Integrand size = 33, antiderivative size = 337 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-\frac {7}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+8 \text {RootSum}\left [625+1000 \text {$\#$1}+300 \text {$\#$1}^2+120 \text {$\#$1}^3+470 \text {$\#$1}^4+24 \text {$\#$1}^5+12 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {25 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}+20 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^2+14 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3+4 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^5}{125+75 \text {$\#$1}+45 \text {$\#$1}^2+235 \text {$\#$1}^3+15 \text {$\#$1}^4+9 \text {$\#$1}^5+7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.25, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {6860, 756, 654, 635, 212, 998, 738, 210} \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=-\frac {(1+i) \arctan \left (\frac {-2 \left ((-2+2 i)+\sqrt {1-i}\right ) \sqrt {x+1}+4 \sqrt {1-i}+(2-2 i)}{4 \sqrt {(1+i)+(1-i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1+i)+(1-i)^{3/2}}}-\frac {(1+i) \arctan \left (\frac {2 \left ((2-2 i)+\sqrt {1-i}\right ) \sqrt {x+1}-4 \sqrt {1-i}+(2-2 i)}{4 \sqrt {(1+i)-(1-i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1+i)-(1-i)^{3/2}}}-\frac {(1-i) \arctan \left (\frac {-2 \left ((-2-2 i)+\sqrt {1+i}\right ) \sqrt {x+1}+4 \sqrt {1+i}+(2+2 i)}{4 \sqrt {(1-i)+(1+i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1-i)+(1+i)^{3/2}}}-\frac {(1-i) \arctan \left (\frac {2 \left ((2+2 i)+\sqrt {1+i}\right ) \sqrt {x+1}-4 \sqrt {1+i}+(2+2 i)}{4 \sqrt {(1-i)-(1+i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1-i)-(1+i)^{3/2}}}+\frac {7}{4} \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )+\sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}} \]
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Rule 210
Rule 212
Rule 635
Rule 654
Rule 738
Rule 756
Rule 998
Rule 6860
Rubi steps \begin{align*} \text {integral}& = 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} \text {arctanh}\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) \arctan \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) \arctan \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) \arctan \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) \arctan \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} \text {arctanh}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-\frac {7}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+\text {RootSum}\left [1-8 \text {$\#$1}+40 \text {$\#$1}^2-48 \text {$\#$1}^3+20 \text {$\#$1}^4+8 \text {$\#$1}^5-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+4 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1+10 \text {$\#$1}-18 \text {$\#$1}^2+10 \text {$\#$1}^3+5 \text {$\#$1}^4-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
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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 | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+4 \textit {\_R}^{3}-2 \textit {\_R}^{2}+2 \textit {\_R} -1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) | \(161\) |
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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 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 11.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x + \sqrt {x + 1}} \left (x^{2} + 1\right )}\, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {{\left (x^{2} - 1\right )} \sqrt {x + 1}}{{\left (x^{2} + 1\right )} \sqrt {x + \sqrt {x + 1}}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {x+1}}{\sqrt {x+\sqrt {x+1}}\,\left (x^2+1\right )} \,d x \]
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