Optimal. Leaf size=65 \[ \frac {1}{8} \tan ^{-1}\left (\frac {x^4+\frac {x^2}{2}-1}{x \sqrt {1-x^4}}\right )-\frac {1}{8} \tanh ^{-1}\left (\frac {x^4-\frac {x^2}{2}-1}{x \sqrt {1-x^4}}\right ) \]
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Rubi [C] time = 0.54, antiderivative size = 155, normalized size of antiderivative = 2.38, number of steps used = 16, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {6728, 406, 221, 409, 1213, 537} \begin {gather*} -\frac {1}{8} \left (1+i \sqrt {15}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{8} \left (1-i \sqrt {15}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \Pi \left (-\frac {2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \Pi \left (\frac {2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \Pi \left (-\frac {2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \Pi \left (\frac {2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 406
Rule 409
Rule 537
Rule 1213
Rule 6728
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{4-7 x^4+4 x^8} \, dx &=\int \left (\frac {\left (1-i \sqrt {15}\right ) \sqrt {1-x^4}}{-7-i \sqrt {15}+8 x^4}+\frac {\left (1+i \sqrt {15}\right ) \sqrt {1-x^4}}{-7+i \sqrt {15}+8 x^4}\right ) \, dx\\ &=\left (1-i \sqrt {15}\right ) \int \frac {\sqrt {1-x^4}}{-7-i \sqrt {15}+8 x^4} \, dx+\left (1+i \sqrt {15}\right ) \int \frac {\sqrt {1-x^4}}{-7+i \sqrt {15}+8 x^4} \, dx\\ &=\frac {1}{8} \left (-1-i \sqrt {15}\right ) \int \frac {1}{\sqrt {1-x^4}} \, dx+\frac {1}{4} \left (-7+i \sqrt {15}\right ) \int \frac {1}{\sqrt {1-x^4} \left (-7+i \sqrt {15}+8 x^4\right )} \, dx+\frac {1}{8} \left (-1+i \sqrt {15}\right ) \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {1}{8} \left (i+\sqrt {15}\right )^2 \int \frac {1}{\sqrt {1-x^4} \left (-7-i \sqrt {15}+8 x^4\right )} \, dx\\ &=-\frac {1}{8} \left (1-i \sqrt {15}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{8} \left (1+i \sqrt {15}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \int \frac {1}{\left (1-\frac {2 x^2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}}\right ) \sqrt {1-x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1+\frac {2 x^2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}}\right ) \sqrt {1-x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1-\frac {2 x^2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}}\right ) \sqrt {1-x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1+\frac {2 x^2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}}\right ) \sqrt {1-x^4}} \, dx\\ &=-\frac {1}{8} \left (1-i \sqrt {15}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{8} \left (1+i \sqrt {15}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {2 x^2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}}\right )} \, dx+\frac {1}{8} \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {2 x^2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}}\right )} \, dx+\frac {1}{8} \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {2 x^2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}}\right )} \, dx+\frac {1}{8} \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {2 x^2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}}\right )} \, dx\\ &=-\frac {1}{8} \left (1-i \sqrt {15}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{8} \left (1+i \sqrt {15}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \Pi \left (-\frac {2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \Pi \left (\frac {2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \Pi \left (-\frac {2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{8} \Pi \left (\frac {2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.63, size = 107, normalized size = 1.65 \begin {gather*} \frac {1}{8} \left (-2 F\left (\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (\frac {1}{\sqrt {\frac {7}{8}-\frac {i \sqrt {15}}{8}}};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (\frac {1}{\sqrt {\frac {7}{8}+\frac {i \sqrt {15}}{8}}};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (-\frac {2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (-\frac {2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}};\left .\sin ^{-1}(x)\right |-1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.26, size = 57, normalized size = 0.88 \begin {gather*} \left (\frac {1}{8}-\frac {i}{8}\right ) \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x}{\sqrt {1-x^4}}\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \tan ^{-1}\left (\frac {(1+i) \sqrt {1-x^4}}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 130, normalized size = 2.00 \begin {gather*} -\frac {1}{8} \, \arctan \left (-\frac {4 \, x^{8} - 7 \, x^{4} - 4 \, {\left (2 \, x^{5} + x^{3} - 2 \, x\right )} \sqrt {-x^{4} + 1} + 4}{4 \, x^{8} + 8 \, x^{6} - 7 \, x^{4} - 8 \, x^{2} + 4}\right ) + \frac {1}{16} \, \log \left (\frac {4 \, x^{8} - 8 \, x^{6} - 7 \, x^{4} + 8 \, x^{2} - 4 \, {\left (2 \, x^{5} - x^{3} - 2 \, x\right )} \sqrt {-x^{4} + 1} + 4}{4 \, x^{8} - 7 \, x^{4} + 4}\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^{4} + 1\right )} \sqrt {-x^{4} + 1}}{4 \, x^{8} - 7 \, x^{4} + 4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 102, normalized size = 1.57 \begin {gather*} \frac {\arctan \left (-\frac {2 \sqrt {-x^{4}+1}}{x}+1\right )}{8}-\frac {\ln \left (\frac {\frac {-x^{4}+1}{2 x^{2}}-\frac {\sqrt {-x^{4}+1}}{2 x}+\frac {1}{4}}{\frac {-x^{4}+1}{2 x^{2}}+\frac {\sqrt {-x^{4}+1}}{2 x}+\frac {1}{4}}\right )}{16}-\frac {\arctan \left (\frac {2 \sqrt {-x^{4}+1}}{x}+1\right )}{8} \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^{4} + 1\right )} \sqrt {-x^{4} + 1}}{4 \, x^{8} - 7 \, x^{4} + 4}\,{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 = -0.02 \begin {gather*} \int \frac {\sqrt {1-x^4}\,\left (x^4+1\right )}{4\,x^8-7\,x^4+4} \,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 {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}{4 x^{8} - 7 x^{4} + 4}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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