3.26.11 \(\int \frac {\sqrt {x+\sqrt {1+x}}}{\sqrt {1+x} (1+x^2)} \, dx\) [2511]

Optimal. Leaf size=209 \[ -\frac {1}{2} \text {RootSum}\left [1-8 \text {$\#$1}+40 \text {$\#$1}^2-48 \text {$\#$1}^3+20 \text {$\#$1}^4+8 \text {$\#$1}^5-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-\log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right )+5 \log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^2-5 \log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^5}{-1+10 \text {$\#$1}-18 \text {$\#$1}^2+10 \text {$\#$1}^3+5 \text {$\#$1}^4-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 0.67, antiderivative size = 365, normalized size of antiderivative = 1.75, number of steps used = 20, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6873, 6860, 1004, 635, 212, 1047, 738, 210} \begin {gather*} -\frac {i \text {ArcTan}\left (\frac {-\left (\left (1-2 \sqrt {1-i}\right ) \sqrt {x+1}\right )+\sqrt {1-i}+2}{2 \sqrt {i+\sqrt {1-i}} \sqrt {x+\sqrt {x+1}}}\right )}{2 \sqrt {\frac {1-i}{i+\sqrt {1-i}}}}+\frac {i \text {ArcTan}\left (\frac {-\left (\left (1-2 \sqrt {1+i}\right ) \sqrt {x+1}\right )+\sqrt {1+i}+2}{2 \sqrt {\sqrt {1+i}-i} \sqrt {x+\sqrt {x+1}}}\right )}{2 \sqrt {-\frac {1+i}{i-\sqrt {1+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 {\sqrt {1-i}-i} \sqrt {x+\sqrt {x+1}}}\right )}{2 \sqrt {-\frac {1-i}{i-\sqrt {1-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 )}{2 \sqrt {\frac {1+i}{i+\sqrt {1+i}}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 210

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

Rule 212

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

Rule 635

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

Rule 738

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

Rule 1004

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

Rule 1047

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

Rule 6860

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

Rule 6873

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

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Mathematica [A]
time = 0.18, size = 217, normalized size = 1.04 \begin {gather*} -\frac {1}{2} \text {RootSum}\left [1-8 \text {$\#$1}+40 \text {$\#$1}^2-48 \text {$\#$1}^3+20 \text {$\#$1}^4+8 \text {$\#$1}^5-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+5 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2-5 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1+10 \text {$\#$1}-18 \text {$\#$1}^2+10 \text {$\#$1}^3+5 \text {$\#$1}^4-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*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] + 5*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2 - 5*Log[-Sqrt[1 + x] + Sqrt[x + Sq
rt[1 + x]] - #1]*#1^4 + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5)/(-1 + 10*#1 - 18*#1^2 + 10*#1^3
+ 5*#1^4 - 3*#1^5 + #1^7) & ]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 0.11, size = 109, normalized size = 0.52

method	result	size
derivativedivides	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-5 \textit {\_R}^{4}+5 \textit {\_R}^{2}-1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )}{2}\)	\(109\)
default	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-5 \textit {\_R}^{4}+5 \textit {\_R}^{2}-1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )}{2}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 6.97, size = 5235, normalized size = 25.05 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(sqrt(1/8*I + 1/8) + sqrt(-1/8*I + 1/8) - 2*sqrt(-3/64*(4*sqrt(1/8*I + 1/8) - I + 1)^2 - 3/64*(4*sqrt
(-1/8*I + 1/8) + I + 1)^2 - 1/32*(4*sqrt(1/8*I + 1/8) - I + 1)*(4*sqrt(-1/8*I + 1/8) + I - 3) + 1/2*sqrt(-1/8*
I + 1/8) + 1/8*I + 1/8) - 1/2)*log(1/4*(2*(((3*x - 1)*sqrt(x + 1) + 4*x - 3)*(4*sqrt(-1/8*I + 1/8) + I + 1) -
3*(3*x - 1)*sqrt(x + 1) - 7*x + 9)*sqrt(x + sqrt(x + 1))*(4*sqrt(1/8*I + 1/8) - I + 1)^2 + 2*(((3*x - 1)*sqrt(
x + 1) + 4*x - 3)*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 4*((3*x - 1)*sqrt(x + 1) + 4*x - 3)*(4*sqrt(-1/8*I + 1/8)
+ I + 1) + 2*(7*x + 1)*sqrt(x + 1) + 12*x - 14)*sqrt(x + sqrt(x + 1))*(4*sqrt(1/8*I + 1/8) - I + 1) + 16*((((
3*x - 1)*sqrt(x + 1) + 4*x - 3)*(4*sqrt(-1/8*I + 1/8) + I + 1) - 3*(3*x - 1)*sqrt(x + 1) - 7*x + 9)*sqrt(x + s
qrt(x + 1))*(4*sqrt(1/8*I + 1/8) - I + 1) - ((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(4*sqrt(-1/8*I + 1/8) + I + 1
) - 2*(11*x - 7)*sqrt(x + 1) - 16*x + 22)*sqrt(x + sqrt(x + 1)))*sqrt(-3/64*(4*sqrt(1/8*I + 1/8) - I + 1)^2 -
3/64*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 1/32*(4*sqrt(1/8*I + 1/8) - I + 1)*(4*sqrt(-1/8*I + 1/8) + I - 3) + 1/
2*sqrt(-1/8*I + 1/8) + 1/8*I + 1/8) - 2*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(4*sqrt(-1/8*I + 1/8) + I + 1)^2
- 2*((7*x + 1)*sqrt(x + 1) + 6*x - 7)*(4*sqrt(-1/8*I + 1/8) + I + 1) - 4*sqrt(x + 1)*(x - 7) - 12*x + 44)*sqrt
(x + sqrt(x + 1)) + ((3*x^2 - (x^2 + 2*(3*x + 4)*sqrt(x + 1) + 8*x + 5)*(4*sqrt(-1/8*I + 1/8) + I + 1) + 2*(9*
x + 7)*sqrt(x + 1) + 24*x + 15)*(4*sqrt(1/8*I + 1/8) - I + 1)^2 + (3*x^2 + 2*(9*x + 7)*sqrt(x + 1) + 24*x + 15
)*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 68*x^2 - ((x^2 + 2*(3*x + 4)*sqrt(x + 1) + 8*x + 5)*(4*sqrt(-1/8*I + 1/8)
+ I + 1)^2 - 2*x^2 - 4*(x^2 + 2*(3*x + 4)*sqrt(x + 1) + 8*x + 5)*(4*sqrt(-1/8*I + 1/8) + I + 1) + 4*(7*x + 6)
*sqrt(x + 1) + 24*x + 30)*(4*sqrt(1/8*I + 1/8) - I + 1) + 2*(x^2 - 2*(7*x + 6)*sqrt(x + 1) - 12*x - 15)*(4*sqr
t(-1/8*I + 1/8) + I + 1) - 8*(14*x^2 - (3*x^2 - (x^2 + 2*(3*x + 4)*sqrt(x + 1) + 8*x + 5)*(4*sqrt(-1/8*I + 1/8
) + I + 1) + 2*(9*x + 7)*sqrt(x + 1) + 24*x + 15)*(4*sqrt(1/8*I + 1/8) - I + 1) - (3*x^2 + 2*(9*x + 7)*sqrt(x
+ 1) + 24*x + 15)*(4*sqrt(-1/8*I + 1/8) + I + 1) + 4*(11*x + 8)*sqrt(x + 1) + 72*x + 30)*sqrt(-3/64*(4*sqrt(1/
8*I + 1/8) - I + 1)^2 - 3/64*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 1/32*(4*sqrt(1/8*I + 1/8) - I + 1)*(4*sqrt(-1/
8*I + 1/8) + I - 3) + 1/2*sqrt(-1/8*I + 1/8) + 1/8*I + 1/8) - 8*(11*x + 3)*sqrt(x + 1) - 64*x - 20)*sqrt(sqrt(
1/8*I + 1/8) + sqrt(-1/8*I + 1/8) - 2*sqrt(-3/64*(4*sqrt(1/8*I + 1/8) - I + 1)^2 - 3/64*(4*sqrt(-1/8*I + 1/8)
+ I + 1)^2 - 1/32*(4*sqrt(1/8*I + 1/8) - I + 1)*(4*sqrt(-1/8*I + 1/8) + I - 3) + 1/2*sqrt(-1/8*I + 1/8) + 1/8*
I + 1/8) - 1/2))/(x^2 + 1)) + 1/4*sqrt(sqrt(1/8*I + 1/8) + sqrt(-1/8*I + 1/8) - 2*sqrt(-3/64*(4*sqrt(1/8*I + 1
/8) - I + 1)^2 - 3/64*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 1/32*(4*sqrt(1/8*I + 1/8) - I + 1)*(4*sqrt(-1/8*I + 1
/8) + I - 3) + 1/2*sqrt(-1/8*I + 1/8) + 1/8*I + 1/8) - 1/2)*log(1/4*(2*(((3*x - 1)*sqrt(x + 1) + 4*x - 3)*(4*s
qrt(-1/8*I + 1/8) + I + 1) - 3*(3*x - 1)*sqrt(x + 1) - 7*x + 9)*sqrt(x + sqrt(x + 1))*(4*sqrt(1/8*I + 1/8) - I
+ 1)^2 + 2*(((3*x - 1)*sqrt(x + 1) + 4*x - 3)*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 4*((3*x - 1)*sqrt(x + 1) + 4
*x - 3)*(4*sqrt(-1/8*I + 1/8) + I + 1) + 2*(7*x + 1)*sqrt(x + 1) + 12*x - 14)*sqrt(x + sqrt(x + 1))*(4*sqrt(1/
8*I + 1/8) - I + 1) + 16*((((3*x - 1)*sqrt(x + 1) + 4*x - 3)*(4*sqrt(-1/8*I + 1/8) + I + 1) - 3*(3*x - 1)*sqrt
(x + 1) - 7*x + 9)*sqrt(x + sqrt(x + 1))*(4*sqrt(1/8*I + 1/8) - I + 1) - ((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*
(4*sqrt(-1/8*I + 1/8) + I + 1) - 2*(11*x - 7)*sqrt(x + 1) - 16*x + 22)*sqrt(x + sqrt(x + 1)))*sqrt(-3/64*(4*sq
rt(1/8*I + 1/8) - I + 1)^2 - 3/64*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 1/32*(4*sqrt(1/8*I + 1/8) - I + 1)*(4*sqr
t(-1/8*I + 1/8) + I - 3) + 1/2*sqrt(-1/8*I + 1/8) + 1/8*I + 1/8) - 2*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(4*s
qrt(-1/8*I + 1/8) + I + 1)^2 - 2*((7*x + 1)*sqrt(x + 1) + 6*x - 7)*(4*sqrt(-1/8*I + 1/8) + I + 1) - 4*sqrt(x +
1)*(x - 7) - 12*x + 44)*sqrt(x + sqrt(x + 1)) - ((3*x^2 - (x^2 + 2*(3*x + 4)*sqrt(x + 1) + 8*x + 5)*(4*sqrt(-
1/8*I + 1/8) + I + 1) + 2*(9*x + 7)*sqrt(x + 1) + 24*x + 15)*(4*sqrt(1/8*I + 1/8) - I + 1)^2 + (3*x^2 + 2*(9*x
+ 7)*sqrt(x + 1) + 24*x + 15)*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 68*x^2 - ((x^2 + 2*(3*x + 4)*sqrt(x + 1) + 8
*x + 5)*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 2*x^2 - 4*(x^2 + 2*(3*x + 4)*sqrt(x + 1) + 8*x + 5)*(4*sqrt(-1/8*I
+ 1/8) + I + 1) + 4*(7*x + 6)*sqrt(x + 1) + 24*x + 30)*(4*sqrt(1/8*I + 1/8) - I + 1) + 2*(x^2 - 2*(7*x + 6)*sq
rt(x + 1) - 12*x - 15)*(4*sqrt(-1/8*I + 1/8) + I + 1) - 8*(14*x^2 - (3*x^2 - (x^2 + 2*(3*x + 4)*sqrt(x + 1) +
8*x + 5)*(4*sqrt(-1/8*I + 1/8) + I + 1) + 2*(9*x + 7)*sqrt(x + 1) + 24*x + 15)*(4*sqrt(1/8*I + 1/8) - I + 1) -
(3*x^2 + 2*(9*x + 7)*sqrt(x + 1) + 24*x + 15)*(4*sqrt(-1/8*I + 1/8) + I + 1) + 4*(11*x + 8)*sqrt(x + 1) + 72*
x + 30)*sqrt(-3/64*(4*sqrt(1/8*I + 1/8) - I + 1)^2 - 3/64*(4*sqrt(-1/8*I + 1/8) + I + 1)^2 - 1/32*(4*sqrt(1/8*
I + 1/8) - I + 1)*(4*sqrt(-1/8*I + 1/8) + I - 3...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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