∫ ∘ _______ x+√ __

3.25.85 $\int \frac {\sqrt {x+\sqrt {1+x}}}{1+x^2} \, dx$

Optimal. Leaf size=204 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^5+20 \text {$\#$1}^4-48 \text {$\#$1}^3+40 \text {$\#$1}^2-8 \text {$\#$1}+1\& ,\frac {\text {$\#$1}^6 \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )-2 \text {$\#$1}^5 \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )+2 \text {$\#$1} \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )+\log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+5 \text {$\#$1}^4+10 \text {$\#$1}^3-18 \text {$\#$1}^2+10 \text {$\#$1}-1}\& \right ] \]

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Rubi [C] time = 0.64, antiderivative size = 337, normalized size of antiderivative = 1.65, number of steps used = 22, number of rules used = 9, integrand size = 21, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.429, Rules used = {6741, 6728, 1021, 1078, 621, 206, 1033, 724, 204} \begin {gather*} \frac {1}{2} i \sqrt {i+\sqrt {1-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 )-\frac {1}{2} i \sqrt {\sqrt {1+i}-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 )+\frac {1}{2} i \sqrt {\sqrt {1-i}-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 )-\frac {1}{2} i \sqrt {i+\sqrt {1+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 ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

+ 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]])]

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 1021

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp

[(h*(a + b*x + c*x^2)^p*(d + 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 + f*x^2)^q*Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(-2*g*f)*(p + q

+ 1))*x + (h*p*(-(b*f)) + c*(-2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ

[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 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 1078

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sym

bol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d

+ e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 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 {\sqrt {x+\sqrt {1+x}}}{1+x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{1+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {i x \sqrt {-1+x+x^2}}{(2+2 i)-2 x^2}+\frac {i x \sqrt {-1+x+x^2}}{(-2+2 i)+2 x^2}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 i \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{(2+2 i)-2 x^2} \, dx,x,\sqrt {1+x}\right )+2 i \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{(-2+2 i)+2 x^2} \, dx,x,\sqrt {1+x}\right )\\ &=i \operatorname {Subst}\left (\int \frac {(1+i)+2 i x+x^2}{\left ((2+2 i)-2 x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-i \operatorname {Subst}\left (\int \frac {(-1+i)+2 i x-x^2}{\sqrt {-1+x+x^2} \left ((-2+2 i)+2 x^2\right )} \, dx,x,\sqrt {1+x}\right )\\ &=-\left (\frac {1}{2} i \operatorname {Subst}\left (\int \frac {(-4-4 i)-4 i x}{\left ((2+2 i)-2 x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {(-4+4 i)+4 i x}{\sqrt {-1+x+x^2} \left ((-2+2 i)+2 x^2\right )} \, dx,x,\sqrt {1+x}\right )\\ &=-\left (\left (-1-i \sqrt {1-i}\right ) \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 )\right )-\left (-1+i \sqrt {1-i}\right ) \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 )+\left (-1-i \sqrt {1+i}\right ) \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 )+\left (-1+i \sqrt {1+i}\right ) \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 )\\ &=-\left (\left (2 \left (1-i \sqrt {1-i}\right )\right ) \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 )\right )-\left (2 \left (1+i \sqrt {1-i}\right )\right ) \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 )+\left (2 \left (1-i \sqrt {1+i}\right )\right ) \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 )+\left (2 \left (1+i \sqrt {1+i}\right )\right ) \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 )\\ &=\frac {1}{2} i \sqrt {i+\sqrt {1-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 )-\frac {1}{2} i \sqrt {-i+\sqrt {1+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 )+\frac {1}{2} i \sqrt {-i+\sqrt {1-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 )-\frac {1}{2} i \sqrt {i+\sqrt {1+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 )\\ \end {align*}

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Mathematica [C] time = 6.43, size = 2581, normalized size = 12.65 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((1 + I) + Sqrt[1 - I])*ArcTan[((2 - 3*I) + (3 - I)*Sqrt[1 - I] - 8*Sqrt[1 + x] - 5*Sqrt[1 - I]*Sqrt[1 + x] +

(2 + 5*I)*(1 + x) + (5*I)*Sqrt[1 - I]*(1 + x) + 4*Sqrt[I - Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] + 2*Sqrt[1 - I]

*Sqrt[I - Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] - (6 + 2*I)*Sqrt[I - Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x

]] - (8*Sqrt[I - Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/Sqrt[1 - I])/((-4 + 7*I) - (6 - 2*I)*Sqrt[1 -

I] + (4 - 2*I)*Sqrt[1 + x] + (6 - 2*I)*Sqrt[1 - I]*Sqrt[1 + x] + (10 + I)*(1 + x) + (8 + 4*I)*Sqrt[1 - I]*(1

+ x))])/(2*Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]) + (((-1 - I) + Sqrt[1 - I])*ArcTan[((-2 + 3*I) + (3 - I)*Sqrt[1

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

Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] + 2*Sqrt[1 - I]*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] + (6 + 2*I)*Sqr

t[I + Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] - (8*Sqrt[I + Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 +

x]])/Sqrt[1 - I])/((4 - 7*I) - (6 - 2*I)*Sqrt[1 - I] - (4 - 2*I)*Sqrt[1 + x] + (6 - 2*I)*Sqrt[1 - I]*Sqrt[1 +

x] - (10 + I)*(1 + x) + (8 + 4*I)*Sqrt[1 - I]*(1 + x))])/(2*Sqrt[1 - I]*Sqrt[I + Sqrt[1 - I]]) - ((I/2)*((-1 +

I) + Sqrt[1 + I])*ArcTan[((1 + 8*I) - 5*(1 + I)^(3/2) - (16 + 8*I)*Sqrt[1 + x] + (10 + 5*I)*Sqrt[1 + I]*Sqrt[

1 + x] + (9 - 8*I)*(1 + x) - (5 - 10*I)*Sqrt[1 + I]*(1 + x) - 4*Sqrt[I - Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] +

(4 - 2*I)*Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] - 8*Sqrt[I - Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x

+ Sqrt[1 + x]] + (8 - 4*I)*Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/((9 + 20*I) -

12*(1 + I)^(3/2) - (14 + 20*I)*Sqrt[1 + x] + (22 + 12*I)*Sqrt[1 + I]*Sqrt[1 + x] + (6 - 15*I)*(1 + x) + (2 +

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

- 8*I) - 5*(1 + I)^(3/2) + (16 + 8*I)*Sqrt[1 + x] + (10 + 5*I)*Sqrt[1 + I]*Sqrt[1 + x] - (9 - 8*I)*(1 + x) - (

5 - 10*I)*Sqrt[1 + I]*(1 + x) + 4*Sqrt[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] + (4 - 2*I)*Sqrt[1 + I]*Sqrt[I +

Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] + 8*Sqrt[I + Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] + (8 - 4*I)*Sq

rt[1 + I]*Sqrt[I + Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/((-9 - 20*I) - 12*(1 + I)^(3/2) + (14 + 20*

I)*Sqrt[1 + x] + (22 + 12*I)*Sqrt[1 + I]*Sqrt[1 + x] - (6 - 15*I)*(1 + x) + (2 + 12*I)*Sqrt[1 + I]*(1 + x))])/

(Sqrt[1 + I]*Sqrt[I + Sqrt[1 + I]]) + ((I/4)*((1 + I) + Sqrt[1 - I])*Log[(Sqrt[1 - I] - Sqrt[1 + x])^2])/(Sqrt

[1 - I]*Sqrt[I - Sqrt[1 - I]]) + (((1 - I) + Sqrt[1 + I])*Log[(Sqrt[1 + I] - Sqrt[1 + x])^2])/(4*Sqrt[1 + I]*S

qrt[I + Sqrt[1 + I]]) + ((I/4)*((-1 - I) + Sqrt[1 - I])*Log[(Sqrt[1 - I] + Sqrt[1 + x])^2])/(Sqrt[1 - I]*Sqrt[

I + Sqrt[1 - I]]) + (((-1 + I) + Sqrt[1 + I])*Log[(Sqrt[1 + I] + Sqrt[1 + x])^2])/(4*Sqrt[1 + I]*Sqrt[I - Sqrt

[1 + I]]) - ((I/4)*((1 + I) + Sqrt[1 - I])*Log[(5 + 17*I) + (14*I)*Sqrt[1 - I] - (10 + 22*I)*Sqrt[1 + x] + (5

- 19*I)*Sqrt[1 - I]*Sqrt[1 + x] - (25 + 2*I)*(1 + x) - (15 + 9*I)*Sqrt[1 - I]*(1 + x) - (4 - 4*I)*Sqrt[I - Sqr

t[1 - I]]*Sqrt[x + Sqrt[1 + x]] - (6 - 2*I)*Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] - (8 - 8*I

)*Sqrt[I - Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] - (12 - 4*I)*Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]*Sqrt[

1 + x]*Sqrt[x + Sqrt[1 + x]]])/(Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]) - ((I/4)*((-1 - I) + Sqrt[1 - I])*Log[(-5 -

17*I) + (14*I)*Sqrt[1 - I] + (10 + 22*I)*Sqrt[1 + x] + (5 - 19*I)*Sqrt[1 - I]*Sqrt[1 + x] + (25 + 2*I)*(1 + x

) - (15 + 9*I)*Sqrt[1 - I]*(1 + x) + (4 - 4*I)*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] - (6 - 2*I)*Sqrt[1

- I]*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] + (8 - 8*I)*Sqrt[I + Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1

+ x]] - (12 - 4*I)*Sqrt[1 - I]*Sqrt[I + Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]]])/(Sqrt[1 - I]*Sqrt[I

+ Sqrt[1 - I]]) - (((-1 + I) + Sqrt[1 + I])*Log[(-3 + 5*I) - (2 + 4*I)*Sqrt[1 + I] + (2 - 2*I)*Sqrt[1 + x] - (

1 - 3*I)*Sqrt[1 + I]*Sqrt[1 + x] - (8 + 7*I)*(1 + x) + (9 + 3*I)*Sqrt[1 + I]*(1 + x) + (4 + 4*I)*Sqrt[I - Sqrt

[1 + I]]*Sqrt[x + Sqrt[1 + x]] - 2*(1 + I)^(3/2)*Sqrt[I - Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] - (8 + 4*I)*Sqrt[

I - Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] + 8*Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x +

Sqrt[1 + x]]])/(4*Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]) - (((1 - I) + Sqrt[1 + I])*Log[(3 - 5*I) - (2 + 4*I)*Sqrt

[1 + I] - (2 - 2*I)*Sqrt[1 + x] - (1 - 3*I)*Sqrt[1 + I]*Sqrt[1 + x] + (8 + 7*I)*(1 + x) + (9 + 3*I)*Sqrt[1 + I

]*(1 + x) - (4 + 4*I)*Sqrt[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] - 2*(1 + I)^(3/2)*Sqrt[I + Sqrt[1 + I]]*Sqrt

[x + Sqrt[1 + x]] + (8 + 4*I)*Sqrt[I + Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] + 8*Sqrt[1 + I]*Sqrt[I +

Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]]])/(4*Sqrt[1 + I]*Sqrt[I + Sqrt[1 + I]])

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IntegrateAlgebraic [A] time = 0.00, size = 212, normalized size = 1.04 \begin {gather*} \frac {1}{2} \text {RootSum}\left [1-8 \text {$\#$1}+40 \text {$\#$1}^2-48 \text {$\#$1}^3+20 \text {$\#$1}^4+8 \text {$\#$1}^5-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^5+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-1+10 \text {$\#$1}-18 \text {$\#$1}^2+10 \text {$\#$1}^3+5 \text {$\#$1}^4-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

RootSum[1 - 8*#1 + 40*#1^2 - 48*#1^3 + 20*#1^4 + 8*#1^5 - 4*#1^6 + #1^8 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[

1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x

]] - #1]*#1^5 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^6)/(-1 + 10*#1 - 18*#1^2 + 10*#1^3 + 5*#1^4

- 3*#1^5 + #1^7) & ]/2

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fricas [B] time = 5.25, size = 4535, normalized size = 22.23

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(sqrt(1/4*I + 1/4) + sqrt(-1/4*I + 1/4) - 2*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*

I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/4) - I)^2))*log(-1/4*(2*(((2*x + 1)*sqrt(x

+ 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I) + 4*(2*x + 1)*sqrt(x + 1) - x - 8)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/

4*I + 1/4) - I)^2 + 2*(((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I)^2 + (3*x - 16)*sqrt(x + 1)

+ 4*x - 3)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I) + 8*((((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/

4*I + 1/4) + I) + 4*(2*x + 1)*sqrt(x + 1) - x - 8)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I) + ((4*(2*x

+ 1)*sqrt(x + 1) - x - 8)*(2*sqrt(1/4*I + 1/4) + I) - (3*x - 16)*sqrt(x + 1) - 4*x + 3)*sqrt(x + sqrt(x + 1))

)*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*

sqrt(-1/4*I + 1/4) - I)^2) + 2*((4*(2*x + 1)*sqrt(x + 1) - x - 8)*(2*sqrt(1/4*I + 1/4) + I)^2 + ((3*x - 16)*sq

rt(x + 1) + 4*x - 3)*(2*sqrt(1/4*I + 1/4) + I) + 12*(2*x + 1)*sqrt(x + 1) + 32*x + 46)*sqrt(x + sqrt(x + 1)) +

((3*x^2 + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(1/4*I + 1/4) + I)^2 + (3*x^2 - 2*(4*x^2 - sqrt(x + 1)*(x

- 2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I) + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(-1/4*I + 1/4) - I)^2 + 4

4*x^2 - 2*(6*x^2 + (16*x + 3)*sqrt(x + 1) + 3*x + 10)*(2*sqrt(1/4*I + 1/4) + I) - 2*((4*x^2 - sqrt(x + 1)*(x -

2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I)^2 + 6*x^2 + (16*x + 3)*sqrt(x + 1) + 3*x + 10)*(2*sqrt(-1/4*I + 1/4)

- I) + 4*(12*x^2 + (3*x^2 + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(1/4*I + 1/4) + I) + (3*x^2 - 2*(4*x^2 -

sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I) + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(-1/4*I +

1/4) - I) + 2*(16*x + 3)*sqrt(x + 1) + 6*x + 20)*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I +

1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/4) - I)^2) + 24*sqrt(x + 1)*(x - 2) + 92*x - 20

)*sqrt(sqrt(1/4*I + 1/4) + sqrt(-1/4*I + 1/4) - 2*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I +

1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/4) - I)^2)))/(x^2 + 1)) + 1/4*sqrt(sqrt(1/4*I

+ 1/4) + sqrt(-1/4*I + 1/4) - 2*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt

(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/4) - I)^2))*log(-1/4*(2*(((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sq

rt(1/4*I + 1/4) + I) + 4*(2*x + 1)*sqrt(x + 1) - x - 8)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I)^2 + 2

*(((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I)^2 + (3*x - 16)*sqrt(x + 1) + 4*x - 3)*sqrt(x + s

qrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I) + 8*((((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I) + 4*(

2*x + 1)*sqrt(x + 1) - x - 8)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I) + ((4*(2*x + 1)*sqrt(x + 1) - x

- 8)*(2*sqrt(1/4*I + 1/4) + I) - (3*x - 16)*sqrt(x + 1) - 4*x + 3)*sqrt(x + sqrt(x + 1)))*sqrt(-3/16*(2*sqrt(

1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/4) -

I)^2) + 2*((4*(2*x + 1)*sqrt(x + 1) - x - 8)*(2*sqrt(1/4*I + 1/4) + I)^2 + ((3*x - 16)*sqrt(x + 1) + 4*x - 3)*

(2*sqrt(1/4*I + 1/4) + I) + 12*(2*x + 1)*sqrt(x + 1) + 32*x + 46)*sqrt(x + sqrt(x + 1)) - ((3*x^2 + 8*sqrt(x +

1)*(x - 2) - 16*x + 5)*(2*sqrt(1/4*I + 1/4) + I)^2 + (3*x^2 - 2*(4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sq

rt(1/4*I + 1/4) + I) + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(-1/4*I + 1/4) - I)^2 + 44*x^2 - 2*(6*x^2 + (1

6*x + 3)*sqrt(x + 1) + 3*x + 10)*(2*sqrt(1/4*I + 1/4) + I) - 2*((4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqr

t(1/4*I + 1/4) + I)^2 + 6*x^2 + (16*x + 3)*sqrt(x + 1) + 3*x + 10)*(2*sqrt(-1/4*I + 1/4) - I) + 4*(12*x^2 + (3

*x^2 + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(1/4*I + 1/4) + I) + (3*x^2 - 2*(4*x^2 - sqrt(x + 1)*(x - 2) +

2*x - 5)*(2*sqrt(1/4*I + 1/4) + I) + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(-1/4*I + 1/4) - I) + 2*(16*x +

3)*sqrt(x + 1) + 6*x + 20)*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/

4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/4) - I)^2) + 24*sqrt(x + 1)*(x - 2) + 92*x - 20)*sqrt(sqrt(1/4*I + 1

/4) + sqrt(-1/4*I + 1/4) - 2*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1

/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/4) - I)^2)))/(x^2 + 1)) - 1/4*sqrt(sqrt(1/4*I + 1/4) + sqrt(-1/4*I

+ 1/4) + 2*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) -

3/16*(2*sqrt(-1/4*I + 1/4) - I)^2))*log(-1/4*(2*(((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I)

+ 4*(2*x + 1)*sqrt(x + 1) - x - 8)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I)^2 + 2*(((2*x + 1)*sqrt(x +

1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I)^2 + (3*x - 16)*sqrt(x + 1) + 4*x - 3)*sqrt(x + sqrt(x + 1))*(2*sqrt(-

1/4*I + 1/4) - I) - 8*((((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I) + 4*(2*x + 1)*sqrt(x + 1)

- x - 8)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I) + ((4*(2*x + 1)*sqrt(x + 1) - x - 8)*(2*sqrt(1/4*I +

1/4) + I) - (3*x - 16)*sqrt(x + 1) - 4*x + 3)*sqrt(x + sqrt(x + 1)))*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 -

1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/4) - I)^2) + 2*((4*(2*x +

1)*sqrt(x + 1) - x - 8)*(2*sqrt(1/4*I + 1/4) + I)^2 + ((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(2*sqrt(1/4*I + 1/4)

+ I) + 12*(2*x + 1)*sqrt(x + 1) + 32*x + 46)*sqrt(x + sqrt(x + 1)) + ((3*x^2 + 8*sqrt(x + 1)*(x - 2) - 16*x +

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

+ 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(-1/4*I + 1/4) - I)^2 + 44*x^2 - 2*(6*x^2 + (16*x + 3)*sqrt(x + 1)

+ 3*x + 10)*(2*sqrt(1/4*I + 1/4) + I) - 2*((4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I)^2

+ 6*x^2 + (16*x + 3)*sqrt(x + 1) + 3*x + 10)*(2*sqrt(-1/4*I + 1/4) - I) - 4*(12*x^2 + (3*x^2 + 8*sqrt(x + 1)*

(x - 2) - 16*x + 5)*(2*sqrt(1/4*I + 1/4) + I) + (3*x^2 - 2*(4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/4

*I + 1/4) + I) + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(-1/4*I + 1/4) - I) + 2*(16*x + 3)*sqrt(x + 1) + 6*x

+ 20)*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/1

6*(2*sqrt(-1/4*I + 1/4) - I)^2) + 24*sqrt(x + 1)*(x - 2) + 92*x - 20)*sqrt(sqrt(1/4*I + 1/4) + sqrt(-1/4*I + 1

/4) + 2*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/

16*(2*sqrt(-1/4*I + 1/4) - I)^2)))/(x^2 + 1)) + 1/4*sqrt(sqrt(1/4*I + 1/4) + sqrt(-1/4*I + 1/4) + 2*sqrt(-3/16

*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I

+ 1/4) - I)^2))*log(-1/4*(2*(((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I) + 4*(2*x + 1)*sqrt(x

+ 1) - x - 8)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I)^2 + 2*(((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqr

t(1/4*I + 1/4) + I)^2 + (3*x - 16)*sqrt(x + 1) + 4*x - 3)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I) - 8

*((((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I) + 4*(2*x + 1)*sqrt(x + 1) - x - 8)*sqrt(x + sqr

t(x + 1))*(2*sqrt(-1/4*I + 1/4) - I) + ((4*(2*x + 1)*sqrt(x + 1) - x - 8)*(2*sqrt(1/4*I + 1/4) + I) - (3*x - 1

6)*sqrt(x + 1) - 4*x + 3)*sqrt(x + sqrt(x + 1)))*sqrt(-3/16*(2*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I +

1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/4) - I)^2) + 2*((4*(2*x + 1)*sqrt(x + 1) - x -

8)*(2*sqrt(1/4*I + 1/4) + I)^2 + ((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(2*sqrt(1/4*I + 1/4) + I) + 12*(2*x + 1)*s

qrt(x + 1) + 32*x + 46)*sqrt(x + sqrt(x + 1)) - ((3*x^2 + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(1/4*I + 1/

4) + I)^2 + (3*x^2 - 2*(4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I) + 8*sqrt(x + 1)*(x -

2) - 16*x + 5)*(2*sqrt(-1/4*I + 1/4) - I)^2 + 44*x^2 - 2*(6*x^2 + (16*x + 3)*sqrt(x + 1) + 3*x + 10)*(2*sqrt(1

/4*I + 1/4) + I) - 2*((4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I)^2 + 6*x^2 + (16*x + 3)

*sqrt(x + 1) + 3*x + 10)*(2*sqrt(-1/4*I + 1/4) - I) - 4*(12*x^2 + (3*x^2 + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(

2*sqrt(1/4*I + 1/4) + I) + (3*x^2 - 2*(4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I) + 8*sq

rt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(-1/4*I + 1/4) - I) + 2*(16*x + 3)*sqrt(x + 1) + 6*x + 20)*sqrt(-3/16*(2*

sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1/

4) - I)^2) + 24*sqrt(x + 1)*(x - 2) + 92*x - 20)*sqrt(sqrt(1/4*I + 1/4) + sqrt(-1/4*I + 1/4) + 2*sqrt(-3/16*(2

*sqrt(1/4*I + 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I + 1/4) + I)*(2*sqrt(-1/4*I + 1/4) - I) - 3/16*(2*sqrt(-1/4*I + 1

/4) - I)^2)))/(x^2 + 1)) + 1/2*sqrt(-1/2*sqrt(-1/4*I + 1/4) + 1/4*I)*log(((((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(

2*sqrt(1/4*I + 1/4) + I) + 4*(2*x + 1)*sqrt(x + 1) - x - 8)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I)^2

+ (((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I)^2 + (3*x - 16)*sqrt(x + 1) + 4*x - 3)*sqrt(x +

sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I) + (((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I)^3 - 6*

(2*x + 1)*sqrt(x + 1) - 16*x - 23)*sqrt(x + sqrt(x + 1)) + (2*(4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(

1/4*I + 1/4) + I)^3 - (3*x^2 - 2*(4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I) + 8*sqrt(x

+ 1)*(x - 2) - 16*x + 5)*(2*sqrt(-1/4*I + 1/4) - I)^2 + 22*x^2 + 2*((4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2

*sqrt(1/4*I + 1/4) + I)^2 + 6*x^2 + (16*x + 3)*sqrt(x + 1) + 3*x + 10)*(2*sqrt(-1/4*I + 1/4) - I) + 12*sqrt(x

+ 1)*(x - 2) + 46*x - 10)*sqrt(-1/2*sqrt(-1/4*I + 1/4) + 1/4*I))/(x^2 + 1)) - 1/2*sqrt(-1/2*sqrt(-1/4*I + 1/4)

+ 1/4*I)*log(((((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I) + 4*(2*x + 1)*sqrt(x + 1) - x - 8)

*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I)^2 + (((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4)

+ I)^2 + (3*x - 16)*sqrt(x + 1) + 4*x - 3)*sqrt(x + sqrt(x + 1))*(2*sqrt(-1/4*I + 1/4) - I) + (((2*x + 1)*sqrt

(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I)^3 - 6*(2*x + 1)*sqrt(x + 1) - 16*x - 23)*sqrt(x + sqrt(x + 1)) -

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

2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I) + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(-1/4*I + 1/4) - I)^2 + 22*

x^2 + 2*((4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I)^2 + 6*x^2 + (16*x + 3)*sqrt(x + 1)

+ 3*x + 10)*(2*sqrt(-1/4*I + 1/4) - I) + 12*sqrt(x + 1)*(x - 2) + 46*x - 10)*sqrt(-1/2*sqrt(-1/4*I + 1/4) + 1/

4*I))/(x^2 + 1)) + 1/2*sqrt(-1/2*sqrt(1/4*I + 1/4) - 1/4*I)*log(-((((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1

/4*I + 1/4) + I)^3 - (4*(2*x + 1)*sqrt(x + 1) - x - 8)*(2*sqrt(1/4*I + 1/4) + I)^2 - ((3*x - 16)*sqrt(x + 1) +

4*x - 3)*(2*sqrt(1/4*I + 1/4) + I) + 10*(2*x + 1)*sqrt(x + 1) - 20*x + 15)*sqrt(x + sqrt(x + 1)) + (2*(4*x^2

- sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/4*I + 1/4) + I)^3 + (3*x^2 + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*s

qrt(1/4*I + 1/4) + I)^2 + 10*x^2 - 2*(6*x^2 + (16*x + 3)*sqrt(x + 1) + 3*x + 10)*(2*sqrt(1/4*I + 1/4) + I) - 2

0*sqrt(x + 1)*(x - 2) - 30*x - 30)*sqrt(-1/2*sqrt(1/4*I + 1/4) - 1/4*I))/(x^2 + 1)) - 1/2*sqrt(-1/2*sqrt(1/4*I

+ 1/4) - 1/4*I)*log(-((((2*x + 1)*sqrt(x + 1) - 9*x - 2)*(2*sqrt(1/4*I + 1/4) + I)^3 - (4*(2*x + 1)*sqrt(x +

1) - x - 8)*(2*sqrt(1/4*I + 1/4) + I)^2 - ((3*x - 16)*sqrt(x + 1) + 4*x - 3)*(2*sqrt(1/4*I + 1/4) + I) + 10*(2

*x + 1)*sqrt(x + 1) - 20*x + 15)*sqrt(x + sqrt(x + 1)) - (2*(4*x^2 - sqrt(x + 1)*(x - 2) + 2*x - 5)*(2*sqrt(1/

4*I + 1/4) + I)^3 + (3*x^2 + 8*sqrt(x + 1)*(x - 2) - 16*x + 5)*(2*sqrt(1/4*I + 1/4) + I)^2 + 10*x^2 - 2*(6*x^2

+ (16*x + 3)*sqrt(x + 1) + 3*x + 10)*(2*sqrt(1/4*I + 1/4) + I) - 20*sqrt(x + 1)*(x - 2) - 30*x - 30)*sqrt(-1/

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

________________________________________________________________________________________

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

[Out]

sage0*x

________________________________________________________________________________________

maple [B] time = 0.12, size = 105, normalized size = 0.51


method	result	size

derivativedivides	$\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{5}+2 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )}{2}$	$105$
default	$\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{5}+2 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )}{2}$	$105$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sum((_R^6-2*_R^5+2*_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 {\sqrt {x + \sqrt {x + 1}}}{x^{2} + 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^2+1),x, algorithm="maxima")

[Out]

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

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

[Out]

int((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 {\sqrt {x + \sqrt {x + 1}}}{x^{2} + 1}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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