Optimal. Leaf size=143 \[ \frac {2 x}{3 \sqrt {\sqrt {x^2+1}+1}}+\sqrt {x^2+1} \left (\frac {2}{3} \sqrt {\sqrt {x^2+1}+1}-\frac {2 x}{3 \sqrt {\sqrt {x^2+1}+1}}\right )-\frac {4}{3} \sqrt {\sqrt {x^2+1}+1}-2 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}-\frac {\sqrt {\sqrt {x^2+1}+1}}{\sqrt {2}}\right ) \]
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Rubi [F] time = 0.21, 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-\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x-\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx &=\int \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}-\frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}}\right ) \, dx\\ &=\int \frac {x}{\sqrt {1+\sqrt {1+x^2}}} \, dx-\int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx,x,1+x^2\right )-\int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ &=-\int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx+\operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,\sqrt {1+x^2}\right )\\ &=-\int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx+\operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,\sqrt {1+x^2}\right )\\ &=-2 \sqrt {1+\sqrt {1+x^2}}+\frac {2}{3} \left (1+\sqrt {1+x^2}\right )^{3/2}-\int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.38, size = 126, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+1} \left (-6 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {\sqrt {x^2+1}}{2}\right )-4 x^2+4 \sqrt {x^2+1} x+8 \sqrt {x^2+1}-3 \sqrt {2} \sqrt {\sqrt {x^2+1}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-1}}{\sqrt {2}}\right )-8 x-2\right )}{6 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 143, normalized size = 1.00 \begin {gather*} \frac {2 x}{3 \sqrt {1+\sqrt {1+x^2}}}-\frac {4}{3} \sqrt {1+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-\frac {2 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2}{3} \sqrt {1+\sqrt {1+x^2}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}-\frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.45, size = 64, normalized size = 0.45 \begin {gather*} \frac {3 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) - 2 \, {\left (x^{2} - \sqrt {x^{2} + 1} {\left (x + 2\right )} + 2 \, x + 2\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, x} \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 - \sqrt {x^{2} + 1}}{\sqrt {\sqrt {x^{2} + 1} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x -\sqrt {x^{2}+1}}{\sqrt {1+\sqrt {x^{2}+1}}}\, dx\]
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*} \frac {2}{3} \, {\left (\sqrt {x^{2} + 1} + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} - \int \frac {\sqrt {x^{2} + 1}}{\sqrt {\sqrt {x^{2} + 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 = -0.01 \begin {gather*} \int \frac {x-\sqrt {x^2+1}}{\sqrt {\sqrt {x^2+1}+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 - \sqrt {x^{2} + 1}}{\sqrt {\sqrt {x^{2} + 1} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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