

Rule 621

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

Rule 724

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

Rule 1019

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[(h*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*f*(p + q + 1)), x] - Dist[1/(2*f*(p + q + 1
)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[h*p*(b*d - a*e) + a*(h*e - 2*g*f)*(p + q + 1) + (2*
h*p*(c*d - a*f) + b*(h*e - 2*g*f)*(p + q + 1))*x + (h*p*(c*e - b*f) + c*(h*e - 2*g*f)*(p + q + 1))*x^2, x], x]
, x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && Ne
Q[p + q + 1, 0]

Rule 1032

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2])
, x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b,
c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]

Rule 1076

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x
_)^2]), x_Symbol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)
/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c
, 0] && NeQ[e^2 - 4*d*f, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {x+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{-1-x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}-2 \operatorname {Subst}\left (\int \frac {-\frac {1}{2}-\frac {x}{2}-\frac {3 x^2}{2}}{\left (-1-x+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}-2 \operatorname {Subst}\left (\int \frac {-2-2 x}{\left (-1-x+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}+6 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )+\frac {1}{5} \left (4 \left (5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+\frac {1}{5} \left (4 \left (5+3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-\sqrt {5}+2 x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}+3 \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )-\frac {1}{5} \left (8 \left (5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-16-8 \left (-1+\sqrt {5}\right )+4 \left (-1+\sqrt {5}\right )^2-x^2} \, dx,x,\frac {-3-\sqrt {5}-2 \left (-2+\sqrt {5}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-\frac {1}{5} \left (8 \left (5+3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-16-8 \left (-1-\sqrt {5}\right )+4 \left (-1-\sqrt {5}\right )^2-x^2} \, dx,x,\frac {-3+\sqrt {5}-\left (-2+2 \left (-1-\sqrt {5}\right )\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}+2 \sqrt {\frac {2}{5} \left (-2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {3+\sqrt {5}-2 \left (2-\sqrt {5}\right ) \sqrt {1+x}}{4 \sqrt {-1+\sqrt {5}} \sqrt {x+\sqrt {1+x}}}\right )+3 \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )+2 \sqrt {\frac {2}{5} \left (2+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {3-\sqrt {5}-2 \left (2+\sqrt {5}\right ) \sqrt {1+x}}{4 \sqrt {1+\sqrt {5}} \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 254, normalized size = 1.30 \begin {gather*} 2 \sqrt {x+\sqrt {x+1}}+\sqrt {\sqrt {5}-1} \tan ^{-1}\left (\frac {2 \left (\sqrt {5}-2\right ) \sqrt {x+1}+\sqrt {5}+3}{4 \sqrt {\sqrt {5}-1} \sqrt {x+\sqrt {x+1}}}\right )-\sqrt {\frac {1}{\sqrt {5}}-\frac {1}{5}} \tan ^{-1}\left (\frac {2 \left (\sqrt {5}-2\right ) \sqrt {x+1}+\sqrt {5}+3}{4 \sqrt {\sqrt {5}-1} \sqrt {x+\sqrt {x+1}}}\right )+3 \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )+\frac {1}{5} \sqrt {1+\sqrt {5}} \left (5+\sqrt {5}\right ) \tanh ^{-1}\left (\frac {-2 \left (2+\sqrt {5}\right ) \sqrt {x+1}-\sqrt {5}+3}{4 \sqrt {1+\sqrt {5}} \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[x + Sqrt[1 + x]] - Sqrt[-1/5 + 1/Sqrt[5]]*ArcTan[(3 + Sqrt[5] + 2*(-2 + Sqrt[5])*Sqrt[1 + x])/(4*Sqrt[-
1 + Sqrt[5]]*Sqrt[x + Sqrt[1 + x]])] + Sqrt[-1 + Sqrt[5]]*ArcTan[(3 + Sqrt[5] + 2*(-2 + Sqrt[5])*Sqrt[1 + x])/
(4*Sqrt[-1 + Sqrt[5]]*Sqrt[x + Sqrt[1 + x]])] + 3*ArcTanh[(1 + 2*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])] + (Sq
rt[1 + Sqrt[5]]*(5 + Sqrt[5])*ArcTanh[(3 - Sqrt[5] - 2*(2 + Sqrt[5])*Sqrt[1 + x])/(4*Sqrt[1 + Sqrt[5]]*Sqrt[x
+ Sqrt[1 + x]])])/5

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IntegrateAlgebraic [A]  time = 0.00, size = 189, normalized size = 0.96 \begin {gather*} 2 \sqrt {x+\sqrt {1+x}}-3 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [-1+6 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{3-3 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[x + Sqrt[1 + x]] - 3*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 4*RootSum[-1 + 6*#1 - 3*#1^2 +
2*#1^3 + #1^4 & , (2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 +
x]] - #1]*#1 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2)/(3 - 3*#1 + 3*#1^2 + 2*#1^3) & ]

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fricas [B]  time = 11.67, size = 422, normalized size = 2.15 \begin {gather*} -\frac {4}{5} \, \sqrt {5} \sqrt {2} \sqrt {\sqrt {5} - 2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (11 \, x^{2} + \sqrt {5} {\left (5 \, x^{2} - 29 \, x - 33\right )} - 4 \, {\left (\sqrt {5} {\left (13 \, x + 11\right )} + 29 \, x + 33\right )} \sqrt {x + 1} - 43 \, x - 55\right )} \sqrt {\sqrt {5} - 1} \sqrt {\sqrt {5} - 2} - 4 \, \sqrt {2} {\left (\sqrt {5} {\left (34 \, x + 33\right )} - {\left (\sqrt {5} {\left (3 \, x + 11\right )} + 7 \, x - 11\right )} \sqrt {x + 1} + 76 \, x + 77\right )} \sqrt {x + \sqrt {x + 1}} \sqrt {\sqrt {5} - 2}}{4 \, {\left (x^{2} - 121 \, x - 121\right )}}\right ) - \frac {1}{5} \, \sqrt {5} \sqrt {2} \sqrt {\sqrt {5} + 2} \log \left (\frac {4 \, {\left (\sqrt {2} {\left (3 \, x^{2} + \sqrt {5} {\left (x^{2} + 3 \, x - 1\right )} + 4 \, {\left (2 \, x + \sqrt {5} - 1\right )} \sqrt {x + 1} + x + 5\right )} \sqrt {\sqrt {5} + 2} + 4 \, {\left ({\left (\sqrt {5} x + x + 2\right )} \sqrt {x + 1} + \sqrt {5} {\left (x + 1\right )} + 3 \, x + 1\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x^{2} - x - 1}\right ) + \frac {1}{5} \, \sqrt {5} \sqrt {2} \sqrt {\sqrt {5} + 2} \log \left (-\frac {4 \, {\left (\sqrt {2} {\left (3 \, x^{2} + \sqrt {5} {\left (x^{2} + 3 \, x - 1\right )} + 4 \, {\left (2 \, x + \sqrt {5} - 1\right )} \sqrt {x + 1} + x + 5\right )} \sqrt {\sqrt {5} + 2} - 4 \, {\left ({\left (\sqrt {5} x + x + 2\right )} \sqrt {x + 1} + \sqrt {5} {\left (x + 1\right )} + 3 \, x + 1\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x^{2} - x - 1}\right ) + 2 \, \sqrt {x + \sqrt {x + 1}} + \frac {3}{2} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x + 1} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/5*sqrt(5)*sqrt(2)*sqrt(sqrt(5) - 2)*arctan(1/4*(sqrt(5)*sqrt(2)*(11*x^2 + sqrt(5)*(5*x^2 - 29*x - 33) - 4*(
sqrt(5)*(13*x + 11) + 29*x + 33)*sqrt(x + 1) - 43*x - 55)*sqrt(sqrt(5) - 1)*sqrt(sqrt(5) - 2) - 4*sqrt(2)*(sqr
t(5)*(34*x + 33) - (sqrt(5)*(3*x + 11) + 7*x - 11)*sqrt(x + 1) + 76*x + 77)*sqrt(x + sqrt(x + 1))*sqrt(sqrt(5)
- 2))/(x^2 - 121*x - 121)) - 1/5*sqrt(5)*sqrt(2)*sqrt(sqrt(5) + 2)*log(4*(sqrt(2)*(3*x^2 + sqrt(5)*(x^2 + 3*x
- 1) + 4*(2*x + sqrt(5) - 1)*sqrt(x + 1) + x + 5)*sqrt(sqrt(5) + 2) + 4*((sqrt(5)*x + x + 2)*sqrt(x + 1) + sq
rt(5)*(x + 1) + 3*x + 1)*sqrt(x + sqrt(x + 1)))/(x^2 - x - 1)) + 1/5*sqrt(5)*sqrt(2)*sqrt(sqrt(5) + 2)*log(-4*
(sqrt(2)*(3*x^2 + sqrt(5)*(x^2 + 3*x - 1) + 4*(2*x + sqrt(5) - 1)*sqrt(x + 1) + x + 5)*sqrt(sqrt(5) + 2) - 4*(
(sqrt(5)*x + x + 2)*sqrt(x + 1) + sqrt(5)*(x + 1) + 3*x + 1)*sqrt(x + sqrt(x + 1)))/(x^2 - x - 1)) + 2*sqrt(x
+ sqrt(x + 1)) + 3/2*log(4*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) + 1) + 8*x + 8*sqrt(x + 1) + 5)

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giac [B]  time = 0.60, size = 212, normalized size = 1.08 \begin {gather*} -8 \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{5}} {\left (\arctan \relax (2) + \arctan \left (\frac {1}{2} \, \sqrt {\sqrt {5} + 1} {\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )} - \frac {1}{2} \, \sqrt {\sqrt {5} - 1}\right )\right )} + 4 \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{5}} \log \left ({\left | 4 \, \sqrt {5} \sqrt {10 \, \sqrt {5} + 20} + 20 \, \sqrt {5} + 40 \, \sqrt {x + \sqrt {x + 1}} - 40 \, \sqrt {x + 1} - 20 \, \sqrt {10 \, \sqrt {5} + 20} + 20 \right |}\right ) - 4 \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{5}} \log \left ({\left | -4 \, \sqrt {5} \sqrt {10 \, \sqrt {5} + 20} + 20 \, \sqrt {5} + 40 \, \sqrt {x + \sqrt {x + 1}} - 40 \, \sqrt {x + 1} + 20 \, \sqrt {10 \, \sqrt {5} + 20} + 20 \right |}\right ) + 2 \, \sqrt {x + \sqrt {x + 1}} - 3 \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*sqrt(1/10*sqrt(5) - 1/5)*(arctan(2) + arctan(1/2*sqrt(sqrt(5) + 1)*(sqrt(x + sqrt(x + 1)) - sqrt(x + 1)) -
1/2*sqrt(sqrt(5) - 1))) + 4*sqrt(1/10*sqrt(5) + 1/5)*log(abs(4*sqrt(5)*sqrt(10*sqrt(5) + 20) + 20*sqrt(5) + 40
*sqrt(x + sqrt(x + 1)) - 40*sqrt(x + 1) - 20*sqrt(10*sqrt(5) + 20) + 20)) - 4*sqrt(1/10*sqrt(5) + 1/5)*log(abs
(-4*sqrt(5)*sqrt(10*sqrt(5) + 20) + 20*sqrt(5) + 40*sqrt(x + sqrt(x + 1)) - 40*sqrt(x + 1) + 20*sqrt(10*sqrt(5
) + 20) + 20)) + 2*sqrt(x + sqrt(x + 1)) - 3*log(-2*sqrt(x + sqrt(x + 1)) + 2*sqrt(x + 1) + 1)

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maple [B]  time = 0.15, size = 403, normalized size = 2.06

method	result	size
derivativedivides	\(\frac {2 \left (\sqrt {5}+1\right ) \sqrt {5}\, \left (\frac {\sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}}{2}+\frac {\left (2+\sqrt {5}\right ) \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}\right )}{4}-\frac {\sqrt {\sqrt {5}+1}\, \arctanh \left (\frac {2 \sqrt {5}+2+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}{2 \sqrt {\sqrt {5}+1}\, \sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}}\right )}{2}\right )}{5}+\frac {2 \left (\sqrt {5}-1\right ) \sqrt {5}\, \left (\frac {\sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}}{2}+\frac {\left (2-\sqrt {5}\right ) \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}\right )}{4}+\frac {\left (-\sqrt {5}+1\right ) \arctan \left (\frac {-2 \sqrt {5}+2+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}{2 \sqrt {\sqrt {5}-1}\, \sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}}\right )}{2 \sqrt {\sqrt {5}-1}}\right )}{5}\)	\(403\)
default	\(\frac {2 \left (\sqrt {5}+1\right ) \sqrt {5}\, \left (\frac {\sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}}{2}+\frac {\left (2+\sqrt {5}\right ) \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}\right )}{4}-\frac {\sqrt {\sqrt {5}+1}\, \arctanh \left (\frac {2 \sqrt {5}+2+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}{2 \sqrt {\sqrt {5}+1}\, \sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}}\right )}{2}\right )}{5}+\frac {2 \left (\sqrt {5}-1\right ) \sqrt {5}\, \left (\frac {\sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}}{2}+\frac {\left (2-\sqrt {5}\right ) \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}\right )}{4}+\frac {\left (-\sqrt {5}+1\right ) \arctan \left (\frac {-2 \sqrt {5}+2+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}{2 \sqrt {\sqrt {5}-1}\, \sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}}\right )}{2 \sqrt {\sqrt {5}-1}}\right )}{5}\)	\(403\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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