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

Optimal. Leaf size=157 \[ 128 \text {RootSum}\left [\text {$\#$1}^8+8 \text {$\#$1}^7+12 \text {$\#$1}^6+24 \text {$\#$1}^5+470 \text {$\#$1}^4+120 \text {$\#$1}^3+300 \text {$\#$1}^2+1000 \text {$\#$1}+625\& ,\frac {\text {$\#$1}^3 \log \left (\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )}{\text {$\#$1}^7+7 \text {$\#$1}^6+9 \text {$\#$1}^5+15 \text {$\#$1}^4+235 \text {$\#$1}^3+45 \text {$\#$1}^2+75 \text {$\#$1}+125}\& \right ]-2 \log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]

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Rubi [C]  time = 1.30, antiderivative size = 383, normalized size of antiderivative = 2.44, number of steps used = 19, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1586, 6741, 6728, 621, 206, 984, 724, 204} \begin {gather*} -\frac {i \tan ^{-1}\left (\frac {-2 \left ((-2+2 i)+\sqrt {1-i}\right ) \sqrt {x+1}+4 \sqrt {1-i}+(2-2 i)}{4 \sqrt {(1+i)+(1-i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1+i)+(1-i)^{3/2}}}-\frac {i \tan ^{-1}\left (\frac {2 \left ((2-2 i)+\sqrt {1-i}\right ) \sqrt {x+1}-4 \sqrt {1-i}+(2-2 i)}{4 \sqrt {(1+i)-(1-i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1+i)-(1-i)^{3/2}}}-\frac {i \tan ^{-1}\left (\frac {2 \left ((-2-2 i)+\sqrt {1+i}\right ) \sqrt {x+1}-(2-2 i) \left (i+(1+i)^{3/2}\right )}{4 \sqrt {(1-i)+(1+i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1-i)+(1+i)^{3/2}}}-\frac {i \tan ^{-1}\left (\frac {-2 \left ((2+2 i)+\sqrt {1+i}\right ) \sqrt {x+1}+4 \sqrt {1+i}-(2+2 i)}{4 \sqrt {(1-i)-(1+i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1-i)-(1+i)^{3/2}}}+2 \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*ArcTan[((2 - 2*I) + 4*Sqrt[1 - I] - 2*((-2 + 2*I) + Sqrt[1 - I])*Sqrt[1 + x])/(4*Sqrt[(1 + I) + (1 - I)^
(3/2)]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[(1 + I) + (1 - I)^(3/2)] - (I*ArcTan[((2 - 2*I) - 4*Sqrt[1 - I] + 2*((2 -
 2*I) + Sqrt[1 - I])*Sqrt[1 + x])/(4*Sqrt[(1 + I) - (1 - I)^(3/2)]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[(1 + I) - (1
- I)^(3/2)] - (I*ArcTan[((-2 + 2*I)*(I + (1 + I)^(3/2)) + 2*((-2 - 2*I) + Sqrt[1 + I])*Sqrt[1 + x])/(4*Sqrt[(1
 - I) + (1 + I)^(3/2)]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[(1 - I) + (1 + I)^(3/2)] - (I*ArcTan[((-2 - 2*I) + 4*Sqrt
[1 + I] - 2*((2 + 2*I) + Sqrt[1 + I])*Sqrt[1 + x])/(4*Sqrt[(1 - I) - (1 + I)^(3/2)]*Sqrt[x + Sqrt[1 + x]])])/S
qrt[(1 - I) - (1 + I)^(3/2)] + 2*ArcTanh[(1 + 2*Sqrt[1 + x])/(2*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 984

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

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

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]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

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

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Mathematica [C]  time = 0.69, size = 360, normalized size = 2.29 \begin {gather*} -\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 {(1+i)+(1-i)^{3/2}}}+\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 {(1-i)+(1+i)^{3/2}}}+2 \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 {(-1-i)+(1-i)^{3/2}}}+\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 {(-1+i)+(1+i)^{3/2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-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[(1 + I) + (1 - I)^(3/2)] + (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[(1 - I) + (1 + I)^(3/2)] + 2*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[(-1 - I) + (1 - I)^(3/2)] + (I*ArcTanh[(2 - Sqrt[1 + I] - (1 + 2*Sqrt[1 + I])*Sq
rt[1 + x])/(2*Sqrt[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[(-1 + I) + (1 + I)^(3/2)]

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IntegrateAlgebraic [A]  time = 0.00, size = 159, normalized size = 1.01 \begin {gather*} -2 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+128 \text {RootSum}\left [625+1000 \text {$\#$1}+300 \text {$\#$1}^2+120 \text {$\#$1}^3+470 \text {$\#$1}^4+24 \text {$\#$1}^5+12 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{125+75 \text {$\#$1}+45 \text {$\#$1}^2+235 \text {$\#$1}^3+15 \text {$\#$1}^4+9 \text {$\#$1}^5+7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 128*RootSum[625 + 1000*#1 + 300*#1^2 + 120*#1^3 + 470*#
1^4 + 24*#1^5 + 12*#1^6 + 8*#1^7 + #1^8 & , (Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] - #1]*#1^3)/(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 = 12.94, size = 5738, normalized size = 36.55

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*sqrt(5*sqrt(14/25*I - 2/25) + 5*sqrt(-14/25*I - 2/25) - 2*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I -
 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25)
- I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6) + 6)*log(-1/100*(20*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt
(14/25*I - 2/25) + I - 3) + 5*(x + 3)*sqrt(x + 1) - 35*x - 5)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) -
 I - 3)^2 + 20*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 12*((3*x - 1)*sqrt(x +
1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x + 3)*sqrt(x + 1) - 320*x - 10)*sqrt(x + sqrt(x + 1))*(5
*sqrt(-14/25*I - 2/25) - I - 3) + 40*((((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*
(x + 3)*sqrt(x + 1) - 35*x - 5)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) - I - 3) + 5*(((x + 3)*sqrt(x +
 1) - 7*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x + 3)*sqrt(x + 1) - 20*x - 10)*sqrt(x + sqrt(x + 1)))*s
qrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) -
I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6) + 100*(((x + 3)*sqrt(x +
 1) - 7*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 2*((x + 3)*sqrt(x + 1) - 32*x - 1)*(5*sqrt(14/25*I - 2/25)
 + I - 3) + 20*(x + 3)*sqrt(x + 1) - 140*x + 180)*sqrt(x + sqrt(x + 1)) + (5*(6*sqrt(5)*(3*x - 1)*sqrt(x + 1)
+ sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + (30*sqrt(5)*(3*x - 1)*sqrt(x + 1) + (2*sqrt
(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*sqrt(5)*(3*x^2 +
 4*x + 15))*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 + 200*sqrt(5)*(9*x - 13)*sqrt(x + 1) + 10*(2*sqrt(5)*(34*x - 3
)*sqrt(x + 1) + sqrt(5)*(3*x^2 + 54*x + 65))*(5*sqrt(14/25*I - 2/25) + I - 3) + ((2*sqrt(5)*(2*x + 1)*sqrt(x +
 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 20*sqrt(5)*(34*x - 3)*sqrt(x + 1) + 12
*(2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*sqrt(5)
*(3*x^2 + 54*x + 65))*(5*sqrt(-14/25*I - 2/25) - I - 3) + 100*sqrt(5)*(13*x^2 + 24*x + 5) + 2*(100*sqrt(5)*(4*
x - 3)*sqrt(x + 1) + 5*(6*sqrt(5)*(3*x - 1)*sqrt(x + 1) + sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(14/25*I - 2/25)
+ I - 3) + (30*sqrt(5)*(3*x - 1)*sqrt(x + 1) + (2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))
*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(-14/25*I - 2/25) - I - 3) + 50*sqrt(
5)*(3*x^2 - 6*x + 5))*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*s
qrt(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6))*
sqrt(5*sqrt(14/25*I - 2/25) + 5*sqrt(-14/25*I - 2/25) - 2*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(
5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 -
30*sqrt(14/25*I - 2/25) - 6*I - 6) + 6))/(x^2 + 1)) - 1/20*sqrt(5)*sqrt(5*sqrt(14/25*I - 2/25) + 5*sqrt(-14/25
*I - 2/25) - 2*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14
/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6) + 6)*log
(-1/100*(20*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*(x + 3)*sqrt(x + 1) - 35*x
 - 5)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 + 20*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqr
t(14/25*I - 2/25) + I - 3)^2 + 12*((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x +
 3)*sqrt(x + 1) - 320*x - 10)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) - I - 3) + 40*((((3*x - 1)*sqrt(x
 + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*(x + 3)*sqrt(x + 1) - 35*x - 5)*sqrt(x + sqrt(x + 1))*(5
*sqrt(-14/25*I - 2/25) - I - 3) + 5*(((x + 3)*sqrt(x + 1) - 7*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x
+ 3)*sqrt(x + 1) - 20*x - 10)*sqrt(x + sqrt(x + 1)))*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqr
t(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sq
rt(14/25*I - 2/25) - 6*I - 6) + 100*(((x + 3)*sqrt(x + 1) - 7*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 2*((
x + 3)*sqrt(x + 1) - 32*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3) + 20*(x + 3)*sqrt(x + 1) - 140*x + 180)*sqrt(x
 + sqrt(x + 1)) - (5*(6*sqrt(5)*(3*x - 1)*sqrt(x + 1) + sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(14/25*I - 2/25) +
I - 3)^2 + (30*sqrt(5)*(3*x - 1)*sqrt(x + 1) + (2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))
*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 + 200*sq
rt(5)*(9*x - 13)*sqrt(x + 1) + 10*(2*sqrt(5)*(34*x - 3)*sqrt(x + 1) + sqrt(5)*(3*x^2 + 54*x + 65))*(5*sqrt(14/
25*I - 2/25) + I - 3) + ((2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/2
5) + I - 3)^2 + 20*sqrt(5)*(34*x - 3)*sqrt(x + 1) + 12*(2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*
x - 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*sqrt(5)*(3*x^2 + 54*x + 65))*(5*sqrt(-14/25*I - 2/25) - I - 3)
+ 100*sqrt(5)*(13*x^2 + 24*x + 5) + 2*(100*sqrt(5)*(4*x - 3)*sqrt(x + 1) + 5*(6*sqrt(5)*(3*x - 1)*sqrt(x + 1)
+ sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + (30*sqrt(5)*(3*x - 1)*sqrt(x + 1) + (2*sqrt(5
)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*sqrt(5)*(3*x^2 + 4
*x + 15))*(5*sqrt(-14/25*I - 2/25) - I - 3) + 50*sqrt(5)*(3*x^2 - 6*x + 5))*sqrt(-3/4*(5*sqrt(14/25*I - 2/25)
+ I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2
/25) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6))*sqrt(5*sqrt(14/25*I - 2/25) + 5*sqrt(-14/25*I - 2/25) -
2*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25)
 - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6) + 6))/(x^2 + 1)) + 1/
20*sqrt(5)*sqrt(5*sqrt(14/25*I - 2/25) + 5*sqrt(-14/25*I - 2/25) + 2*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3
)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) -
I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6) + 6)*log(-1/100*(20*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(1
4/25*I - 2/25) + I - 3) + 5*(x + 3)*sqrt(x + 1) - 35*x - 5)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) - I
 - 3)^2 + 20*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 12*((3*x - 1)*sqrt(x + 1)
 - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x + 3)*sqrt(x + 1) - 320*x - 10)*sqrt(x + sqrt(x + 1))*(5*s
qrt(-14/25*I - 2/25) - I - 3) - 40*((((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*(x
 + 3)*sqrt(x + 1) - 35*x - 5)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) - I - 3) + 5*(((x + 3)*sqrt(x + 1
) - 7*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x + 3)*sqrt(x + 1) - 20*x - 10)*sqrt(x + sqrt(x + 1)))*sqr
t(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) - I
- 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6) + 100*(((x + 3)*sqrt(x + 1
) - 7*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 2*((x + 3)*sqrt(x + 1) - 32*x - 1)*(5*sqrt(14/25*I - 2/25) +
 I - 3) + 20*(x + 3)*sqrt(x + 1) - 140*x + 180)*sqrt(x + sqrt(x + 1)) + (5*(6*sqrt(5)*(3*x - 1)*sqrt(x + 1) +
sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + (30*sqrt(5)*(3*x - 1)*sqrt(x + 1) + (2*sqrt(5
)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*sqrt(5)*(3*x^2 + 4
*x + 15))*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 + 200*sqrt(5)*(9*x - 13)*sqrt(x + 1) + 10*(2*sqrt(5)*(34*x - 3)*
sqrt(x + 1) + sqrt(5)*(3*x^2 + 54*x + 65))*(5*sqrt(14/25*I - 2/25) + I - 3) + ((2*sqrt(5)*(2*x + 1)*sqrt(x + 1
) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 20*sqrt(5)*(34*x - 3)*sqrt(x + 1) + 12*(
2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*sqrt(5)*(
3*x^2 + 54*x + 65))*(5*sqrt(-14/25*I - 2/25) - I - 3) + 100*sqrt(5)*(13*x^2 + 24*x + 5) - 2*(100*sqrt(5)*(4*x
- 3)*sqrt(x + 1) + 5*(6*sqrt(5)*(3*x - 1)*sqrt(x + 1) + sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(14/25*I - 2/25) +
I - 3) + (30*sqrt(5)*(3*x - 1)*sqrt(x + 1) + (2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(
5*sqrt(14/25*I - 2/25) + I - 3) + 5*sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(-14/25*I - 2/25) - I - 3) + 50*sqrt(5)
*(3*x^2 - 6*x + 5))*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqr
t(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6))*sq
rt(5*sqrt(14/25*I - 2/25) + 5*sqrt(-14/25*I - 2/25) + 2*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*
sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30
*sqrt(14/25*I - 2/25) - 6*I - 6) + 6))/(x^2 + 1)) - 1/20*sqrt(5)*sqrt(5*sqrt(14/25*I - 2/25) + 5*sqrt(-14/25*I
 - 2/25) + 2*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/2
5*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6) + 6)*log(-
1/100*(20*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*(x + 3)*sqrt(x + 1) - 35*x -
 5)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 + 20*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(
14/25*I - 2/25) + I - 3)^2 + 12*((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x + 3
)*sqrt(x + 1) - 320*x - 10)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) - I - 3) - 40*((((3*x - 1)*sqrt(x +
 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*(x + 3)*sqrt(x + 1) - 35*x - 5)*sqrt(x + sqrt(x + 1))*(5*s
qrt(-14/25*I - 2/25) - I - 3) + 5*(((x + 3)*sqrt(x + 1) - 7*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x +
3)*sqrt(x + 1) - 20*x - 10)*sqrt(x + sqrt(x + 1)))*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqrt(
14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sqrt
(14/25*I - 2/25) - 6*I - 6) + 100*(((x + 3)*sqrt(x + 1) - 7*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 2*((x
+ 3)*sqrt(x + 1) - 32*x - 1)*(5*sqrt(14/25*I - 2/25) + I - 3) + 20*(x + 3)*sqrt(x + 1) - 140*x + 180)*sqrt(x +
 sqrt(x + 1)) - (5*(6*sqrt(5)*(3*x - 1)*sqrt(x + 1) + sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(14/25*I - 2/25) + I
- 3)^2 + (30*sqrt(5)*(3*x - 1)*sqrt(x + 1) + (2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(
5*sqrt(14/25*I - 2/25) + I - 3) + 5*sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 + 200*sqrt
(5)*(9*x - 13)*sqrt(x + 1) + 10*(2*sqrt(5)*(34*x - 3)*sqrt(x + 1) + sqrt(5)*(3*x^2 + 54*x + 65))*(5*sqrt(14/25
*I - 2/25) + I - 3) + ((2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/25)
 + I - 3)^2 + 20*sqrt(5)*(34*x - 3)*sqrt(x + 1) + 12*(2*sqrt(5)*(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x
- 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*sqrt(5)*(3*x^2 + 54*x + 65))*(5*sqrt(-14/25*I - 2/25) - I - 3) +
100*sqrt(5)*(13*x^2 + 24*x + 5) - 2*(100*sqrt(5)*(4*x - 3)*sqrt(x + 1) + 5*(6*sqrt(5)*(3*x - 1)*sqrt(x + 1) +
sqrt(5)*(3*x^2 + 4*x + 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + (30*sqrt(5)*(3*x - 1)*sqrt(x + 1) + (2*sqrt(5)*
(2*x + 1)*sqrt(x + 1) - sqrt(5)*(11*x^2 - 2*x - 15))*(5*sqrt(14/25*I - 2/25) + I - 3) + 5*sqrt(5)*(3*x^2 + 4*x
 + 15))*(5*sqrt(-14/25*I - 2/25) - I - 3) + 50*sqrt(5)*(3*x^2 - 6*x + 5))*sqrt(-3/4*(5*sqrt(14/25*I - 2/25) +
I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) - I - 3) - 3/4*(5*sqrt(-14/25*I - 2/2
5) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6))*sqrt(5*sqrt(14/25*I - 2/25) + 5*sqrt(-14/25*I - 2/25) + 2*
sqrt(-3/4*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 1/2*(5*sqrt(14/25*I - 2/25) + I + 9)*(5*sqrt(-14/25*I - 2/25) -
 I - 3) - 3/4*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 - 30*sqrt(14/25*I - 2/25) - 6*I - 6) + 6))/(x^2 + 1)) - 1/2*
sqrt(-1/2*sqrt(-14/25*I - 2/25) + 1/10*I + 3/10)*log(1/25*(2*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*
I - 2/25) + I - 3) + 5*(x + 3)*sqrt(x + 1) - 35*x - 5)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) - I - 3)
^2 + 2*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 12*((3*x - 1)*sqrt(x + 1) - 6*x
 - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x + 3)*sqrt(x + 1) - 320*x - 10)*sqrt(x + sqrt(x + 1))*(5*sqrt(-1
4/25*I - 2/25) - I - 3) + 2*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3)^3 + 12*((3*x -
 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 60*((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(1
4/25*I - 2/25) + I - 3) - 50*(x + 3)*sqrt(x + 1) - 1150*x - 450)*sqrt(x + sqrt(x + 1)) + ((11*x^2 - 2*(2*x + 1
)*sqrt(x + 1) - 2*x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3)^3 + 12*(11*x^2 - 2*(2*x + 1)*sqrt(x + 1) - 2*x - 15
)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - (15*x^2 - (11*x^2 - 2*(2*x + 1)*sqrt(x + 1) - 2*x - 15)*(5*sqrt(14/25*I
 - 2/25) + I - 3) + 30*(3*x - 1)*sqrt(x + 1) + 20*x + 75)*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 + 650*x^2 + 60*(
11*x^2 - 2*(2*x + 1)*sqrt(x + 1) - 2*x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3) + ((11*x^2 - 2*(2*x + 1)*sqrt(x
+ 1) - 2*x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 30*x^2 + 12*(11*x^2 - 2*(2*x + 1)*sqrt(x + 1) - 2*x - 15
)*(5*sqrt(14/25*I - 2/25) + I - 3) - 20*(34*x - 3)*sqrt(x + 1) - 540*x - 650)*(5*sqrt(-14/25*I - 2/25) - I - 3
) - 100*(21*x + 13)*sqrt(x + 1) - 1800*x - 2750)*sqrt(-1/2*sqrt(-14/25*I - 2/25) + 1/10*I + 3/10))/(x^2 + 1))
+ 1/2*sqrt(-1/2*sqrt(-14/25*I - 2/25) + 1/10*I + 3/10)*log(1/25*(2*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(
14/25*I - 2/25) + I - 3) + 5*(x + 3)*sqrt(x + 1) - 35*x - 5)*sqrt(x + sqrt(x + 1))*(5*sqrt(-14/25*I - 2/25) -
I - 3)^2 + 2*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 12*((3*x - 1)*sqrt(x + 1)
 - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3) + 10*(x + 3)*sqrt(x + 1) - 320*x - 10)*sqrt(x + sqrt(x + 1))*(5*s
qrt(-14/25*I - 2/25) - I - 3) + 2*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3)^3 + 12*(
(3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 60*((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*
sqrt(14/25*I - 2/25) + I - 3) - 50*(x + 3)*sqrt(x + 1) - 1150*x - 450)*sqrt(x + sqrt(x + 1)) - ((11*x^2 - 2*(2
*x + 1)*sqrt(x + 1) - 2*x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3)^3 + 12*(11*x^2 - 2*(2*x + 1)*sqrt(x + 1) - 2*
x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - (15*x^2 - (11*x^2 - 2*(2*x + 1)*sqrt(x + 1) - 2*x - 15)*(5*sqrt(1
4/25*I - 2/25) + I - 3) + 30*(3*x - 1)*sqrt(x + 1) + 20*x + 75)*(5*sqrt(-14/25*I - 2/25) - I - 3)^2 + 650*x^2
+ 60*(11*x^2 - 2*(2*x + 1)*sqrt(x + 1) - 2*x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3) + ((11*x^2 - 2*(2*x + 1)*s
qrt(x + 1) - 2*x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 - 30*x^2 + 12*(11*x^2 - 2*(2*x + 1)*sqrt(x + 1) - 2*
x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3) - 20*(34*x - 3)*sqrt(x + 1) - 540*x - 650)*(5*sqrt(-14/25*I - 2/25) -
 I - 3) - 100*(21*x + 13)*sqrt(x + 1) - 1800*x - 2750)*sqrt(-1/2*sqrt(-14/25*I - 2/25) + 1/10*I + 3/10))/(x^2
+ 1)) - 1/2*sqrt(-1/2*sqrt(14/25*I - 2/25) - 1/10*I + 3/10)*log(-1/25*(2*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5
*sqrt(14/25*I - 2/25) + I - 3)^3 + ((31*x - 27)*sqrt(x + 1) - 37*x - 31)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 +
10*((17*x - 9)*sqrt(x + 1) - 4*x - 17)*(5*sqrt(14/25*I - 2/25) + I - 3) + 50*(13*x - 1)*sqrt(x + 1) - 50*x - 1
50)*sqrt(x + sqrt(x + 1)) + ((11*x^2 - 2*(2*x + 1)*sqrt(x + 1) - 2*x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3)^3
+ (147*x^2 + 6*(7*x - 9)*sqrt(x + 1) - 4*x - 105)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 1550*x^2 + 10*(69*x^2 +
 2*(22*x - 9)*sqrt(x + 1) + 42*x - 25)*(5*sqrt(14/25*I - 2/25) + I - 3) + 100*(13*x + 9)*sqrt(x + 1) + 1400*x
- 250)*sqrt(-1/2*sqrt(14/25*I - 2/25) - 1/10*I + 3/10))/(x^2 + 1)) + 1/2*sqrt(-1/2*sqrt(14/25*I - 2/25) - 1/10
*I + 3/10)*log(-1/25*(2*(((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(14/25*I - 2/25) + I - 3)^3 + ((31*x - 27)*s
qrt(x + 1) - 37*x - 31)*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 10*((17*x - 9)*sqrt(x + 1) - 4*x - 17)*(5*sqrt(14
/25*I - 2/25) + I - 3) + 50*(13*x - 1)*sqrt(x + 1) - 50*x - 150)*sqrt(x + sqrt(x + 1)) - ((11*x^2 - 2*(2*x + 1
)*sqrt(x + 1) - 2*x - 15)*(5*sqrt(14/25*I - 2/25) + I - 3)^3 + (147*x^2 + 6*(7*x - 9)*sqrt(x + 1) - 4*x - 105)
*(5*sqrt(14/25*I - 2/25) + I - 3)^2 + 1550*x^2 + 10*(69*x^2 + 2*(22*x - 9)*sqrt(x + 1) + 42*x - 25)*(5*sqrt(14
/25*I - 2/25) + I - 3) + 100*(13*x + 9)*sqrt(x + 1) + 1400*x - 250)*sqrt(-1/2*sqrt(14/25*I - 2/25) - 1/10*I +
3/10))/(x^2 + 1)) + log(4*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) + 1) + 8*x + 8*sqrt(x + 1) + 5)

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

[Out]

sage0*x

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maple [B]  time = 0.10, size = 125, normalized size = 0.80

method result size
derivativedivides \(2 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (8 \textit {\_R}^{3}-12 \textit {\_R}^{2}+6 \textit {\_R} -1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) \(125\)
default \(2 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (8 \textit {\_R}^{3}-12 \textit {\_R}^{2}+6 \textit {\_R} -1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) \(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*ln(1/2+(1+x)^(1/2)+(x+(1+x)^(1/2))^(1/2))+sum((8*_R^3-12*_R^2+6*_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}} \sqrt {x + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2-1}{\sqrt {x+\sqrt {x+1}}\,\left (x^2+1\right )\,\sqrt {x+1}} \,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 + 1)^(1/2)),x)

[Out]

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

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

[Out]

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

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