Optimal. Leaf size=61 \[ \frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d^2 \sqrt {e}} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1150, 382, 377, 208} \begin {gather*} \frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d^2 \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 377
Rule 382
Rule 1150
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx &=\int \frac {1}{\left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\int \frac {1}{\left (d-e x^2\right ) \sqrt {d+e x^2}} \, dx}{2 d}\\ &=\frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{d-2 d e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{2 d}\\ &=\frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d^2 \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 108, normalized size = 1.77 \begin {gather*} \frac {\frac {4 x}{\sqrt {d+e x^2}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {d}-\sqrt {e} x}{\sqrt {2} \sqrt {d+e x^2}}\right )}{\sqrt {e}}+\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {e} x}{\sqrt {2} \sqrt {d+e x^2}}\right )}{\sqrt {e}}}{8 d^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 84, normalized size = 1.38 \begin {gather*} \frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (-\frac {e x^2}{\sqrt {2} d}+\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {2} d}+\frac {1}{\sqrt {2}}\right )}{2 \sqrt {2} d^2 \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 209, normalized size = 3.43 \begin {gather*} \left [\frac {\sqrt {2} {\left (e x^{2} + d\right )} \sqrt {e} \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 \, \sqrt {e x^{2} + d} e x}{16 \, {\left (d^{2} e^{2} x^{2} + d^{3} e\right )}}, -\frac {\sqrt {2} {\left (e x^{2} + d\right )} \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 \, \sqrt {e x^{2} + d} e x}{8 \, {\left (d^{2} e^{2} x^{2} + d^{3} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 1, normalized size = 0.02 \begin {gather*} +\infty \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 441, normalized size = 7.23 \begin {gather*} -\frac {\sqrt {2}\, 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 )}{4 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {d}}+\frac {\sqrt {2}\, 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 )}{4 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {d}}-\frac {\sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{2 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \left (x -\frac {\sqrt {-d e}}{e}\right ) d}-\frac {\sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{2 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \left (x +\frac {\sqrt {-d e}}{e}\right ) d} \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 {1}{{\left (e^{2} x^{4} - d^{2}\right )} \sqrt {e x^{2} + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\left (d^2-e^2\,x^4\right )\,\sqrt {e\,x^2+d}} \,d x \end {gather*}
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 \begin {gather*} - \int \frac {1}{- d^{2} \sqrt {d + e x^{2}} + e^{2} x^{4} \sqrt {d + e x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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