3.28.13 \(\int \frac {-1+x^2}{(1+x^2) \sqrt {x+\sqrt {1+x}}} \, dx\)
Optimal. Leaf size=248 \[ 32 \text {RootSum}\left [\text {$\#$1}^8-8 \text {$\#$1}^7+12 \text {$\#$1}^6-24 \text {$\#$1}^5+470 \text {$\#$1}^4-120 \text {$\#$1}^3+300 \text {$\#$1}^2-1000 \text {$\#$1}+625\& ,\frac {\text {$\#$1}^4 \log \left (-\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )-2 \text {$\#$1}^3 \log \left (-\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )+5 \text {$\#$1}^2 \log \left (-\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )}{\text {$\#$1}^7-7 \text {$\#$1}^6+9 \text {$\#$1}^5-15 \text {$\#$1}^4+235 \text {$\#$1}^3-45 \text {$\#$1}^2+75 \text {$\#$1}-125}\& \right ]+2 \sqrt {x+\sqrt {x+1}}+\log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]
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Rubi [C] time = 0.82, 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
= {6728, 640, 621, 206, 1033, 724, 204} \begin {gather*} 2 \sqrt {x+\sqrt {x+1}}+\frac {i \tan ^{-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}}}-\frac {i \tan ^{-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}}-\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 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 640
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 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 1033
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 6728
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 \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-4 \operatorname {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 \operatorname {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 )-\operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}-4 i \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 [C] time = 0.18, size = 554, normalized size = 2.23 \begin {gather*} \frac {1}{2} \left (4 \sqrt {x+\sqrt {x+1}}+\frac {2 i \log \left (-\sqrt {x+1}+\sqrt {1+i}\right )}{\sqrt {i+\sqrt {1+i}}}+\frac {2 i \log \left (\sqrt {x+1}+\sqrt {1+i}\right )}{\sqrt {i-\sqrt {1+i}}}+\frac {\log \left (x-2 \sqrt {1-i} \sqrt {x+1}+(2-i)\right )}{\sqrt {i-\sqrt {1-i}}}-\frac {\log \left (-x+2 \sqrt {1-i} \sqrt {x+1}-(2-i)\right )}{\sqrt {i-\sqrt {1-i}}}-2 \log \left (2 \sqrt {x+1}+2 \sqrt {x+\sqrt {x+1}}+1\right )-\frac {2 i \log \left (2 \sqrt {1+i} \sqrt {x+1}-\sqrt {x+1}-2 \sqrt {i-\sqrt {1+i}} \sqrt {x+\sqrt {x+1}}+\sqrt {1+i}+2\right )}{\sqrt {i-\sqrt {1+i}}}-\frac {2 i \log \left (-2 \sqrt {1+i} \sqrt {x+1}-\sqrt {x+1}-2 \sqrt {i+\sqrt {1+i}} \sqrt {x+\sqrt {x+1}}-\sqrt {1+i}+2\right )}{\sqrt {i+\sqrt {1+i}}}-\frac {2 i \tan ^{-1}\left (\frac {2 \sqrt {i+\sqrt {1-i}} \sqrt {x+\sqrt {x+1}}}{2 \sqrt {1-i} \sqrt {x+1}-\sqrt {x+1}+\sqrt {1-i}+2}\right )}{\sqrt {i+\sqrt {1-i}}}+\frac {2 i \tan ^{-1}\left (\frac {2 \sqrt {i-\sqrt {1-i}} \sqrt {x+\sqrt {x+1}}}{2 \sqrt {1-i} \sqrt {x+1}+\sqrt {x+1}+\sqrt {1-i}-2}\right )}{\sqrt {i-\sqrt {1-i}}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-1 + x^2)/((1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]
[Out]
(4*Sqrt[x + Sqrt[1 + x]] - ((2*I)*ArcTan[(2*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]])/(2 + Sqrt[1 - I] - Sq
rt[1 + x] + 2*Sqrt[1 - I]*Sqrt[1 + x])])/Sqrt[I + Sqrt[1 - I]] + ((2*I)*ArcTan[(2*Sqrt[I - Sqrt[1 - I]]*Sqrt[x
+ Sqrt[1 + x]])/(-2 + Sqrt[1 - I] + Sqrt[1 + x] + 2*Sqrt[1 - I]*Sqrt[1 + x])])/Sqrt[I - Sqrt[1 - I]] + ((2*I)
*Log[Sqrt[1 + I] - Sqrt[1 + x]])/Sqrt[I + Sqrt[1 + I]] + ((2*I)*Log[Sqrt[1 + I] + Sqrt[1 + x]])/Sqrt[I - Sqrt[
1 + I]] + Log[(2 - I) + x - 2*Sqrt[1 - I]*Sqrt[1 + x]]/Sqrt[I - Sqrt[1 - I]] - Log[(-2 + I) - x + 2*Sqrt[1 - I
]*Sqrt[1 + x]]/Sqrt[I - Sqrt[1 - I]] - 2*Log[1 + 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] - ((2*I)*Log[2 + Sqr
t[1 + I] - Sqrt[1 + x] + 2*Sqrt[1 + I]*Sqrt[1 + x] - 2*Sqrt[I - Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]]])/Sqrt[I -
Sqrt[1 + I]] - ((2*I)*Log[2 - Sqrt[1 + I] - Sqrt[1 + x] - 2*Sqrt[1 + I]*Sqrt[1 + x] - 2*Sqrt[I + Sqrt[1 + I]]*
Sqrt[x + Sqrt[1 + x]]])/Sqrt[I + Sqrt[1 + I]])/2
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IntegrateAlgebraic [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]
IntegrateAlgebraic[(-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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fricas [B] time = 10.47, size = 5633, normalized size = 22.71
result too large to display
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^2 + 104*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*sq
rt(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))*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))/(x^2 + 1)) + 1/20*s
qrt(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)*sq
rt(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 + 2
8/25) + 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(4/2
5*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*(1
1*(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 - 7
60)*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 + 104*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*s
qrt(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) + sqr
t(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)*(1
6*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 + 28/25) - 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))*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))/(x^2 + 1)) - 1/20*sqrt(5)*sqrt(5*sqrt(4/25*I + 28/25) + 5*sqrt(-4/25*I + 28/2
5) + 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)*l
og(-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 + 28/2
5) + 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 + 104*x + 55))*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - ((6*sqrt(5)*sqr
t(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)*(1
1*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(4/2
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))*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))/(x^2 + 1)) - 1/2*sqrt(-1/
2*sqrt(-4/25*I + 28/25) + 2/5*I + 1/5)*log(1/50*((((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
+ (((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*sqr
t(-4/25*I + 28/25) - 4*I - 2) + (((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^3 + 8*(
(3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - 1200*(2*x + 1)*sqrt(x + 1) - 3200*x -
4600)*sqrt(x + sqrt(x + 1)) + ((11*x^2 + 6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)
^3 + 8*(11*x^2 + 6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - (60*x^2 - (11*x^2 +
6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) + 10*(16*x + 3)*sqrt(x + 1) + 30*x + 100)
*(5*sqrt(-4/25*I + 28/25) - 4*I - 2)^2 + 5200*x^2 + ((11*x^2 + 6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*
I + 28/25) + 4*I - 2)^2 - 330*x^2 + 8*(11*x^2 + 6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*
I - 2) - 80*(11*x + 13)*sqrt(x + 1) - 1040*x - 550)*(5*sqrt(-4/25*I + 28/25) - 4*I - 2) + 2200*sqrt(x + 1)*(x
- 2) - 4400*x - 3000)*sqrt(-1/2*sqrt(-4/25*I + 28/25) + 2/5*I + 1/5))/(x^2 + 1)) + 1/2*sqrt(-1/2*sqrt(-4/25*I
+ 28/25) + 2/5*I + 1/5)*log(1/50*((((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 + (((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(4/2
5*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) + (((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^3 + 8*((3*x - 16)*sqrt
(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - 1200*(2*x + 1)*sqrt(x + 1) - 3200*x - 4600)*sqrt(x +
sqrt(x + 1)) - ((11*x^2 + 6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^3 + 8*(11*x^2
+ 6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - (60*x^2 - (11*x^2 + 6*sqrt(x + 1)*(
x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) + 10*(16*x + 3)*sqrt(x + 1) + 30*x + 100)*(5*sqrt(-4/25*
I + 28/25) - 4*I - 2)^2 + 5200*x^2 + ((11*x^2 + 6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*
I - 2)^2 - 330*x^2 + 8*(11*x^2 + 6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - 80*(11
*x + 13)*sqrt(x + 1) - 1040*x - 550)*(5*sqrt(-4/25*I + 28/25) - 4*I - 2) + 2200*sqrt(x + 1)*(x - 2) - 4400*x -
3000)*sqrt(-1/2*sqrt(-4/25*I + 28/25) + 2/5*I + 1/5))/(x^2 + 1)) - 1/2*sqrt(-1/2*sqrt(4/25*I + 28/25) - 2/5*I
+ 1/5)*log(-1/50*((((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^3 + 2*(4*(13*x - 11)
*sqrt(x + 1) + 11*x - 52)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 + 20*(11*(2*x + 1)*sqrt(x + 1) + 41*x - 22)*(5*
sqrt(4/25*I + 28/25) + 4*I - 2) + 1200*(2*x + 1)*sqrt(x + 1) + 3200*x + 4600)*sqrt(x + sqrt(x + 1)) + ((11*x^2
+ 6*sqrt(x + 1)*(x - 2) + 23*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^3 + 2*(74*x^2 + (104*x - 33)*sqrt(x +
1) + 107*x + 30)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^2 - 5200*x^2 + 10*(33*x^2 + 8*(11*x + 13)*sqrt(x + 1) + 10
4*x + 55)*(5*sqrt(4/25*I + 28/25) + 4*I - 2) - 2200*sqrt(x + 1)*(x - 2) + 4400*x + 3000)*sqrt(-1/2*sqrt(4/25*I
+ 28/25) - 2/5*I + 1/5))/(x^2 + 1)) + 1/2*sqrt(-1/2*sqrt(4/25*I + 28/25) - 2/5*I + 1/5)*log(-1/50*((((3*x - 1
6)*sqrt(x + 1) + 4*x - 3)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^3 + 2*(4*(13*x - 11)*sqrt(x + 1) + 11*x - 52)*(5*
sqrt(4/25*I + 28/25) + 4*I - 2)^2 + 20*(11*(2*x + 1)*sqrt(x + 1) + 41*x - 22)*(5*sqrt(4/25*I + 28/25) + 4*I -
2) + 1200*(2*x + 1)*sqrt(x + 1) + 3200*x + 4600)*sqrt(x + sqrt(x + 1)) - ((11*x^2 + 6*sqrt(x + 1)*(x - 2) + 23
*x - 5)*(5*sqrt(4/25*I + 28/25) + 4*I - 2)^3 + 2*(74*x^2 + (104*x - 33)*sqrt(x + 1) + 107*x + 30)*(5*sqrt(4/25
*I + 28/25) + 4*I - 2)^2 - 5200*x^2 + 10*(33*x^2 + 8*(11*x + 13)*sqrt(x + 1) + 104*x + 55)*(5*sqrt(4/25*I + 28
/25) + 4*I - 2) - 2200*sqrt(x + 1)*(x - 2) + 4400*x + 3000)*sqrt(-1/2*sqrt(4/25*I + 28/25) - 2/5*I + 1/5))/(x^
2 + 1)) + 2*sqrt(x + sqrt(x + 1)) + 1/2*log(4*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) + 1) - 8*x - 8*sqrt(x + 1)
- 5)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \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]
sage0*x
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maple [B] time = 0.13, size = 143, normalized size = 0.58
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| method |
result |
size |
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| 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 |
\(2 \sqrt {x +\sqrt {1+x}}-\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (4 \textit {\_R}^{4}-4 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) |
\(143\) |
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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*} \int \frac {x^{2} - 1}{{\left (x^{2} + 1\right )} \sqrt {x + \sqrt {x + 1}}}\,{d x} \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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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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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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