Optimal. Leaf size=155 \[ 4 \text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^3-5 \text {$\#$1}^2+3 \text {$\#$1}+1\& ,\frac {\text {$\#$1}^3 \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )+\text {$\#$1} \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )}{4 \text {$\#$1}^3+6 \text {$\#$1}^2-10 \text {$\#$1}+3}\& \right ]+x-2 \sqrt {x+\sqrt {x+1}}-\log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]
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Rubi [F] time = 1.18, 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+\sqrt {x+\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+\sqrt {x+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \left (-1+x^2\right )}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {x}{-1+x^2+\sqrt {-1+x+x^2}}+\frac {x^3}{-1+x^2+\sqrt {-1+x+x^2}}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {x}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\right )+2 \operatorname {Subst}\left (\int \frac {x^3}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \left (\frac {x \left (-1+x^2\right )}{2-x-3 x^2+x^4}-\frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )\right )+2 \operatorname {Subst}\left (\int \left (x-\frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}+\frac {x \left (-2+x+2 x^2\right )}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=x-2 \operatorname {Subst}\left (\int \frac {x \left (-1+x^2\right )}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \left (-2+x+2 x^2\right )}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=x+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+2 x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {2+4 x+4 x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=x+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{2-x-3 x^2+x^4}+\frac {2 x}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {2}{2-x-3 x^2+x^4}+\frac {4 x}{2-x-3 x^2+x^4}+\frac {4 x^2}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=x+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {1}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-\operatorname {Subst}\left (\int \frac {x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ \end {align*}
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Mathematica [B] time = 14.41, size = 4996, normalized size = 32.23 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.37, size = 160, normalized size = 1.03 \begin {gather*} 1+x-2 \sqrt {x+\sqrt {1+x}}-\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\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 + \sqrt {x + \sqrt {x + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 502, normalized size = 3.24
method | result | size |
derivativedivides | \(-\sqrt {x +\sqrt {1+x}}+\sqrt {1+x}-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}-3 \textit {\_R}^{2}-\textit {\_R} +6\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+\frac {5}{2 \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )}+\ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+1+x +2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R}^{2}+2\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )\) | \(502\) |
default | \(-\sqrt {x +\sqrt {1+x}}+\sqrt {1+x}-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}-3 \textit {\_R}^{2}-\textit {\_R} +6\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+\frac {5}{2 \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )}+\ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+1+x +2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R}^{2}+2\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )\) | \(502\) |
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 + \sqrt {x + \sqrt {x + 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 = -0.01 \begin {gather*} \int \frac {x}{x+\sqrt {x+\sqrt {x+1}}} \,d x \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 {x}{x + \sqrt {x + \sqrt {x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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