Optimal. Leaf size=35 \[ -\frac {\text {Li}_3\left (\frac {b c-a d}{d (a+b x)}+1\right )}{b c-a d} \]
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Rubi [A] time = 0.06, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6610} \[ -\frac {\text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{b c-a d} \]
Antiderivative was successfully verified.
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Rule 6610
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx &=-\frac {\text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b c-a d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.86 \[ \frac {\text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{a d-b c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (\frac {b c - a d}{b d x + a d} + 1\right )}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left (\frac {b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 1.03 \[ \frac {\polylog \left (3, -\frac {a d -b c}{\left (b x +a \right ) d}+1\right )}{a d -b c} \]
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 \[ \int \frac {{\rm Li}_2\left (\frac {b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,1-\frac {a\,d-b\,c}{d\,\left (a+b\,x\right )}\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\left (- \frac {a d}{a d + b d x} + \frac {b c}{a d + b d x} + 1\right )}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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