Optimal. Leaf size=26 \[ \frac {\sqrt {x^4-1}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt {x^4-1}}\right ) \]
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Rubi [C] time = 0.98, antiderivative size = 291, normalized size of antiderivative = 11.19, number of steps used = 32, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {6728, 277, 306, 222, 1185, 1209, 1188, 1215, 1457, 540, 253, 538, 537} \begin {gather*} \frac {\sqrt {x^4-1}}{x}+\frac {\left (1+\sqrt {5}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4-1}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4-1}}-\frac {\sqrt {2} \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{\sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{2} \left (1-\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {x^4-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 253
Rule 277
Rule 306
Rule 537
Rule 538
Rule 540
Rule 1185
Rule 1188
Rule 1209
Rule 1215
Rule 1457
Rule 6728
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx &=\int \left (-\frac {\sqrt {-1+x^4}}{x^2}+\frac {\left (1+2 x^2\right ) \sqrt {-1+x^4}}{-1+x^2+x^4}\right ) \, dx\\ &=-\int \frac {\sqrt {-1+x^4}}{x^2} \, dx+\int \frac {\left (1+2 x^2\right ) \sqrt {-1+x^4}}{-1+x^2+x^4} \, dx\\ &=\frac {\sqrt {-1+x^4}}{x}-2 \int \frac {x^2}{\sqrt {-1+x^4}} \, dx+\int \left (\frac {2 \sqrt {-1+x^4}}{1-\sqrt {5}+2 x^2}+\frac {2 \sqrt {-1+x^4}}{1+\sqrt {5}+2 x^2}\right ) \, dx\\ &=\frac {\sqrt {-1+x^4}}{x}-2 \int \frac {1}{\sqrt {-1+x^4}} \, dx+2 \int \frac {1-x^2}{\sqrt {-1+x^4}} \, dx+2 \int \frac {\sqrt {-1+x^4}}{1-\sqrt {5}+2 x^2} \, dx+2 \int \frac {\sqrt {-1+x^4}}{1+\sqrt {5}+2 x^2} \, dx\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {1}{2} \int \frac {1-\sqrt {5}-2 x^2}{\sqrt {-1+x^4}} \, dx-\frac {1}{2} \int \frac {1+\sqrt {5}-2 x^2}{\sqrt {-1+x^4}} \, dx+\left (1-\sqrt {5}\right ) \int \frac {1}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx+\left (1+\sqrt {5}\right ) \int \frac {1}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {1}{2} \left (-1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3-\sqrt {5}}+\frac {\left (2 \left (1-\sqrt {5}\right )\right ) \int \frac {1-x^2}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx}{3-\sqrt {5}}-\frac {1}{2} \left (-1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {\left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3+\sqrt {5}}+\frac {\left (2 \left (1+\sqrt {5}\right )\right ) \int \frac {1-x^2}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx}{3+\sqrt {5}}-2 \int \frac {1-x^2}{\sqrt {-1+x^4}} \, dx\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (2 \left (1-\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (2 \left (1+\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}-\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}-\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}-\frac {\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3-\sqrt {5}}-\frac {\left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3+\sqrt {5}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {1}{2} \left (1-\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {-1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.42, size = 92, normalized size = 3.54 \begin {gather*} \frac {x^4-\sqrt {1-x^4} x F\left (\left .\sin ^{-1}(x)\right |-1\right )+\sqrt {1-x^4} x \Pi \left (-\frac {2}{1+\sqrt {5}};\left .\sin ^{-1}(x)\right |-1\right )+\sqrt {1-x^4} x \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )-1}{x \sqrt {x^4-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 26, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^4}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt {-1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 40, normalized size = 1.54 \begin {gather*} \frac {x \arctan \left (\frac {2 \, \sqrt {x^{4} - 1} x}{x^{4} - x^{2} - 1}\right ) + 2 \, \sqrt {x^{4} - 1}}{2 \, 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^{4} + 1\right )} \sqrt {x^{4} - 1}}{{\left (x^{4} + x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 38, normalized size = 1.46
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}-\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right )\right ) \sqrt {2}}{2}\) | \(38\) |
trager | \(\frac {\sqrt {x^{4}-1}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 x \sqrt {x^{4}-1}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}-1}\right )}{2}\) | \(73\) |
default | \(\frac {\sqrt {x^{4}-1}}{x}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{\sqrt {x^{4}-1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-1\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}-1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 i \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , -\underline {\hspace {1.25 ex}}\alpha ^{2}-1, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{4}\) | \(146\) |
risch | \(\frac {\sqrt {x^{4}-1}}{x}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{\sqrt {x^{4}-1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-1\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}-1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 i \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , -\underline {\hspace {1.25 ex}}\alpha ^{2}-1, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{4}\) | \(146\) |
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}}{{\left (x^{4} + x^{2} - 1\right )} x^{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 = -0.04 \begin {gather*} \int \frac {\sqrt {x^4-1}\,\left (x^4+1\right )}{x^2\,\left (x^4+x^2-1\right )} \,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 )}{x^{2} \left (x^{4} + x^{2} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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