Optimal. Leaf size=85 \[ \frac {2 \left (x^2+2\right ) x}{5 \sqrt {\sqrt {x^2+1}+1}}-\frac {4 \sqrt {x^2+1} x}{5 \sqrt {\sqrt {x^2+1}+1}}-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right ) \]
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Rubi [F] time = 0.28, 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^2-\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^2-\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx &=\int \left (\frac {x^2}{\sqrt {1+\sqrt {1+x^2}}}-\frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}}\right ) \, dx\\ &=\int \frac {x^2}{\sqrt {1+\sqrt {1+x^2}}} \, dx-\int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.21, size = 125, normalized size = 1.47 \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+1} \left (-10 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {\sqrt {x^2+1}}{2}\right )+4 \sqrt {x^2+1} x^2-12 x^2+16 \sqrt {x^2+1}-5 \sqrt {2} \sqrt {\sqrt {x^2+1}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-1}}{\sqrt {2}}\right )-6\right )}{10 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 85, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2+2\right ) x}{5 \sqrt {\sqrt {x^2+1}+1}}-\frac {4 \sqrt {x^2+1} x}{5 \sqrt {\sqrt {x^2+1}+1}}-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 65, normalized size = 0.76 \begin {gather*} \frac {5 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) - 2 \, {\left (3 \, x^{2} - {\left (x^{2} + 4\right )} \sqrt {x^{2} + 1} + 4\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{5 \, 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^{2} - \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 = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}-\sqrt {x^{2}+1}}{\sqrt {1+\sqrt {x^{2}+1}}}\, dx \end {gather*}
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} - \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 {\sqrt {x^2+1}-x^2}{\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^{2} - \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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