Optimal. Leaf size=245 \[ \frac {4}{11} \text {RootSum}\left [\text {$\#$1}^5+\text {$\#$1}^4-2 \text {$\#$1}^3-\text {$\#$1}^2+\text {$\#$1}-1\& ,\frac {8 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )+6 \text {$\#$1}^3 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-12 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-5 \text {$\#$1} \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )+\log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )}{5 \text {$\#$1}^4+4 \text {$\#$1}^3-6 \text {$\#$1}^2-2 \text {$\#$1}+1}\& \right ]+\frac {2}{55} \left (15+\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right )-\frac {2}{55} \left (\sqrt {5}-15\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}-1\right ) \]
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Rubi [F] time = 0.76, 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}{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}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \left (-1+x^2\right )}{-x \sqrt {1+x}+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x^2 \left (2-3 x^2+x^4\right )}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{7} \log \left (\sqrt {1+x}-4 \left (1+\sqrt {1+x}\right )^{3/2}+4 \left (1+\sqrt {1+x}\right )^{5/2}-\left (1+\sqrt {1+x}\right )^{7/2}\right )+\frac {4}{7} \operatorname {Subst}\left (\int \frac {2 x+2 x^2-x^4}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{7} \log \left (\sqrt {1+x}-4 \left (1+\sqrt {1+x}\right )^{3/2}+4 \left (1+\sqrt {1+x}\right )^{5/2}-\left (1+\sqrt {1+x}\right )^{7/2}\right )+\frac {4}{7} \operatorname {Subst}\left (\int \frac {x \left (2+2 x-x^3\right )}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{7} \log \left (\sqrt {1+x}-4 \left (1+\sqrt {1+x}\right )^{3/2}+4 \left (1+\sqrt {1+x}\right )^{5/2}-\left (1+\sqrt {1+x}\right )^{7/2}\right )+\frac {4}{7} \operatorname {Subst}\left (\int \left (\frac {4-x}{11 \left (-1-x+x^2\right )}+\frac {-4-13 x-18 x^2-2 x^3+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 )\\ &=\frac {4}{7} \log \left (\sqrt {1+x}-4 \left (1+\sqrt {1+x}\right )^{3/2}+4 \left (1+\sqrt {1+x}\right )^{5/2}-\left (1+\sqrt {1+x}\right )^{7/2}\right )+\frac {4}{77} \operatorname {Subst}\left (\int \frac {4-x}{-1-x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {4}{77} \operatorname {Subst}\left (\int \frac {-4-13 x-18 x^2-2 x^3+x^4}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{385} \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}{7} \log \left (\sqrt {1+x}-4 \left (1+\sqrt {1+x}\right )^{3/2}+4 \left (1+\sqrt {1+x}\right )^{5/2}-\left (1+\sqrt {1+x}\right )^{7/2}\right )+\frac {4}{385} \operatorname {Subst}\left (\int \frac {-21-63 x-84 x^2-14 x^3}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {1}{385} \left (2 \left (5-7 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {1}{385} \left (2 \left (5+7 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-\frac {2}{385} \left (5+7 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{385} \left (5-7 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{385} \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}{7} \log \left (\sqrt {1+x}-4 \left (1+\sqrt {1+x}\right )^{3/2}+4 \left (1+\sqrt {1+x}\right )^{5/2}-\left (1+\sqrt {1+x}\right )^{7/2}\right )+\frac {4}{385} \operatorname {Subst}\left (\int \left (-\frac {21}{-1+x-x^2-2 x^3+x^4+x^5}-\frac {63 x}{-1+x-x^2-2 x^3+x^4+x^5}-\frac {84 x^2}{-1+x-x^2-2 x^3+x^4+x^5}-\frac {14 x^3}{-1+x-x^2-2 x^3+x^4+x^5}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-\frac {2}{385} \left (5+7 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{385} \left (5-7 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{385} \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}{7} \log \left (\sqrt {1+x}-4 \left (1+\sqrt {1+x}\right )^{3/2}+4 \left (1+\sqrt {1+x}\right )^{5/2}-\left (1+\sqrt {1+x}\right )^{7/2}\right )-\frac {8}{55} \operatorname {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 {12}{55} \operatorname {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 {36}{55} \operatorname {Subst}\left (\int \frac {x}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {48}{55} \operatorname {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 )\\ \end {align*}
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Mathematica [B] time = 3.74, size = 1881, normalized size = 7.68
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.00, size = 245, normalized size = 1.00 \begin {gather*} \frac {2}{55} \left (15+\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{55} \left (-15+\sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{11} \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )-5 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-12 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+6 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+8 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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*} \int \frac {x}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 128, normalized size = 0.52
method | result | size |
derivativedivides | \(\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (8 \textit {\_R}^{4}+6 \textit {\_R}^{3}-12 \textit {\_R}^{2}-5 \textit {\_R} +1\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}+\frac {6 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {4 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}\) | \(128\) |
default | \(\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (8 \textit {\_R}^{4}+6 \textit {\_R}^{3}-12 \textit {\_R}^{2}-5 \textit {\_R} +1\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}+\frac {6 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {4 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}\) | \(128\) |
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*} \int \frac {x}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}\,{d x} \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}{\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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sympy [F(-1)] 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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