Optimal. Leaf size=78 \[ \frac {\log \left (x \sqrt {2 d e-b}+d+e x^2\right )}{2 \sqrt {2 d e-b}}-\frac {\log \left (-x \sqrt {2 d e-b}+d+e x^2\right )}{2 \sqrt {2 d e-b}} \]
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Rubi [A] time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1164, 628} \begin {gather*} \frac {\log \left (x \sqrt {2 d e-b}+d+e x^2\right )}{2 \sqrt {2 d e-b}}-\frac {\log \left (-x \sqrt {2 d e-b}+d+e x^2\right )}{2 \sqrt {2 d e-b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1164
Rubi steps
\begin {align*} \int \frac {d-e x^2}{d^2+b x^2+e^2 x^4} \, dx &=-\frac {\int \frac {\frac {\sqrt {-b+2 d e}}{e}+2 x}{-\frac {d}{e}-\frac {\sqrt {-b+2 d e} x}{e}-x^2} \, dx}{2 \sqrt {-b+2 d e}}-\frac {\int \frac {\frac {\sqrt {-b+2 d e}}{e}-2 x}{-\frac {d}{e}+\frac {\sqrt {-b+2 d e} x}{e}-x^2} \, dx}{2 \sqrt {-b+2 d e}}\\ &=-\frac {\log \left (d-\sqrt {-b+2 d e} x+e x^2\right )}{2 \sqrt {-b+2 d e}}+\frac {\log \left (d+\sqrt {-b+2 d e} x+e x^2\right )}{2 \sqrt {-b+2 d e}}\\ \end {align*}
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Mathematica [B] time = 0.12, size = 182, normalized size = 2.33 \begin {gather*} \frac {\frac {\left (-\sqrt {b^2-4 d^2 e^2}+b+2 d e\right ) \tan ^{-1}\left (\frac {\sqrt {2} e x}{\sqrt {b-\sqrt {b^2-4 d^2 e^2}}}\right )}{\sqrt {b-\sqrt {b^2-4 d^2 e^2}}}-\frac {\left (\sqrt {b^2-4 d^2 e^2}+b+2 d e\right ) \tan ^{-1}\left (\frac {\sqrt {2} e x}{\sqrt {\sqrt {b^2-4 d^2 e^2}+b}}\right )}{\sqrt {\sqrt {b^2-4 d^2 e^2}+b}}}{\sqrt {2} \sqrt {b^2-4 d^2 e^2}} \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+b x^2+e^2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.70, size = 172, normalized size = 2.21 \begin {gather*} \left [\frac {\log \left (\frac {e^{2} x^{4} + {\left (4 \, d e - b\right )} x^{2} + d^{2} + 2 \, {\left (e x^{3} + d x\right )} \sqrt {2 \, d e - b}}{e^{2} x^{4} + b x^{2} + d^{2}}\right )}{2 \, \sqrt {2 \, d e - b}}, -\frac {\sqrt {-2 \, d e + b} \arctan \left (\frac {\sqrt {-2 \, d e + b} e x}{2 \, d e - b}\right ) - \sqrt {-2 \, d e + b} \arctan \left (\frac {{\left (e^{2} x^{3} - {\left (d e - b\right )} x\right )} \sqrt {-2 \, d e + b}}{2 \, d^{2} e - b d}\right )}{2 \, d e - b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 1642, normalized size = 21.05
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 88, normalized size = 1.13 \begin {gather*} -\frac {\sqrt {2 d e -b}\, \ln \left (e \,x^{2}+d +\sqrt {2 d e -b}\, x \right )}{-4 d e +2 b}+\frac {\sqrt {2 d e -b}\, \ln \left (-e \,x^{2}-d +\sqrt {2 d e -b}\, x \right )}{-4 d e +2 b} \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 {e x^{2} - d}{e^{2} x^{4} + b x^{2} + d^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 99, normalized size = 1.27 \begin {gather*} \frac {\mathrm {atan}\left (\frac {b\,x\,\left (b-2\,d\,e\right )+2\,b\,e^2\,x^3+4\,d^2\,e^2\,x-e^2\,x^3\,\left (b-2\,d\,e\right )+3\,d\,e\,x\,\left (b-2\,d\,e\right )}{\left (2\,e\,d^2+b\,d\right )\,\sqrt {b-2\,d\,e}}\right )-\mathrm {atan}\left (\frac {e\,x}{\sqrt {b-2\,d\,e}}\right )}{\sqrt {b-2\,d\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 121, normalized size = 1.55 \begin {gather*} \frac {\sqrt {- \frac {1}{b - 2 d e}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (- b \sqrt {- \frac {1}{b - 2 d e}} + 2 d e \sqrt {- \frac {1}{b - 2 d e}}\right )}{e} \right )}}{2} - \frac {\sqrt {- \frac {1}{b - 2 d e}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (b \sqrt {- \frac {1}{b - 2 d e}} - 2 d e \sqrt {- \frac {1}{b - 2 d e}}\right )}{e} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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