Optimal. Leaf size=199 \[ \sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}}+2 \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {2}-1}}\right )-2 \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {1+\sqrt {2}}}\right )+2 \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {2}-1}}\right )-2 \sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {1+\sqrt {2}}}\right ) \]
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Rubi [A] time = 0.56, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6725, 2117, 14, 2119, 1628, 828, 826, 1166, 204, 206, 207, 203} \begin {gather*} \sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}}-\frac {2 \tan ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {2 \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}}-\frac {2 \tanh ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {2 \tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 203
Rule 204
Rule 206
Rule 207
Rule 826
Rule 828
Rule 1166
Rule 1628
Rule 2117
Rule 2119
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {2}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=2 \int \frac {1}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx+\int \frac {1}{\sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{x^{5/2}} \, dx,x,x+\sqrt {1+x^2}\right )+2 \int \left (-\frac {1}{2 (1-x) \sqrt {x+\sqrt {1+x^2}}}-\frac {1}{2 (1+x) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,x+\sqrt {1+x^2}\right )-\int \frac {1}{(1-x) \sqrt {x+\sqrt {1+x^2}}} \, dx-\int \frac {1}{(1+x) \sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\operatorname {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (1+2 x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\operatorname {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (-1+2 x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\operatorname {Subst}\left (\int \left (-\frac {1}{x^{3/2}}+\frac {2 (1+x)}{x^{3/2} \left (1+2 x-x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )-\operatorname {Subst}\left (\int \left (\frac {1}{x^{3/2}}+\frac {2 (1-x)}{x^{3/2} \left (-1+2 x+x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-2 \operatorname {Subst}\left (\int \frac {1+x}{x^{3/2} \left (1+2 x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-2 \operatorname {Subst}\left (\int \frac {1-x}{x^{3/2} \left (-1+2 x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-2 \operatorname {Subst}\left (\int \frac {-1+x}{\sqrt {x} \left (1+2 x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+2 \operatorname {Subst}\left (\int \frac {-1-x}{\sqrt {x} \left (-1+2 x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-4 \operatorname {Subst}\left (\int \frac {-1+x^2}{1+2 x^2-x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+4 \operatorname {Subst}\left (\int \frac {-1-x^2}{-1+2 x^2+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-2 \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{\sqrt {-1+\sqrt {2}}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{\sqrt {-1+\sqrt {2}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 233, normalized size = 1.17 \begin {gather*} \sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}}-\left (\left (\sqrt {2}-2\right ) \sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {1}{\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}}\right )\right )-\sqrt {2 \left (\sqrt {2}-1\right )} \left (2+\sqrt {2}\right ) \tan ^{-1}\left (\frac {1}{\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}}\right )+\left (\sqrt {2}-2\right ) \sqrt {2 \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {1}{\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}}\right )+\sqrt {2 \left (\sqrt {2}-1\right )} \left (2+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {1}{\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.29, size = 199, normalized size = 1.00 \begin {gather*} -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-2 \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+2 \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-2 \sqrt {-1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+2 \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 287, normalized size = 1.44 \begin {gather*} -\frac {2}{3} \, {\left (x^{2} - \sqrt {x^{2} + 1} x - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}} - 4 \, \sqrt {\sqrt {2} + 1} \arctan \left (\sqrt {x + \sqrt {2} + \sqrt {x^{2} + 1} - 1} \sqrt {\sqrt {2} + 1} - \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {2} + 1}\right ) + 4 \, \sqrt {\sqrt {2} - 1} \arctan \left (\sqrt {x + \sqrt {2} + \sqrt {x^{2} + 1} + 1} \sqrt {\sqrt {2} - 1} - \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {2} - 1}\right ) - \sqrt {\sqrt {2} - 1} \log \left (2 \, {\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {\sqrt {2} - 1} \log \left (-2 \, {\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {\sqrt {2} + 1} \log \left (2 \, \sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {\sqrt {2} + 1} \log \left (-2 \, \sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) \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^{2} + 1}{{\left (x^{2} - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{2}+1}{\left (x^{2}-1\right ) \sqrt {x +\sqrt {x^{2}+1}}}\, dx\]
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^{2} + 1}{{\left (x^{2} - 1\right )} \sqrt {x + \sqrt {x^{2} + 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^2+1}{\left (x^2-1\right )\,\sqrt {x+\sqrt {x^2+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^{2} + 1}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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