Integrand size = 33, antiderivative size = 337 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-\frac {7}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+8 \text {RootSum}\left [625+1000 \text {$\#$1}+300 \text {$\#$1}^2+120 \text {$\#$1}^3+470 \text {$\#$1}^4+24 \text {$\#$1}^5+12 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {25 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}+20 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^2+14 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3+4 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^5}{125+75 \text {$\#$1}+45 \text {$\#$1}^2+235 \text {$\#$1}^3+15 \text {$\#$1}^4+9 \text {$\#$1}^5+7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-\frac {7}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+\text {RootSum}\left [1-8 \text {$\#$1}+40 \text {$\#$1}^2-48 \text {$\#$1}^3+20 \text {$\#$1}^4+8 \text {$\#$1}^5-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+4 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1+10 \text {$\#$1}-18 \text {$\#$1}^2+10 \text {$\#$1}^3+5 \text {$\#$1}^4-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
(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 + 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[-S qrt[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) & ]
Result contains complex when optimal does not.
Time = 1.51 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.31, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2003, 7267, 25, 7292, 7279, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {x+1} \left (x^2-1\right )}{\left (x^2+1\right ) \sqrt {x+\sqrt {x+1}}} \, dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {(x-1) (x+1)^{3/2}}{\left (x^2+1\right ) \sqrt {x+\sqrt {x+1}}}dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int -\frac {(1-x) (x+1)^2}{\left (x^2+1\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {(1-x) (x+1)^2}{\left (x^2+1\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -2 \int \frac {(1-x) (x+1)^2}{\sqrt {x+\sqrt {x+1}} \left ((x+1)^2-2 (x+1)+2\right )}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -2 \int \left (\frac {2 (x+1)}{\sqrt {x+\sqrt {x+1}} \left ((x+1)^2-2 (x+1)+2\right )}-\frac {x+1}{\sqrt {x+\sqrt {x+1}}}\right )d\sqrt {x+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \arctan \left (\frac {-2 \left ((-2+2 i)+\sqrt {1-i}\right ) \sqrt {x+1}+4 \sqrt {1-i}+(2-2 i)}{4 \sqrt {(1+i)+(1-i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1+i)+(1-i)^{3/2}}}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \arctan \left (\frac {2 \left ((2-2 i)+\sqrt {1-i}\right ) \sqrt {x+1}-4 \sqrt {1-i}+(2-2 i)}{4 \sqrt {(1+i)-(1-i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1+i)-(1-i)^{3/2}}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \arctan \left (\frac {-2 \left ((-2-2 i)+\sqrt {1+i}\right ) \sqrt {x+1}+4 \sqrt {1+i}+(2+2 i)}{4 \sqrt {(1-i)+(1+i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1-i)+(1+i)^{3/2}}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \arctan \left (\frac {2 \left ((2+2 i)+\sqrt {1+i}\right ) \sqrt {x+1}-4 \sqrt {1+i}+(2+2 i)}{4 \sqrt {(1-i)-(1+i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1-i)-(1+i)^{3/2}}}+\frac {7}{8} \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )+\frac {1}{2} \sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{4} \sqrt {x+\sqrt {x+1}}\right )\) |
2*((-3*Sqrt[x + Sqrt[1 + x]])/4 + (Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/2 - ((1/2 + I/2)*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)] - ((1/2 + I/2)*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)] - ((1/2 - I /2)*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)] - ((1/2 - I/2)*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)]*S qrt[x + Sqrt[1 + x]])])/Sqrt[(1 - I) - (1 + I)^(3/2)] + (7*ArcTanh[(1 + 2* Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])])/8)
3.30.25.3.1 Defintions of rubi rules used
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
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] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.16 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.48
| method | result | size |
| derivativedivides | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+4 \textit {\_R}^{3}-2 \textit {\_R}^{2}+2 \textit {\_R} -1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) | \(161\) |
| default | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+4 \textit {\_R}^{3}-2 \textit {\_R}^{2}+2 \textit {\_R} -1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) | \(161\) |
(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))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 9.87 (sec) , antiderivative size = 5812, normalized size of antiderivative = 17.25 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\text {Too large to display} \]
Not integrable
Time = 11.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x + \sqrt {x + 1}} \left (x^{2} + 1\right )}\, dx \]
Not integrable
Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {{\left (x^{2} - 1\right )} \sqrt {x + 1}}{{\left (x^{2} + 1\right )} \sqrt {x + \sqrt {x + 1}}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument ValueInvalid _EXT in replace_ext Error: Bad Argument ValueInv alid _EXT
Not integrable
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {x+1}}{\sqrt {x+\sqrt {x+1}}\,\left (x^2+1\right )} \,d x \]