Optimal. Leaf size=33 \[ -\frac {\text {Li}_2\left (1-\frac {c+d x}{a+b x}\right )}{h (b c-a d)} \]
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Rubi [A] time = 0.10, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2502, 2315} \[ -\frac {\text {PolyLog}\left (2,1-\frac {c+d x}{a+b x}\right )}{h (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2502
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {c+d x}{a+b x}\right )}{(a+b x) ((a-c) h+(b-d) h x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\log (x)}{1-x} \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d) h}\\ &=-\frac {\text {Li}_2\left (1-\frac {c+d x}{a+b x}\right )}{(b c-a d) h}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 298, normalized size = 9.03 \[ \frac {2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )+2 \text {Li}_2\left (-\frac {b (a-c+b x-d x)}{b c-a d}\right )-2 \text {Li}_2\left (-\frac {d (-a+c-b x+d x)}{a d-b c}\right )-\log ^2\left (\frac {a d-b c}{d (a+b x)}\right )-2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log \left (\frac {a d-b c}{d (a+b x)}\right )+2 \log \left (\frac {c+d x}{a+b x}\right ) \log \left (\frac {a d-b c}{d (a+b x)}\right )+2 \log \left (\frac {(b-d) (a+b x)}{b c-a d}\right ) \log (a+b x-c-d x)-2 \log (a+b x-c-d x) \log \left (\frac {(b-d) (c+d x)}{b c-a d}\right )+2 \log (a+b x-c-d x) \log \left (\frac {c+d x}{a+b x}\right )}{h (2 b c-2 a d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 32, normalized size = 0.97 \[ -\frac {{\rm Li}_2\left (-\frac {d x + c}{b x + a} + 1\right )}{{\left (b c - a d\right )} h} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 1.27 \[ \frac {\dilog \left (\frac {d}{b}-\frac {a d -b c}{\left (b x +a \right ) b}\right )}{\left (a d -b c \right ) h} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.85, size = 357, normalized size = 10.82 \[ {\left (\frac {\log \left (-{\left (b - d\right )} x - a + c\right )}{{\left (b c - a d\right )} h} - \frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} h}\right )} \log \left (\frac {d x + c}{b x + a}\right ) + \frac {2 \, \log \left (-{\left (b - d\right )} x - a + c\right ) \log \left (b x + a\right ) - \log \left (b x + a\right )^{2}}{2 \, {\left (b c h - a d h\right )}} + \frac {\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )}{b c h - a d h} - \frac {\log \left (b x + a\right ) \log \left (-\frac {a {\left (b - d\right )} + {\left (b^{2} - b d\right )} x}{b c - a d} + 1\right ) + {\rm Li}_2\left (\frac {a {\left (b - d\right )} + {\left (b^{2} - b d\right )} x}{b c - a d}\right )}{b c h - a d h} - \frac {\log \left (-{\left (b - d\right )} x - a + c\right ) \log \left (\frac {a d - c d + {\left (b d - d^{2}\right )} x}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {a d - c d + {\left (b d - d^{2}\right )} x}{b c - a d}\right )}{b c h - a d h} \]
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 \[ \int \frac {\ln \left (\frac {c+d\,x}{a+b\,x}\right )}{\left (h\,\left (a-c\right )+h\,x\,\left (b-d\right )\right )\,\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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