Optimal. Leaf size=53 \[ -\frac {\text {Li}_2\left (1+\frac {2 \sqrt {e} x \left (\sqrt {-d}-\sqrt {e} x\right )}{d+e x^2}\right )}{2 \sqrt {-d} \sqrt {e}} \]
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Rubi [A]
time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2497}
\begin {gather*} -\frac {\text {PolyLog}\left (2,\frac {2 \sqrt {e} x \left (\sqrt {-d}-\sqrt {e} x\right )}{d+e x^2}+1\right )}{2 \sqrt {-d} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2497
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {2 x \left (\frac {d \sqrt {e}}{\sqrt {-d}}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx &=-\frac {\text {Li}_2\left (1+\frac {2 \sqrt {e} x \left (\sqrt {-d}-\sqrt {e} x\right )}{d+e x^2}\right )}{2 \sqrt {-d} \sqrt {e}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(320\) vs. \(2(53)=106\).
time = 0.15, size = 320, normalized size = 6.04 \begin {gather*} \frac {-2 \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )-\log ^2\left (\sqrt {-d}+\sqrt {e} x\right )+2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right )+2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {2 \left (-\sqrt {-d} \sqrt {e} x+e x^2\right )}{d+e x^2}\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {2 \left (-\sqrt {-d} \sqrt {e} x+e x^2\right )}{d+e x^2}\right )+2 \text {Li}_2\left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+2 \text {Li}_2\left (\frac {d+\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \text {Li}_2\left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{4 \sqrt {-d} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (\frac {2 x \left (e x +\frac {d \sqrt {e}}{\sqrt {-d}}\right )}{e \,x^{2}+d}\right )}{e \,x^{2}+d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 44, normalized size = 0.83 \begin {gather*} \frac {\sqrt {-d} {\rm Li}_2\left (-\frac {2 \, {\left (x^{2} e - \sqrt {-d} x e^{\frac {1}{2}}\right )}}{x^{2} e + d} + 1\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\ln \left (\frac {2\,x\,\left (e\,x-\sqrt {-d}\,\sqrt {e}\right )}{e\,x^2+d}\right )}{e\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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