Optimal. Leaf size=248 \[ 2 \sqrt {x+\sqrt {1+x}}+\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+32 \text {RootSum}\left [625-1000 \text {$\#$1}+300 \text {$\#$1}^2-120 \text {$\#$1}^3+470 \text {$\#$1}^4-24 \text {$\#$1}^5+12 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\& ,\frac {5 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2-2 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-125+75 \text {$\#$1}-45 \text {$\#$1}^2+235 \text {$\#$1}^3-15 \text {$\#$1}^4+9 \text {$\#$1}^5-7 \text {$\#$1}^6+\text {$\#$1}^7}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.53, antiderivative size = 375, normalized size of antiderivative = 1.51, number of steps
used = 18, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6860, 654,
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 )}{\sqrt {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 )}{\sqrt {\sqrt {1+i}-i}}+2 \sqrt {x+\sqrt {x+1}}-\tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-\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 )}{\sqrt {\sqrt {1-i}-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 )}{\sqrt {i+\sqrt {1+i}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 635
Rule 654
Rule 738
Rule 1047
Rule 6860
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx &=2 \text {Subst}\left (\int \frac {x^3 \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}{\sqrt {-1+x+x^2}}-\frac {2 x}{\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}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-4 \text {Subst}\left (\int \frac {x}{\sqrt {-1+x+x^2} \left (2-2 x^2+x^4\right )} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}-4 \text {Subst}\left (\int \left (\frac {i x}{\left ((2+2 i)-2 x^2\right ) \sqrt {-1+x+x^2}}+\frac {i x}{\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}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}-4 i \text {Subst}\left (\int \frac {x}{\left ((2+2 i)-2 x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-4 i \text {Subst}\left (\int \frac {x}{\sqrt {-1+x+x^2} \left ((-2+2 i)+2 x^2\right )} \, dx,x,\sqrt {1+x}\right )-2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}-\tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )-2 i \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 )-2 i \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 )-2 i \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 )-2 i \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 )\\ &=2 \sqrt {x+\sqrt {1+x}}-\tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )+4 i \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 )+4 i \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 )+4 i \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 )+4 i \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 )\\ &=2 \sqrt {x+\sqrt {1+x}}+\frac {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 )}{\sqrt {i+\sqrt {1-i}}}-\frac {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 )}{\sqrt {-i+\sqrt {1+i}}}-\tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )-\frac {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 )}{\sqrt {-i+\sqrt {1-i}}}+\frac {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 )}{\sqrt {i+\sqrt {1+i}}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 242, normalized size = 0.98 \begin {gather*} 2 \sqrt {x+\sqrt {1+x}}+\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+32 \text {RootSum}\left [625-1000 \text {$\#$1}+300 \text {$\#$1}^2-120 \text {$\#$1}^3+470 \text {$\#$1}^4-24 \text {$\#$1}^5+12 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {5 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^2-2 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^4}{-125+75 \text {$\#$1}-45 \text {$\#$1}^2+235 \text {$\#$1}^3-15 \text {$\#$1}^4+9 \text {$\#$1}^5-7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 0.10, size = 143, normalized size = 0.58
method | result | size |
derivativedivides | \(2 \sqrt {x +\sqrt {1+x}}-\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (4 \textit {\_R}^{4}-4 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) | \(143\) |
default | \(2 \sqrt {x +\sqrt {1+x}}-\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (4 \textit {\_R}^{4}-4 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 8.51, size = 5633, normalized size = 22.71 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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 )}{\sqrt {x + \sqrt {x + 1}} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2-1}{\sqrt {x+\sqrt {x+1}}\,\left (x^2+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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