Optimal. Leaf size=70 \[ \frac {2 \sqrt {x^2+1} x}{3 \sqrt {\sqrt {x^2+1}+1}}+\frac {4 x}{3 \sqrt {\sqrt {x^2+1}+1}}-4 \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {x^2+1}+1}}\right ) \]
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Rubi [F] time = 0.57, 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 {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx &=\int \left (\sqrt {1+\sqrt {1+x^2}}-\frac {2 \sqrt {1+\sqrt {1+x^2}}}{1+x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx\right )+\int \sqrt {1+\sqrt {1+x^2}} \, dx\\ &=\frac {2 x^3}{3 \left (1+\sqrt {1+x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-2 \int \left (\frac {i \sqrt {1+\sqrt {1+x^2}}}{2 (i-x)}+\frac {i \sqrt {1+\sqrt {1+x^2}}}{2 (i+x)}\right ) \, dx\\ &=\frac {2 x^3}{3 \left (1+\sqrt {1+x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-i \int \frac {\sqrt {1+\sqrt {1+x^2}}}{i-x} \, dx-i \int \frac {\sqrt {1+\sqrt {1+x^2}}}{i+x} \, dx\\ \end {align*}
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Mathematica [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.13, size = 70, normalized size = 1.00 \begin {gather*} \frac {4 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}-4 \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.57, size = 87, normalized size = 1.24 \begin {gather*} \frac {3 \, x \arctan \left (\frac {4 \, {\left (x^{4} - 12 \, x^{2} + {\left (5 \, x^{2} - 3\right )} \sqrt {x^{2} + 1} + 3\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{5} - 46 \, x^{3} + 17 \, x}\right ) + 2 \, {\left (x^{2} + \sqrt {x^{2} + 1} - 1\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 {{\left (x^{2} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-1\right ) \sqrt {1+\sqrt {x^{2}+1}}}{x^{2}+1}\, dx\]
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 {{\left (x^{2} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{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 {\left (x^2-1\right )\,\sqrt {\sqrt {x^2+1}+1}}{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 {\left (x - 1\right ) \left (x + 1\right ) \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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