Optimal. Leaf size=90 \[ \frac {\sqrt {1-2 x^4} \left (-2 x^2-1\right )}{4 x^4}+\frac {i \log \left (\sqrt {1-2 x^4}+i \sqrt {2} x^2\right )}{\sqrt {2}}+i \tan ^{-1}\left (\sqrt {2} x^2-i \sqrt {1-2 x^4}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 0.66, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1252, 811, 844, 216, 266, 63, 206} \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\sqrt {1-2 x^4}\right )-\frac {\sin ^{-1}\left (\sqrt {2} x^2\right )}{\sqrt {2}}-\frac {\sqrt {1-2 x^4} \left (2 x^2+1\right )}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 216
Rule 266
Rule 811
Rule 844
Rule 1252
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt {1-2 x^4}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1+x) \sqrt {1-2 x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (1+2 x^2\right ) \sqrt {1-2 x^4}}{4 x^4}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {4+8 x}{x \sqrt {1-2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (1+2 x^2\right ) \sqrt {1-2 x^4}}{4 x^4}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-2 x^2}} \, dx,x,x^2\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (1+2 x^2\right ) \sqrt {1-2 x^4}}{4 x^4}-\frac {\sin ^{-1}\left (\sqrt {2} x^2\right )}{\sqrt {2}}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x} x} \, dx,x,x^4\right )\\ &=-\frac {\left (1+2 x^2\right ) \sqrt {1-2 x^4}}{4 x^4}-\frac {\sin ^{-1}\left (\sqrt {2} x^2\right )}{\sqrt {2}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}-\frac {x^2}{2}} \, dx,x,\sqrt {1-2 x^4}\right )\\ &=-\frac {\left (1+2 x^2\right ) \sqrt {1-2 x^4}}{4 x^4}-\frac {\sin ^{-1}\left (\sqrt {2} x^2\right )}{\sqrt {2}}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-2 x^4}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 0.66 \begin {gather*} \frac {1}{4} \left (2 \tanh ^{-1}\left (\sqrt {1-2 x^4}\right )-2 \sqrt {2} \sin ^{-1}\left (\sqrt {2} x^2\right )-\frac {\sqrt {1-2 x^4} \left (2 x^2+1\right )}{x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 84, normalized size = 0.93 \begin {gather*} \frac {\left (-1-2 x^2\right ) \sqrt {1-2 x^4}}{4 x^4}-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x^2}{-1+\sqrt {1-2 x^4}}\right )+\frac {\log \left (x^2\right )}{2}-\frac {1}{2} \log \left (-1+\sqrt {1-2 x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 78, normalized size = 0.87 \begin {gather*} \frac {4 \, \sqrt {2} x^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, x^{4} + 1} - \sqrt {2}}{2 \, x^{2}}\right ) - 2 \, x^{4} \log \left (\frac {\sqrt {-2 \, x^{4} + 1} - 1}{x^{2}}\right ) - \sqrt {-2 \, x^{4} + 1} {\left (2 \, x^{2} + 1\right )}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 148, normalized size = 1.64 \begin {gather*} \frac {x^{4} {\left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {-2 \, x^{4} + 1} - \sqrt {2}\right )}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {2} \sqrt {-2 \, x^{4} + 1} - \sqrt {2}\right )}^{2}} - \frac {1}{2} \, \sqrt {2} \arcsin \left (\sqrt {2} x^{2}\right ) - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {-2 \, x^{4} + 1} - \sqrt {2}\right )}}{8 \, x^{2}} + \frac {{\left (\sqrt {2} \sqrt {-2 \, x^{4} + 1} - \sqrt {2}\right )}^{2}}{32 \, x^{4}} - \frac {1}{2} \, \log \left (-\frac {\sqrt {2} \sqrt {-2 \, x^{4} + 1} - \sqrt {2}}{2 \, x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 55, normalized size = 0.61
method | result | size |
elliptic | \(-\frac {\sqrt {2}\, \arcsin \left (\sqrt {2}\, x^{2}\right )}{2}-\frac {\sqrt {-2 x^{4}+1}}{4 x^{4}}+\frac {\arctanh \left (\frac {1}{\sqrt {-2 x^{4}+1}}\right )}{2}-\frac {\sqrt {-2 x^{4}+1}}{2 x^{2}}\) | \(55\) |
risch | \(\frac {4 x^{6}+2 x^{4}-2 x^{2}-1}{4 x^{4} \sqrt {-2 x^{4}+1}}-\frac {\sqrt {2}\, \arcsin \left (\sqrt {2}\, x^{2}\right )}{2}+\frac {\arctanh \left (\frac {1}{\sqrt {-2 x^{4}+1}}\right )}{2}\) | \(58\) |
trager | \(-\frac {\left (2 x^{2}+1\right ) \sqrt {-2 x^{4}+1}}{4 x^{4}}+\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}-\sqrt {-2 x^{4}+1}\right )}{2}-\frac {\ln \left (\frac {\sqrt {-2 x^{4}+1}-1}{x^{2}}\right )}{2}\) | \(73\) |
default | \(-\frac {\left (-2 x^{4}+1\right )^{\frac {3}{2}}}{4 x^{4}}-\frac {\sqrt {-2 x^{4}+1}}{2}+\frac {\arctanh \left (\frac {1}{\sqrt {-2 x^{4}+1}}\right )}{2}-\frac {\left (-2 x^{4}+1\right )^{\frac {3}{2}}}{2 x^{2}}-x^{2} \sqrt {-2 x^{4}+1}-\frac {\sqrt {2}\, \arcsin \left (\sqrt {2}\, x^{2}\right )}{2}\) | \(80\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \left (-8 x^{4}+8\right )}{8 x^{4}}-\frac {\sqrt {\pi }\, \sqrt {-2 x^{4}+1}}{x^{4}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-2 x^{4}+1}}{2}\right )-\left (-\ln \relax (2)-1+4 \ln \relax (x )+i \pi \right ) \sqrt {\pi }-\frac {\sqrt {\pi }}{x^{4}}}{4 \sqrt {\pi }}-\frac {i \sqrt {2}\, \left (-\frac {2 i \sqrt {\pi }\, \sqrt {2}\, \sqrt {-2 x^{4}+1}}{x^{2}}-4 i \sqrt {\pi }\, \arcsin \left (\sqrt {2}\, x^{2}\right )\right )}{8 \sqrt {\pi }}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 80, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, x^{4} + 1}}{2 \, x^{2}}\right ) - \frac {\sqrt {-2 \, x^{4} + 1}}{2 \, x^{2}} - \frac {\sqrt {-2 \, x^{4} + 1}}{4 \, x^{4}} + \frac {1}{4} \, \log \left (\sqrt {-2 \, x^{4} + 1} + 1\right ) - \frac {1}{4} \, \log \left (\sqrt {-2 \, x^{4} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 71, normalized size = 0.79 \begin {gather*} -\frac {\ln \left (\sqrt {\frac {1}{2\,x^4}-1}-\sqrt {\frac {1}{2\,x^4}}\right )}{2}-\frac {\sqrt {2}\,\mathrm {asin}\left (\sqrt {2}\,x^2\right )}{2}-\frac {\sqrt {2}\,\sqrt {\frac {1}{2}-x^4}}{2\,x^2}-\frac {\sqrt {2}\,\sqrt {\frac {1}{2}-x^4}}{4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.12, size = 230, normalized size = 2.56 \begin {gather*} \begin {cases} - \frac {i x^{2}}{\sqrt {2 x^{4} - 1}} + \frac {\sqrt {2} i \operatorname {acosh}{\left (\sqrt {2} x^{2} \right )}}{2} + \frac {i}{2 x^{2} \sqrt {2 x^{4} - 1}} & \text {for}\: 2 \left |{x^{4}}\right | > 1 \\\frac {x^{2}}{\sqrt {1 - 2 x^{4}}} - \frac {\sqrt {2} \operatorname {asin}{\left (\sqrt {2} x^{2} \right )}}{2} - \frac {1}{2 x^{2} \sqrt {1 - 2 x^{4}}} & \text {otherwise} \end {cases} + \begin {cases} \frac {\operatorname {acosh}{\left (\frac {\sqrt {2}}{2 x^{2}} \right )}}{2} + \frac {\sqrt {2}}{4 x^{2} \sqrt {-1 + \frac {1}{2 x^{4}}}} - \frac {\sqrt {2}}{8 x^{6} \sqrt {-1 + \frac {1}{2 x^{4}}}} & \text {for}\: \frac {1}{2 \left |{x^{4}}\right |} > 1 \\- \frac {i \operatorname {asin}{\left (\frac {\sqrt {2}}{2 x^{2}} \right )}}{2} - \frac {\sqrt {2} i}{4 x^{2} \sqrt {1 - \frac {1}{2 x^{4}}}} + \frac {\sqrt {2} i}{8 x^{6} \sqrt {1 - \frac {1}{2 x^{4}}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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