Optimal. Leaf size=50 \[ \frac {\sqrt {-\frac {e}{d}} \text {Li}_2\left (\frac {2 x \left (d \sqrt {-\frac {e}{d}}-e x\right )}{e x^2+d}+1\right )}{2 e} \]
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Rubi [A] time = 0.09, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2447} \[ \frac {\sqrt {-\frac {e}{d}} \text {PolyLog}\left (2,\frac {2 x \left (d \sqrt {-\frac {e}{d}}-e x\right )}{d+e x^2}+1\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 2447
Rubi steps
\begin {align*} \int \frac {\log \left (-\frac {2 x \left (d \sqrt {-\frac {e}{d}}-e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx &=\frac {\sqrt {-\frac {e}{d}} \text {Li}_2\left (1+\frac {2 x \left (d \sqrt {-\frac {e}{d}}-e x\right )}{d+e x^2}\right )}{2 e}\\ \end {align*}
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Mathematica [B] time = 0.38, size = 642, normalized size = 12.84 \[ \frac {-2 \text {Li}_2\left (\frac {\sqrt {-\frac {e}{d}} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e}+\sqrt {-d} \sqrt {-\frac {e}{d}}}\right )+2 \text {Li}_2\left (\frac {\sqrt {-\frac {e}{d}} \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {-d} \sqrt {-\frac {e}{d}}-\sqrt {e}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )-2 \text {Li}_2\left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right )+2 \text {Li}_2\left (\frac {d+\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right )+2 \log \left (\frac {2 e x \left (\frac {1}{\sqrt {-\frac {e}{d}}}+x\right )}{d+e x^2}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {2 e x \left (\frac {1}{\sqrt {-\frac {e}{d}}}+x\right )}{d+e x^2}\right )+\log ^2\left (\sqrt {-d}-\sqrt {e} x\right )-\log ^2\left (\sqrt {-d}+\sqrt {e} x\right )-2 \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-2 \log \left (\frac {\sqrt {e} \left (x \sqrt {-\frac {e}{d}}+1\right )}{\sqrt {-d} \sqrt {-\frac {e}{d}}+\sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {-d} \sqrt {e} x+d}{2 d}\right )+2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (x \sqrt {-\frac {e}{d}}+1\right )}{\sqrt {e}-\sqrt {-d} \sqrt {-\frac {e}{d}}}\right )}{4 \sqrt {-d} \sqrt {e}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 45, normalized size = 0.90 \[ \frac {\sqrt {-\frac {e}{d}} {\rm Li}_2\left (-\frac {2 \, {\left (e x^{2} - d x \sqrt {-\frac {e}{d}}\right )}}{e x^{2} + d} + 1\right )}{2 \, e} \]
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 \[ \int \frac {\log \left (\frac {2 \, {\left (e x - d \sqrt {-\frac {e}{d}}\right )} x}{e x^{2} + d}\right )}{e x^{2} + d}\,{d x} \]
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 \[ \int \frac {\ln \left (-\frac {2 \left (-e x +\sqrt {-\frac {e}{d}}\, d \right ) x}{e \,x^{2}+d}\right )}{e \,x^{2}+d}\, dx \]
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 \[ \int \frac {\ln \left (\frac {2\,x\,\left (e\,x-d\,\sqrt {-\frac {e}{d}}\right )}{e\,x^2+d}\right )}{e\,x^2+d} \,d x \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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