Optimal. Leaf size=50 \[ -\frac {\text {Li}_2\left (\frac {2 x \left (\sqrt {d} \sqrt {-e}-e x\right )}{e x^2+d}+1\right )}{2 \sqrt {d} \sqrt {-e}} \]
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Rubi [A] time = 0.08, antiderivative size = 50, 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 = {2447} \[ -\frac {\text {PolyLog}\left (2,\frac {2 x \left (\sqrt {d} \sqrt {-e}-e x\right )}{d+e x^2}+1\right )}{2 \sqrt {d} \sqrt {-e}} \]
Antiderivative was successfully verified.
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Rule 2447
Rubi steps
\begin {align*} \int \frac {\log \left (-\frac {2 x \left (\sqrt {d} \sqrt {-e}-e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx &=-\frac {\text {Li}_2\left (1+\frac {2 x \left (\sqrt {d} \sqrt {-e}-e x\right )}{d+e x^2}\right )}{2 \sqrt {d} \sqrt {-e}}\\ \end {align*}
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Mathematica [B] time = 0.26, size = 645, normalized size = 12.90 \[ \frac {2 \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )-2 \text {Li}_2\left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right )+2 \text {Li}_2\left (\frac {d+\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right )-2 \text {Li}_2\left (\frac {e x-\sqrt {-d} \sqrt {e}}{\sqrt {d} \sqrt {-e}-\sqrt {-d} \sqrt {e}}\right )+2 \text {Li}_2\left (\frac {e x+\sqrt {-d} \sqrt {e}}{\sqrt {d} \sqrt {-e}+\sqrt {-d} \sqrt {e}}\right )+2 \log \left (\frac {2 x \left (e x-\sqrt {d} \sqrt {-e}\right )}{d+e x^2}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {2 x \left (e x-\sqrt {d} \sqrt {-e}\right )}{d+e x^2}\right )+\log ^2\left (\sqrt {-d}-\sqrt {e} x\right )-\log ^2\left (\sqrt {-d}+\sqrt {e} x\right )-2 \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-2 \log \left (\frac {\sqrt {d} \sqrt {-e}-e x}{\sqrt {d} \sqrt {-e}-\sqrt {-d} \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {-d} \sqrt {e} x+d}{2 d}\right )+2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {d} \sqrt {-e}-e x}{\sqrt {d} \sqrt {-e}+\sqrt {-d} \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 44, normalized size = 0.88 \[ \frac {\sqrt {-e} {\rm Li}_2\left (-\frac {2 \, {\left (e x^{2} - \sqrt {d} \sqrt {-e} x\right )}}{e x^{2} + d} + 1\right )}{2 \, \sqrt {d} e} \]
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 \[ \int \frac {\log \left (\frac {2 \, {\left (e x - \sqrt {d} \sqrt {-e}\right )} x}{e x^{2} + d}\right )}{e x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (-\frac {2 \left (-e x +\sqrt {-e}\, \sqrt {d}\right ) x}{e \,x^{2}+d}\right )}{e \,x^{2}+d}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {\ln \left (\frac {2\,x\,\left (e\,x-\sqrt {d}\,\sqrt {-e}\right )}{e\,x^2+d}\right )}{e\,x^2+d} \,d x \]
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 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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