Optimal. Leaf size=24 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1150, 208} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 1150
Rubi steps
\begin {align*} \int \frac {d+e x^2}{d^2-e^2 x^4} \, dx &=\int \frac {1}{d-e x^2} \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\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 = 1.09, size = 68, normalized size = 2.83 \begin {gather*} \left [\frac {\sqrt {d e} \log \left (\frac {e x^{2} + 2 \, \sqrt {d e} x + d}{e x^{2} - d}\right )}{2 \, d e}, -\frac {\sqrt {-d e} \arctan \left (\frac {\sqrt {-d e} x}{d}\right )}{d e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 116, normalized size = 4.83 \begin {gather*} \frac {{\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {7}{2}} - {\left (d^{2}\right )}^{\frac {1}{4}} {\left | d \right |} e^{\frac {7}{2}}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{{\left (d^{2}\right )}^{\frac {1}{4}}}\right ) e^{\left (-4\right )}}{2 \, d^{2}} + \frac {{\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 ({\left | {\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right )}{4 \, d^{2}} - \frac {{\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {7}{2}} + {\left (d^{2}\right )}^{\frac {1}{4}} {\left | d \right |} e^{\frac {7}{2}}\right )} e^{\left (-4\right )} \log \left ({\left | -{\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right )}{4 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.67 \begin {gather*} \frac {\arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.35, size = 31, normalized size = 1.29 \begin {gather*} -\frac {\log \left (\frac {e x - \sqrt {d e}}{e x + \sqrt {d e}}\right )}{2 \, \sqrt {d e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 16, normalized size = 0.67 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{\sqrt {d}\,\sqrt {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 46, normalized size = 1.92 \begin {gather*} - \frac {\sqrt {\frac {1}{d e}} \log {\left (- d \sqrt {\frac {1}{d e}} + x \right )}}{2} + \frac {\sqrt {\frac {1}{d e}} \log {\left (d \sqrt {\frac {1}{d e}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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