Optimal. Leaf size=38 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {2} d \sqrt {e}} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1150, 377, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {2} d \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 377
Rule 1150
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx &=\int \frac {1}{\left (d-e x^2\right ) \sqrt {d+e x^2}} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{d-2 d e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {2} d \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 38, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {2} d \sqrt {e}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 138, normalized size = 3.63 \[ \left [\frac {\sqrt {2} \log \left (\frac {17 \, e^{2} x^{4} + 14 \, d e x^{2} + 4 \, \sqrt {2} {\left (3 \, e x^{3} + d x\right )} \sqrt {e x^{2} + d} \sqrt {e} + d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right )}{8 \, d \sqrt {e}}, -\frac {\sqrt {2} \sqrt {-e} \arctan \left (\frac {\sqrt {2} {\left (3 \, e x^{2} + d\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{4 \, {\left (e^{2} x^{3} + d e x\right )}}\right )}{4 \, d e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 131, normalized size = 3.45 \[ -\frac {{\left (\sqrt {2} i \arctan \left (\frac {e^{\frac {1}{2}}}{\sqrt {-\frac {d e + \sqrt {d^{2}} e}{d}}}\right ) e^{\frac {1}{2}} - \sqrt {2} i \arctan \left (\frac {e^{\frac {1}{2}}}{\sqrt {-\frac {d e - \sqrt {d^{2}} e}{d}}}\right ) e^{\frac {1}{2}}\right )} e^{\left (-1\right )} \mathrm {sgn}\relax (x)}{4 \, {\left | d \right |}} + \frac {\sqrt {2} i \arctan \left (\frac {\sqrt {\frac {d}{x^{2}} + e}}{\sqrt {-\frac {d e \mathrm {sgn}\relax (x) + \sqrt {d^{2}} e}{d \mathrm {sgn}\relax (x)}}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, {\left | d \right |} {\left | \mathrm {sgn}\relax (x) \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 986, normalized size = 25.95 \[ -\frac {\sqrt {2}\, \sqrt {d}\, e \ln \left (\frac {4 d +2 \sqrt {2}\, \sqrt {2 d +\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}\, \sqrt {d}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}{x -\frac {\sqrt {d e}}{e}}\right )}{2 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right )}+\frac {\sqrt {2}\, \sqrt {d}\, e \ln \left (\frac {4 d +2 \sqrt {2}\, \sqrt {2 d +\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}\, \sqrt {d}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}{x +\frac {\sqrt {d e}}{e}}\right )}{2 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right )}-\frac {\sqrt {e}\, \ln \left (\frac {\left (x -\frac {\sqrt {-d e}}{e}\right ) e +\sqrt {-d e}}{\sqrt {e}}+\sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}\right )}{2 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right )}-\frac {\sqrt {e}\, \ln \left (\frac {\left (x +\frac {\sqrt {-d e}}{e}\right ) e -\sqrt {-d e}}{\sqrt {e}}+\sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}\right )}{2 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right )}+\frac {\sqrt {e}\, \ln \left (\frac {\left (x -\frac {\sqrt {d e}}{e}\right ) e +\sqrt {d e}}{\sqrt {e}}+\sqrt {2 d +\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}\right )}{2 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right )}+\frac {\sqrt {e}\, \ln \left (\frac {\left (x +\frac {\sqrt {d e}}{e}\right ) e -\sqrt {d e}}{\sqrt {e}}+\sqrt {2 d +\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}\right )}{2 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right )}-\frac {\sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}\, e}{2 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {-d e}}+\frac {\sqrt {2 d +\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}\, e}{2 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right )}+\frac {\sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}\, e}{2 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {-d e}}-\frac {\sqrt {2 d +\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}\, e}{2 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right )} \]
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 \[ -\int \frac {\sqrt {e x^{2} + d}}{e^{2} x^{4} - d^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {e\,x^2+d}}{d^2-e^2\,x^4} \,d x \]
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 \[ - \int \frac {1}{- d \sqrt {d + e x^{2}} + e x^{2} \sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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