Optimal. Leaf size=262 \[ \frac {x \left (x^2+5\right )}{2 \left (x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+4 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {2 \left (7+5 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{\sqrt {2}}-\sqrt {2 \left (5 \sqrt {2}-7\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
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Rubi [F] time = 1.21, 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 {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \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 {x^2+\sqrt {1+x^4}}} \, dx &=\int \left (\frac {1}{\sqrt {x^2+\sqrt {1+x^4}}}+\frac {4}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {4}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=4 \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-4 \int \frac {1}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (4 \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 )+4 \int \left (-\frac {1}{4 (i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{4 (i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{2 \left (-1-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (2 i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )-2 i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-2 \int \frac {1}{\left (-1-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (2 i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )-2 i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-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+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-2 i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-2 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-\int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.95, size = 342, normalized size = 1.31 \begin {gather*} \frac {x \left (5+x^2\right )}{2 \left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+4 \sqrt {2} \tan ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {14+10 \sqrt {2}} \tan ^{-1}\left (\frac {-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-\sqrt {-14+10 \sqrt {2}} \tanh ^{-1}\left (\frac {-\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 6.03, size = 517, normalized size = 1.97 \begin {gather*} \frac {8 \, {\left (x^{2} + 1\right )} \sqrt {10 \, \sqrt {2} + 14} \arctan \left (-\frac {{\left (4 \, x^{2} - 2 \, \sqrt {2} {\left (x^{2} - 3\right )} - \sqrt {x^{4} + 1} {\left ({\left (\sqrt {2} - 2\right )} \sqrt {-8 \, \sqrt {2} + 12} - 2 \, \sqrt {2} + 4\right )} - {\left (2 \, x^{2} - \sqrt {2} {\left (x^{2} - 1\right )}\right )} \sqrt {-8 \, \sqrt {2} + 12} - 8\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} + 14}}{8 \, x}\right ) - 16 \, \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 ) + \sqrt {2} {\left (x^{2} + 1\right )} \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 ) - 2 \, {\left (x^{2} + 1\right )} \sqrt {10 \, \sqrt {2} - 14} \log \left (\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} + {\left (4 \, x^{3} + \sqrt {2} {\left (3 \, x^{3} + 7 \, x\right )} - \sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} + 10 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} - 14} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) + 2 \, {\left (x^{2} + 1\right )} \sqrt {10 \, \sqrt {2} - 14} \log \left (\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} - {\left (4 \, x^{3} + \sqrt {2} {\left (3 \, x^{3} + 7 \, x\right )} - \sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} + 10 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} - 14} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - 4 \, {\left (x^{5} + 5 \, x^{3} - \sqrt {x^{4} + 1} {\left (x^{3} + 5 \, x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\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^{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 = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-1\right )^{2}}{\left (x^{2}+1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]
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*} \int \frac {{\left (x^{2} - 1\right )}^{2}}{\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 {\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}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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