Optimal. Leaf size=223 \[ \frac {\left (x^2+1\right ) x^2+\sqrt {x^4+1} x^2}{x \left (x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
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Rubi [F] time = 3.36, 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 )^2}{\left (1+x^2\right )^2 \sqrt {1+x^4} \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 {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx &=\int \left (\frac {1}{\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}+\frac {4}{\left (1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}-\frac {4}{\left (1+x^2\right ) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=4 \int \frac {1}{\left (1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-4 \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (4 \int \left (\frac {i}{2 (i-x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}+\frac {i}{2 (i+x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\right )+4 \int \left (-\frac {1}{4 (i-x)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{4 (i+x)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{2 \left (-1-x^2\right ) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \frac {1}{\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (2 i \int \frac {1}{(i-x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )-2 i \int \frac {1}{(i+x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-2 \int \frac {1}{\left (-1-x^2\right ) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i-x)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i+x)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (2 i \int \frac {1}{(i-x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )-2 i \int \frac {1}{(i+x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-2 \int \left (-\frac {i}{2 (i-x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}-\frac {i}{2 (i+x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \frac {1}{\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i-x)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i+x)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=i \int \frac {1}{(i-x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx+i \int \frac {1}{(i+x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-2 i \int \frac {1}{(i-x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-2 i \int \frac {1}{(i+x) \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i-x)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i+x)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.09, size = 294, normalized size = 1.32 \begin {gather*} \frac {\left (x^2+1\right ) x^2+\sqrt {x^4+1} x^2}{x \left (x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {x^4+1}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {\sqrt {x^4+1}+x^2}}\right )-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {x^4+1}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}}{x \sqrt {\sqrt {x^4+1}+x^2}}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}} \sqrt {x^4+1}+\sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}} x^2-\sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}}}{x \sqrt {\sqrt {x^4+1}+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.29, size = 385, normalized size = 1.73 \begin {gather*} -\frac {4 \, {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} + 1} \arctan \left (\frac {{\left (x^{2} - {\left (x^{2} + \sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 3} + \sqrt {x^{4} + 1} {\left (\sqrt {-2 \, \sqrt {2} + 3} - 1\right )} + \sqrt {2} - 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1}}{2 \, x}\right ) + 2 \, \sqrt {2} {\left (x^{2} + 1\right )} \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 ) + {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (\frac {\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 )}}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (\frac {\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 )}}{x^{2} + 1}\right ) + 4 \, {\left (x^{3} - \sqrt {x^{4} + 1} x - x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, {\left (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 {{\left (x^{2} - 1\right )}^{2}}{\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2}-1\right )^{2}}{\left (x^{2}+1\right )^{2} \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+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 {{\left (x^{2} - 1\right )}^{2}}{\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}^{2}}\,{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 = -0.00 \begin {gather*} \int \frac {{\left (x^2-1\right )}^2}{{\left (x^2+1\right )}^2\,\sqrt {x^4+1}\,\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right )^{2} \left (x + 1\right )^{2}}{\left (x^{2} + 1\right )^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {x^{4} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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