3.28.13 \(\int \frac {-1+x^2}{(1+x^2) \sqrt {x+\sqrt {1+x}}} \, dx\) [2713]

Optimal. Leaf size=248 \[ 2 \sqrt {x+\sqrt {1+x}}+\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+32 \text {RootSum}\left [625-1000 \text {$\#$1}+300 \text {$\#$1}^2-120 \text {$\#$1}^3+470 \text {$\#$1}^4-24 \text {$\#$1}^5+12 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\& ,\frac {5 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2-2 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-125+75 \text {$\#$1}-45 \text {$\#$1}^2+235 \text {$\#$1}^3-15 \text {$\#$1}^4+9 \text {$\#$1}^5-7 \text {$\#$1}^6+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 0.53, antiderivative size = 375, normalized size of antiderivative = 1.51, number of steps used = 18, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6860, 654, 635, 212, 1047, 738, 210} \begin {gather*} \frac {i \text {ArcTan}\left (\frac {-\left (\left (1-2 \sqrt {1-i}\right ) \sqrt {x+1}\right )+\sqrt {1-i}+2}{2 \sqrt {i+\sqrt {1-i}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {i+\sqrt {1-i}}}-\frac {i \text {ArcTan}\left (\frac {-\left (\left (1-2 \sqrt {1+i}\right ) \sqrt {x+1}\right )+\sqrt {1+i}+2}{2 \sqrt {\sqrt {1+i}-i} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {\sqrt {1+i}-i}}+2 \sqrt {x+\sqrt {x+1}}-\tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-\frac {i \tanh ^{-1}\left (\frac {-\left (\left (1+2 \sqrt {1-i}\right ) \sqrt {x+1}\right )-\sqrt {1-i}+2}{2 \sqrt {\sqrt {1-i}-i} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {\sqrt {1-i}-i}}+\frac {i \tanh ^{-1}\left (\frac {-\left (\left (1+2 \sqrt {1+i}\right ) \sqrt {x+1}\right )-\sqrt {1+i}+2}{2 \sqrt {i+\sqrt {1+i}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {i+\sqrt {1+i}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^2)/((1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]

[Out]

2*Sqrt[x + Sqrt[1 + x]] + (I*ArcTan[(2 + Sqrt[1 - I] - (1 - 2*Sqrt[1 - I])*Sqrt[1 + x])/(2*Sqrt[I + Sqrt[1 - I
]]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[I + Sqrt[1 - I]] - (I*ArcTan[(2 + Sqrt[1 + I] - (1 - 2*Sqrt[1 + I])*Sqrt[1 +
x])/(2*Sqrt[-I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[-I + Sqrt[1 + I]] - ArcTanh[(1 + 2*Sqrt[1 + x])/(2
*Sqrt[x + Sqrt[1 + x]])] - (I*ArcTanh[(2 - Sqrt[1 - I] - (1 + 2*Sqrt[1 - I])*Sqrt[1 + x])/(2*Sqrt[-I + Sqrt[1
- I]]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[-I + Sqrt[1 - I]] + (I*ArcTanh[(2 - Sqrt[1 + I] - (1 + 2*Sqrt[1 + I])*Sqrt
[1 + x])/(2*Sqrt[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[I + Sqrt[1 + I]]

Rule 210

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

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 635

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 654

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

Rule 738

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 1047

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

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 242, normalized size = 0.98 \begin {gather*} 2 \sqrt {x+\sqrt {1+x}}+\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+32 \text {RootSum}\left [625-1000 \text {$\#$1}+300 \text {$\#$1}^2-120 \text {$\#$1}^3+470 \text {$\#$1}^4-24 \text {$\#$1}^5+12 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {5 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^2-2 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^4}{-125+75 \text {$\#$1}-45 \text {$\#$1}^2+235 \text {$\#$1}^3-15 \text {$\#$1}^4+9 \text {$\#$1}^5-7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[x + Sqrt[1 + x]] + Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 32*RootSum[625 - 1000*#1 + 300*#
1^2 - 120*#1^3 + 470*#1^4 - 24*#1^5 + 12*#1^6 - 8*#1^7 + #1^8 & , (5*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[
1 + x]] + #1]*#1^2 - 2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] + #1]*#1^3 + Log[-1 - 2*Sqrt[1 + x] +
2*Sqrt[x + Sqrt[1 + x]] + #1]*#1^4)/(-125 + 75*#1 - 45*#1^2 + 235*#1^3 - 15*#1^4 + 9*#1^5 - 7*#1^6 + #1^7) & ]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 0.10, size = 143, normalized size = 0.58

method result size
derivativedivides \(2 \sqrt {x +\sqrt {1+x}}-\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (4 \textit {\_R}^{4}-4 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) \(143\)
default \(2 \sqrt {x +\sqrt {1+x}}-\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (4 \textit {\_R}^{4}-4 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) \(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x+(1+x)^(1/2))^(1/2)-ln(1/2+(1+x)^(1/2)+(x+(1+x)^(1/2))^(1/2))-sum((4*_R^4-4*_R^3+5*_R^2-4*_R+1)/(_R^7-3*_R
^5+5*_R^4+10*_R^3-18*_R^2+10*_R-1)*ln((x+(1+x)^(1/2))^(1/2)-(1+x)^(1/2)-_R),_R=RootOf(_Z^8-4*_Z^6+8*_Z^5+20*_Z
^4-48*_Z^3+40*_Z^2-8*_Z+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 8.51, size = 5633, normalized size = 22.71 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*sqrt(5*sqrt(4/25*I + 28/25) + 5*sqrt(-4/25*I + 28/25) - 2*sqrt(-3/4*(5*sqrt(4/25*I + 28/25) + 4*I
 - 2)^2 - 1/2*(5*sqrt(4/25*I + 28/25) + 4*I + 6)*(5*sqrt(-4/25*I + 28/25) - 4*I - 2) - 3/4*(5*sqrt(-4/25*I + 2
8/25) - 4*I - 2)^2 - 20*sqrt(4/25*I + 28/25) - 16*I + 24) + 4)*log(-1/100*(10*(((3*x - 16)*sqrt(x + 1) + 4*x -
 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - 40*(2*x + 1)*sqrt(x + 1) + 10*x + 80)*sqrt(x + sqrt(x + 1))*(5*sqrt(-
4/25*I + 28/25) - 4*I - 2)^2 + 10*(((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 + 8
*((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - 220*(2*x + 1)*sqrt(x + 1) - 820*x + 4
40)*sqrt(x + sqrt(x + 1))*(5*sqrt(-4/25*I + 28/25) - 4*I - 2) + 20*((((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqr
t(4/25*I + 28/25) + 4*I - 2) - 40*(2*x + 1)*sqrt(x + 1) + 10*x + 80)*sqrt(x + sqrt(x + 1))*(5*sqrt(-4/25*I + 2
8/25) - 4*I - 2) - 10*((4*(2*x + 1)*sqrt(x + 1) - x - 8)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) + 10*(2*x + 1)*sqr
t(x + 1) - 90*x - 20)*sqrt(x + sqrt(x + 1)))*sqrt(-3/4*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - 1/2*(5*sqrt(4/25
*I + 28/25) + 4*I + 6)*(5*sqrt(-4/25*I + 28/25) - 4*I - 2) - 3/4*(5*sqrt(-4/25*I + 28/25) - 4*I - 2)^2 - 20*sq
rt(4/25*I + 28/25) - 16*I + 24) - 100*((4*(2*x + 1)*sqrt(x + 1) - x - 8)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2
+ 2*(11*(2*x + 1)*sqrt(x + 1) + 41*x - 22)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - 320*(2*x + 1)*sqrt(x + 1) + 80
*x - 760)*sqrt(x + sqrt(x + 1)) + (10*(sqrt(5)*(16*x + 3)*sqrt(x + 1) + sqrt(5)*(6*x^2 + 3*x + 10))*(5*sqrt(4/
25*I + 28/25) + 4*I - 2)^2 + (10*sqrt(5)*(16*x + 3)*sqrt(x + 1) - (6*sqrt(5)*sqrt(x + 1)*(x - 2) + sqrt(5)*(11
*x^2 + 23*x - 5))*(5*sqrt(4/25*I + 28/25) + 4*I - 2) + 10*sqrt(5)*(6*x^2 + 3*x + 10))*(5*sqrt(-4/25*I + 28/25)
 - 4*I - 2)^2 + 1200*sqrt(5)*sqrt(x + 1)*(x - 2) + 10*(8*sqrt(5)*(11*x + 13)*sqrt(x + 1) + sqrt(5)*(33*x^2 + 1
04*x + 55))*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - ((6*sqrt(5)*sqrt(x + 1)*(x - 2) + sqrt(5)*(11*x^2 + 23*x - 5)
)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - 80*sqrt(5)*(11*x + 13)*sqrt(x + 1) + 8*(6*sqrt(5)*sqrt(x + 1)*(x - 2)
 + sqrt(5)*(11*x^2 + 23*x - 5))*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - 10*sqrt(5)*(33*x^2 + 104*x + 55))*(5*sqrt
(-4/25*I + 28/25) - 4*I - 2) + 400*sqrt(5)*(23*x^2 - 6*x - 20) + 2*(400*sqrt(5)*sqrt(x + 1)*(x - 2) + 10*(sqrt
(5)*(16*x + 3)*sqrt(x + 1) + sqrt(5)*(6*x^2 + 3*x + 10))*(5*sqrt(4/25*I + 28/25) + 4*I - 2) + (10*sqrt(5)*(16*
x + 3)*sqrt(x + 1) - (6*sqrt(5)*sqrt(x + 1)*(x - 2) + sqrt(5)*(11*x^2 + 23*x - 5))*(5*sqrt(4/25*I + 28/25) + 4
*I - 2) + 10*sqrt(5)*(6*x^2 + 3*x + 10))*(5*sqrt(-4/25*I + 28/25) - 4*I - 2) + 50*sqrt(5)*(3*x^2 - 16*x + 5))*
sqrt(-3/4*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - 1/2*(5*sqrt(4/25*I + 28/25) + 4*I + 6)*(5*sqrt(-4/25*I + 28/2
5) - 4*I - 2) - 3/4*(5*sqrt(-4/25*I + 28/25) - 4*I - 2)^2 - 20*sqrt(4/25*I + 28/25) - 16*I + 24))*sqrt(5*sqrt(
4/25*I + 28/25) + 5*sqrt(-4/25*I + 28/25) - 2*sqrt(-3/4*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - 1/2*(5*sqrt(4/2
5*I + 28/25) + 4*I + 6)*(5*sqrt(-4/25*I + 28/25) - 4*I - 2) - 3/4*(5*sqrt(-4/25*I + 28/25) - 4*I - 2)^2 - 20*s
qrt(4/25*I + 28/25) - 16*I + 24) + 4))/(x^2 + 1)) - 1/20*sqrt(5)*sqrt(5*sqrt(4/25*I + 28/25) + 5*sqrt(-4/25*I
+ 28/25) - 2*sqrt(-3/4*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - 1/2*(5*sqrt(4/25*I + 28/25) + 4*I + 6)*(5*sqrt(-
4/25*I + 28/25) - 4*I - 2) - 3/4*(5*sqrt(-4/25*I + 28/25) - 4*I - 2)^2 - 20*sqrt(4/25*I + 28/25) - 16*I + 24)
+ 4)*log(-1/100*(10*(((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - 40*(2*x + 1)*sqrt
(x + 1) + 10*x + 80)*sqrt(x + sqrt(x + 1))*(5*sqrt(-4/25*I + 28/25) - 4*I - 2)^2 + 10*(((3*x - 16)*sqrt(x + 1)
 + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 + 8*((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25
) + 4*I - 2) - 220*(2*x + 1)*sqrt(x + 1) - 820*x + 440)*sqrt(x + sqrt(x + 1))*(5*sqrt(-4/25*I + 28/25) - 4*I -
 2) + 20*((((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - 40*(2*x + 1)*sqrt(x + 1) +
10*x + 80)*sqrt(x + sqrt(x + 1))*(5*sqrt(-4/25*I + 28/25) - 4*I - 2) - 10*((4*(2*x + 1)*sqrt(x + 1) - x - 8)*(
5*sqrt(4/25*I + 28/25) + 4*I - 2) + 10*(2*x + 1)*sqrt(x + 1) - 90*x - 20)*sqrt(x + sqrt(x + 1)))*sqrt(-3/4*(5*
sqrt(4/25*I + 28/25) + 4*I - 2)^2 - 1/2*(5*sqrt(4/25*I + 28/25) + 4*I + 6)*(5*sqrt(-4/25*I + 28/25) - 4*I - 2)
 - 3/4*(5*sqrt(-4/25*I + 28/25) - 4*I - 2)^2 - 20*sqrt(4/25*I + 28/25) - 16*I + 24) - 100*((4*(2*x + 1)*sqrt(x
 + 1) - x - 8)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 + 2*(11*(2*x + 1)*sqrt(x + 1) + 41*x - 22)*(5*sqrt(4/25*I
+ 28/25) + 4*I - 2) - 320*(2*x + 1)*sqrt(x + 1) + 80*x - 760)*sqrt(x + sqrt(x + 1)) - (10*(sqrt(5)*(16*x + 3)*
sqrt(x + 1) + sqrt(5)*(6*x^2 + 3*x + 10))*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 + (10*sqrt(5)*(16*x + 3)*sqrt(x
 + 1) - (6*sqrt(5)*sqrt(x + 1)*(x - 2) + sqrt(5)*(11*x^2 + 23*x - 5))*(5*sqrt(4/25*I + 28/25) + 4*I - 2) + 10*
sqrt(5)*(6*x^2 + 3*x + 10))*(5*sqrt(-4/25*I + 28/25) - 4*I - 2)^2 + 1200*sqrt(5)*sqrt(x + 1)*(x - 2) + 10*(8*s
qrt(5)*(11*x + 13)*sqrt(x + 1) + sqrt(5)*(33*x^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt {x + \sqrt {x + 1}} \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x - 1)*(x + 1)/(sqrt(x + sqrt(x + 1))*(x**2 + 1)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Invalid _EXT in replace_ext Error: Bad Argument ValueWarning, integration of abs or sign assumes constant s
ign by inte

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2-1}{\sqrt {x+\sqrt {x+1}}\,\left (x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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