∫ ∘ _______ x+√ ___ 1+x x2−√ __

Optimal. Leaf size=204 \[ 2 \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+16 \text {$\#$1}^5-8 \text {$\#$1}^4-2 \text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}-1\& ,\frac {\text {$\#$1}^6 \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )-2 \text {$\#$1}^5 \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )+2 \text {$\#$1} \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )+\log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )}{4 \text {$\#$1}^7-12 \text {$\#$1}^5+40 \text {$\#$1}^4-16 \text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+3}\& \right ] \]

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Rubi [F] time = 0.27, 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 {\sqrt {x+\sqrt {1+x}}}{x^2-\sqrt {1+x}} \, dx \end {gather*}

\begin {align*} \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2-\sqrt {1+x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{-x+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{1-x-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ \end {align*}

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Mathematica [B] time = 7.23, size = 4139, normalized size = 20.29 \begin {gather*} \text {Result too large to show} \end {gather*}

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IntegrateAlgebraic [A] time = 0.00, size = 212, normalized size = 1.04 \begin {gather*} 2 \text {RootSum}\left [-1+6 \text {$\#$1}+3 \text {$\#$1}^2-2 \text {$\#$1}^3-8 \text {$\#$1}^4+16 \text {$\#$1}^5-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^5+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{3+3 \text {$\#$1}-3 \text {$\#$1}^2-16 \text {$\#$1}^3+40 \text {$\#$1}^4-12 \text {$\#$1}^5+4 \text {$\#$1}^7}\&\right ] \end {gather*}

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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + \sqrt {x + 1}}}{x^{2} - \sqrt {x + 1}}\,{d x} \end {gather*}

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method	result	size

derivativedivides	$2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+16 \textit {\_Z}^{5}-8 \textit {\_Z}^{4}-2 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}+6 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{5}+2 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-12 \textit {\_R}^{5}+40 \textit {\_R}^{4}-16 \textit {\_R}^{3}-3 \textit {\_R}^{2}+3 \textit {\_R} +3}\right )$	$107$
default	$2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+16 \textit {\_Z}^{5}-8 \textit {\_Z}^{4}-2 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}+6 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{5}+2 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-12 \textit {\_R}^{5}+40 \textit {\_R}^{4}-16 \textit {\_R}^{3}-3 \textit {\_R}^{2}+3 \textit {\_R} +3}\right )$	$107$

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + \sqrt {x + 1}}}{x^{2} - \sqrt {x + 1}}\,{d x} \end {gather*}

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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\sqrt {x+\sqrt {x+1}}}{\sqrt {x+1}-x^2} \,d x \end {gather*}

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + \sqrt {x + 1}}}{x^{2} - \sqrt {x + 1}}\, dx \end {gather*}

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3.25.87 \(\int \frac {\sqrt {x+\sqrt {1+x}}}{x^2-\sqrt {1+x}} \, dx\)