3.30.24 \(\int \frac {\sqrt {1+x} (-1+x^2)}{(1+x^2) \sqrt {x+\sqrt {1+x}}} \, dx\) [2924]

Optimal. Leaf size=337 \[ -\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-\frac {7}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+8 \text {RootSum}\left [625+1000 \text {$\#$1}+300 \text {$\#$1}^2+120 \text {$\#$1}^3+470 \text {$\#$1}^4+24 \text {$\#$1}^5+12 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\& ,\frac {25 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}+20 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^2+14 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3+4 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^5}{125+75 \text {$\#$1}+45 \text {$\#$1}^2+235 \text {$\#$1}^3+15 \text {$\#$1}^4+9 \text {$\#$1}^5+7 \text {$\#$1}^6+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 0.95, antiderivative size = 420, normalized size of antiderivative = 1.25, 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 = {6860, 756, 654, 635, 212, 998, 738, 210} \begin {gather*} -\frac {(1+i) \text {ArcTan}\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 {(1+i) \text {ArcTan}\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 {(1-i) \text {ArcTan}\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 {(1-i) \text {ArcTan}\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}}}+\sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}+\frac {7}{4} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[x + Sqrt[1 + x]])/2 + Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] - ((1 + 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)] - ((1 + 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)] - ((1 - 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 + S
qrt[1 + x]])])/Sqrt[(1 - I) + (1 + I)^(3/2)] - ((1 - I)*ArcTan[((2 + 2*I) - 4*Sqrt[1 + I] + 2*((2 + 2*I) + Sqr
t[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)]
+ (7*ArcTanh[(1 + 2*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])])/4

Rule 210

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

Rule 212

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

Rule 635

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

Rule 654

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

Rule 738

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

Rule 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 998

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 6860

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx &=2 \text {Subst}\left (\int \frac {x^4 \left (-2+x^2\right )}{\sqrt {-1+x+x^2} \left (2-2 x^2+x^4\right )} \, dx,x,\sqrt {1+x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {x^2}{\sqrt {-1+x+x^2}}-\frac {2 x^2}{\sqrt {-1+x+x^2} \left (2-2 x^2+x^4\right )}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-4 \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x+x^2} \left (2-2 x^2+x^4\right )} \, dx,x,\sqrt {1+x}\right )\\ &=\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-4 \text {Subst}\left (\int \left (\frac {1-i}{\sqrt {-1+x+x^2} \left ((-2-2 i)+2 x^2\right )}+\frac {1+i}{\sqrt {-1+x+x^2} \left ((-2+2 i)+2 x^2\right )}\right ) \, dx,x,\sqrt {1+x}\right )+\text {Subst}\left (\int \frac {1-\frac {3 x}{2}}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {7}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-(4-4 i) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2} \left ((-2-2 i)+2 x^2\right )} \, dx,x,\sqrt {1+x}\right )-(4+4 i) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2} \left ((-2+2 i)+2 x^2\right )} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-(2-2 i) \text {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 )-(2-2 i) \text {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+2 i) \text {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+2 i) \text {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 )+\frac {7}{2} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {7}{4} \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )-(-4-4 i) \text {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 )-(-4-4 i) \text {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+4 i) \text {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+4 i) \text {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 {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-\frac {(1+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 {(1+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 {(1-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 {(1-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 {7}{4} \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 339, normalized size = 1.01 \begin {gather*} \frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-\frac {7}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+\text {RootSum}\left [1-8 \text {$\#$1}+40 \text {$\#$1}^2-48 \text {$\#$1}^3+20 \text {$\#$1}^4+8 \text {$\#$1}^5-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+4 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1+10 \text {$\#$1}-18 \text {$\#$1}^2+10 \text {$\#$1}^3+5 \text {$\#$1}^4-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 + Roo
tSum[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]*#1^2 + 4*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^3 - Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[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 = 0.13, size = 161, normalized size = 0.48

method	result	size
derivativedivides	\(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+4 \textit {\_R}^{3}-2 \textit {\_R}^{2}+2 \textit {\_R} -1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\)	\(161\)
default	\(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+4 \textit {\_R}^{3}-2 \textit {\_R}^{2}+2 \textit {\_R} -1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\)	\(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1+x)^(1/2)*(x+(1+x)^(1/2))^(1/2)-3/2*(x+(1+x)^(1/2))^(1/2)+7/4*ln(1/2+(1+x)^(1/2)+(x+(1+x)^(1/2))^(1/2))+sum(
(2*_R^5-_R^4+4*_R^3-2*_R^2+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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 11.69, size = 5812, normalized size = 17.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*sqrt(5*sqrt(56/25*I + 8/25) + 5*sqrt(-56/25*I + 8/25) - 2*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I
+ 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I
+ 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 4)*log(-1/200*(40*(((11*x - 7)*sqrt(x + 1) + 8*x
- 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqr
t(56/25*I + 8/25) - 6*I + 2)^2 + 40*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2
- 8*((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*(3*x - 1)*sqrt(x + 1) - 240*
x - 120)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 80*((((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(
5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25
*I + 8/25) - 6*I + 2) - 10*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 20*(3*x
- 1)*sqrt(x + 1) - 80*x + 60)*sqrt(x + sqrt(x + 1)))*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*s
qrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6)
+ 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 400*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25*I + 8/25) +
6*I + 2)^2 - 4*((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 240*(3*x - 1)*sqrt(x +
1) - 560*x - 80)*sqrt(x + sqrt(x + 1)) + ((40*sqrt(5)*(11*x + 8)*sqrt(x + 1) - (2*sqrt(5)*(33*x + 19)*sqrt(x
+ 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 20*sqrt(5)*(7*x^2 + 36*x + 15))*(5*
sqrt(56/25*I + 8/25) - 6*I + 2)^2 + 20*(2*sqrt(5)*(11*x + 8)*sqrt(x + 1) + sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqr
t(-56/25*I + 8/25) + 6*I + 2)^2 - 3200*sqrt(5)*(6*x + 13)*sqrt(x + 1) - ((2*sqrt(5)*(33*x + 19)*sqrt(x + 1) +
sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 80*sqrt(5)*(x + 3)*sqrt(x + 1) - 8*(2*sq
rt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*sqrt(5)
*(13*x^2 + 24*x - 15))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 40*(2*sqrt(5)*(x + 3)*sqrt(x + 1) - sqrt(5)*(13*x^
2 + 24*x - 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 3200*sqrt(5)*(x^2 + 8*x + 5) - 2*(400*sqrt(5)*(9*x + 7)*
sqrt(x + 1) - (40*sqrt(5)*(11*x + 8)*sqrt(x + 1) - (2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x
+ 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 20*sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(56/25*I + 8/25) - 6*I +
2) - 20*(2*sqrt(5)*(11*x + 8)*sqrt(x + 1) + sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) +
600*sqrt(5)*(x^2 + 8*x + 5))*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) +
6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8
/25) + 24*I - 56))*sqrt(5*sqrt(56/25*I + 8/25) + 5*sqrt(-56/25*I + 8/25) - 2*sqrt(-3/4*(5*sqrt(56/25*I + 8/25)
- 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56
/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 4))/(x^2 + 1)) - 1/20*sqrt(5)*sqrt(5*sqrt(5
6/25*I + 8/25) + 5*sqrt(-56/25*I + 8/25) - 2*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/
25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sq
rt(-56/25*I + 8/25) + 24*I - 56) - 4)*log(-1/200*(40*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8
/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I +
2)^2 + 40*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 8*((11*x - 7)*sqrt(x +
1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*(3*x - 1)*sqrt(x + 1) - 240*x - 120)*sqrt(x + sqrt(x +
1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 80*((((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) +
6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) - 10
*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 20*(3*x - 1)*sqrt(x + 1) - 80*x +
60)*sqrt(x + sqrt(x + 1)))*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I
+ 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25
) + 24*I - 56) - 400*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 4*((3*x - 1)
*sqrt(x + 1) - 6*x - 3)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 240*(3*x - 1)*sqrt(x + 1) - 560*x - 80)*sqrt(x +
sqrt(x + 1)) - ((40*sqrt(5)*(11*x + 8)*sqrt(x + 1) - (2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 8
8*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 20*sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(56/25*I + 8/25) - 6*I
+ 2)^2 + 20*(2*sqrt(5)*(11*x + 8)*sqrt(x + 1) + sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(-56/25*I + 8/25) + 6*I +
2)^2 - 3200*sqrt(5)*(6*x + 13)*sqrt(x + 1) - (...

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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 )^{\frac {3}{2}}}{\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((1+x)**(1/2)*(x**2-1)/(x**2+1)/(x+(1+x)**(1/2))**(1/2),x)

[Out]

Integral((x - 1)*(x + 1)**(3/2)/(sqrt(x + sqrt(x + 1))*(x**2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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