Optimal. Leaf size=252 \[ x+\frac {4}{55} \left (5+4 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{55} \left (-5+4 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\& ,\frac {3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+7 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\& \right ] \]
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Rubi [F]
time = 0.74, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx &=2 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )^2}{-x \sqrt {1+x}+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right )\\ &=4 \text {Subst}\left (\int \frac {x^4 \left (-2+x^2\right )^2 \left (-1+x^2\right )}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \text {Subst}\left (\int \left (-x+x^3+\frac {x \left (1-2 x^2+x^4\right )}{1-x^2+4 x^3-4 x^5+x^7}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2+4 \text {Subst}\left (\int \frac {x \left (1-2 x^2+x^4\right )}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2+4 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )^2}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2+4 \text {Subst}\left (\int \left (\frac {3+2 x}{11 \left (-1-x+x^2\right )}+\frac {-3-7 x+3 x^2+4 x^3-2 x^4}{11 \left (-1+x-x^2-2 x^3+x^4+x^5\right )}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2+\frac {4}{11} \text {Subst}\left (\int \frac {3+2 x}{-1-x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {4}{11} \text {Subst}\left (\int \frac {-3-7 x+3 x^2+4 x^3-2 x^4}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2-\frac {8}{55} \log \left (2+\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}+2 \left (1+\sqrt {1+x}\right )^{3/2}-\left (1+\sqrt {1+x}\right )^2-\left (1+\sqrt {1+x}\right )^{5/2}\right )+\frac {4}{55} \text {Subst}\left (\int \frac {-13-39 x+3 x^2+28 x^3}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{55} \left (4 \left (5-4 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{55} \left (4 \left (5+4 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2+\frac {4}{55} \left (5-4 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{55} \left (5+4 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {8}{55} \log \left (2+\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}+2 \left (1+\sqrt {1+x}\right )^{3/2}-\left (1+\sqrt {1+x}\right )^2-\left (1+\sqrt {1+x}\right )^{5/2}\right )+\frac {4}{55} \text {Subst}\left (\int \left (-\frac {13}{-1+x-x^2-2 x^3+x^4+x^5}-\frac {39 x}{-1+x-x^2-2 x^3+x^4+x^5}+\frac {3 x^2}{-1+x-x^2-2 x^3+x^4+x^5}+\frac {28 x^3}{-1+x-x^2-2 x^3+x^4+x^5}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2+\frac {4}{55} \left (5-4 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{55} \left (5+4 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {8}{55} \log \left (2+\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}+2 \left (1+\sqrt {1+x}\right )^{3/2}-\left (1+\sqrt {1+x}\right )^2-\left (1+\sqrt {1+x}\right )^{5/2}\right )+\frac {12}{55} \text {Subst}\left (\int \frac {x^2}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {52}{55} \text {Subst}\left (\int \frac {1}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {112}{55} \text {Subst}\left (\int \frac {x^3}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {156}{55} \text {Subst}\left (\int \frac {x}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 252, normalized size = 1.00 \begin {gather*} x+\frac {4}{55} \left (5+4 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{55} \left (-5+4 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+7 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\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 = 0.06, size = 145, normalized size = 0.58
method | result | size |
derivativedivides | \(\left (1+\sqrt {1+x}\right )^{2}-2-2 \sqrt {1+x}+\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (-2 \textit {\_R}^{4}+4 \textit {\_R}^{3}+3 \textit {\_R}^{2}-7 \textit {\_R} -3\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}+\frac {4 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {32 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}\) | \(145\) |
default | \(\left (1+\sqrt {1+x}\right )^{2}-2-2 \sqrt {1+x}+\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (-2 \textit {\_R}^{4}+4 \textit {\_R}^{3}+3 \textit {\_R}^{2}-7 \textit {\_R} -3\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}+\frac {4 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {32 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}\) | \(145\) |
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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}{\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}-x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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