Optimal. Leaf size=85 \[ \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b c-a d}-\frac {\text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b c-a d} \]
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Rubi [A]
time = 0.09, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2588, 6745}
\begin {gather*} \frac {\text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right ) \log \left (\frac {e (c+d x)}{a+b x}\right )}{b c-a d}-\frac {\text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{b c-a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2588
Rule 6745
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx &=\frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b c-a d}+\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 {\log \left (\frac {e (c+d x)}{a+b x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b c-a d}-\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 = 0.02, size = 68, normalized size = 0.80 \begin {gather*} \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )-\text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b c-a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs.
\(2(85)=170\).
time = 3.71, size = 186, normalized size = 2.19
method | result | size |
default | \(\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (-\frac {\frac {e \left (d x +c \right ) b}{b x +a}-e d}{e d}\right )}{2 a d -2 c b}-\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (1-\frac {b \left (d x +c \right )}{d \left (b x +a \right )}\right )}{2 \left (a d -c b \right )}-\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \polylog \left (2, \frac {b \left (d x +c \right )}{d \left (b x +a \right )}\right )}{a d -c b}+\frac {\polylog \left (3, \frac {b \left (d x +c \right )}{d \left (b x +a \right )}\right )}{a d -c b}\) | \(186\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {b \int \frac {\log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}^{2}}{a + b x}\, dx}{2 \left (a d - b c\right )} + \frac {\log {\left (\frac {a d - b c}{d \left (a + b x\right )} \right )} \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}^{2}}{2 a d - 2 b c} \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 = -0.01 \begin {gather*} \int \frac {\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\,\ln \left (\frac {a\,d-b\,c}{d\,\left (a+b\,x\right )}\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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