Optimal. Leaf size=242 \[ \frac {x}{2 \sqrt {x^2+\sqrt {1+x^4}}}+2 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-2 \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-2 \sqrt {-1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]
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Rubi [F]
time = 0.38, 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 {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx &=\int \left (\frac {1}{\sqrt {x^2+\sqrt {1+x^4}}}-\frac {2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (2 \int \left (\frac {i}{2 (i-x) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {i}{2 (i+x) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\right )+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )-i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.79, size = 242, normalized size = 1.00 \begin {gather*} \frac {x}{2 \sqrt {x^2+\sqrt {1+x^4}}}+2 \sqrt {2} \text {ArcTan}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-2 \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-2 \sqrt {-1+\sqrt {2}} \tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{2}-1}{\left (x^{2}+1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 420 vs.
\(2 (189) = 378\).
time = 2.81, size = 420, normalized size = 1.74 \begin {gather*} -\frac {1}{2} \, {\left (x^{3} - \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - 2 \, \sqrt {\sqrt {2} + 1} \arctan \left (\frac {{\left (2 \, x^{2} - {\left (x^{2} + \sqrt {2} + 1\right )} \sqrt {-8 \, \sqrt {2} + 12} + \sqrt {x^{4} + 1} {\left (\sqrt {-8 \, \sqrt {2} + 12} - 2\right )} + 2 \, \sqrt {2} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1}}{4 \, x}\right ) - \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} {\left (x^{3} + 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}\right )}}{x^{2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} {\left (x^{3} + 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}\right )}}{x^{2} + 1}\right ) \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 )}{\left (x^{2} + 1\right ) \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\, dx \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*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {\sqrt {x^4+1}+x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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