Optimal. Leaf size=90 \[ \frac {\log \left (\sqrt {2} \sqrt {d} \sqrt {e} x+d+e x^2\right )}{2 \sqrt {2} \sqrt {d} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt {d} \sqrt {e} x+d+e x^2\right )}{2 \sqrt {2} \sqrt {d} \sqrt {e}} \]
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Rubi [A] time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1165, 628} \begin {gather*} \frac {\log \left (\sqrt {2} \sqrt {d} \sqrt {e} x+d+e x^2\right )}{2 \sqrt {2} \sqrt {d} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt {d} \sqrt {e} x+d+e x^2\right )}{2 \sqrt {2} \sqrt {d} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1165
Rubi steps
\begin {align*} \int \frac {d-e x^2}{d^2+e^2 x^4} \, dx &=-\frac {\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt {e}}+2 x}{-\frac {d}{e}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt {e}}-x^2} \, dx}{2 \sqrt {2} \sqrt {d} \sqrt {e}}-\frac {\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt {e}}-2 x}{-\frac {d}{e}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt {e}}-x^2} \, dx}{2 \sqrt {2} \sqrt {d} \sqrt {e}}\\ &=-\frac {\log \left (d-\sqrt {2} \sqrt {d} \sqrt {e} x+e x^2\right )}{2 \sqrt {2} \sqrt {d} \sqrt {e}}+\frac {\log \left (d+\sqrt {2} \sqrt {d} \sqrt {e} x+e x^2\right )}{2 \sqrt {2} \sqrt {d} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 75, normalized size = 0.83 \begin {gather*} \frac {\log \left (\sqrt {2} \sqrt {d} \sqrt {e} x+d+e x^2\right )-\log \left (\sqrt {2} \sqrt {d} \sqrt {e} x-d-e x^2\right )}{2 \sqrt {2} \sqrt {d} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d-e x^2}{d^2+e^2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.53, size = 140, normalized size = 1.56 \begin {gather*} \left [\frac {\sqrt {2} \sqrt {d e} \log \left (\frac {e^{2} x^{4} + 4 \, d e x^{2} + 2 \, \sqrt {2} {\left (e x^{3} + d x\right )} \sqrt {d e} + d^{2}}{e^{2} x^{4} + d^{2}}\right )}{4 \, d e}, -\frac {\sqrt {2} \sqrt {-d e} \arctan \left (\frac {\sqrt {2} \sqrt {-d e} x}{2 \, d}\right ) - \sqrt {2} \sqrt {-d e} \arctan \left (\frac {\sqrt {2} {\left (e x^{3} - d x\right )} \sqrt {-d e}}{2 \, d^{2}}\right )}{2 \, d e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 222, normalized size = 2.47 \begin {gather*} \frac {\sqrt {2} {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} - {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {11}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + 2 \, x\right )} e^{\frac {1}{2}}}{2 \, {\left (d^{2}\right )}^{\frac {1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac {\sqrt {2} {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} - {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {11}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} - 2 \, x\right )} e^{\frac {1}{2}}}{2 \, {\left (d^{2}\right )}^{\frac {1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac {\sqrt {2} {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} + {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {11}{2}}\right )} e^{\left (-6\right )} \log \left (\sqrt {2} {\left (d^{2}\right )}^{\frac {1}{4}} x e^{\left (-\frac {1}{2}\right )} + x^{2} + \sqrt {d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} - \frac {\sqrt {2} {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} + {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {11}{2}}\right )} e^{\left (-6\right )} \log \left (-\sqrt {2} {\left (d^{2}\right )}^{\frac {1}{4}} x e^{\left (-\frac {1}{2}\right )} + x^{2} + \sqrt {d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 290, normalized size = 3.22 \begin {gather*} \frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )}{4 d}+\frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )}{4 d}+\frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}\right )}{8 d}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} e}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} e}-\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}\right )}{8 \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.41, size = 62, normalized size = 0.69 \begin {gather*} \frac {\sqrt {2} \log \left (e x^{2} + \sqrt {2} \sqrt {d} \sqrt {e} x + d\right )}{4 \, \sqrt {d} \sqrt {e}} - \frac {\sqrt {2} \log \left (e x^{2} - \sqrt {2} \sqrt {d} \sqrt {e} x + d\right )}{4 \, \sqrt {d} \sqrt {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 41, normalized size = 0.46 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {2\,\sqrt {2}\,\sqrt {d}\,e^{7/2}\,x}{2\,e^4\,x^2+2\,d\,e^3}\right )}{2\,\sqrt {d}\,\sqrt {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 80, normalized size = 0.89 \begin {gather*} - \frac {\sqrt {2} \sqrt {\frac {1}{d e}} \log {\left (- \sqrt {2} d x \sqrt {\frac {1}{d e}} + \frac {d}{e} + x^{2} \right )}}{4} + \frac {\sqrt {2} \sqrt {\frac {1}{d e}} \log {\left (\sqrt {2} d x \sqrt {\frac {1}{d e}} + \frac {d}{e} + x^{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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