Optimal. Leaf size=29 \[ \frac {\log (x) \log \left (1+\frac {b x}{a}\right )}{b}+\frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2354, 2438}
\begin {gather*} \frac {\text {PolyLog}\left (2,-\frac {b x}{a}\right )}{b}+\frac {\log (x) \log \left (\frac {b x}{a}+1\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2354
Rule 2438
Rubi steps
\begin {align*} \int \frac {\log (x)}{a+b x} \, dx &=\frac {\log (x) \log \left (1+\frac {b x}{a}\right )}{b}-\frac {\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx}{b}\\ &=\frac {\log (x) \log \left (1+\frac {b x}{a}\right )}{b}+\frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.03 \begin {gather*} \frac {\log (x) \log \left (\frac {a+b x}{a}\right )}{b}+\frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 32, normalized size = 1.10
method | result | size |
default | \(\frac {\dilog \left (\frac {b x +a}{a}\right )}{b}+\frac {\ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{b}\) | \(32\) |
risch | \(\frac {\dilog \left (\frac {b x +a}{a}\right )}{b}+\frac {\ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{b}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 25, normalized size = 0.86 \begin {gather*} \frac {\log \left (\frac {b x}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a}\right )}{b} \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 [C] Result contains complex when optimal does not.
time = 3.45, size = 177, normalized size = 6.10 \begin {gather*} \begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {b \left (\frac {a}{b} + x\right )}{a}\right )}{b} & \text {for}\: \frac {1}{\left |{\frac {a}{b} + x}\right |} < 1 \wedge \left |{\frac {a}{b} + x}\right | < 1 \\\frac {\log {\left (\frac {a}{b} \right )} \log {\left (\frac {a}{b} + x \right )}}{b} + \frac {i \pi \log {\left (\frac {a}{b} + x \right )}}{b} - \frac {\operatorname {Li}_{2}\left (\frac {b \left (\frac {a}{b} + x\right )}{a}\right )}{b} & \text {for}\: \left |{\frac {a}{b} + x}\right | < 1 \\- \frac {\log {\left (\frac {a}{b} \right )} \log {\left (\frac {1}{\frac {a}{b} + x} \right )}}{b} - \frac {i \pi \log {\left (\frac {1}{\frac {a}{b} + x} \right )}}{b} - \frac {\operatorname {Li}_{2}\left (\frac {b \left (\frac {a}{b} + x\right )}{a}\right )}{b} & \text {for}\: \frac {1}{\left |{\frac {a}{b} + x}\right |} < 1 \\- \frac {{G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {\frac {a}{b} + x} \right )} \log {\left (\frac {a}{b} \right )}}{b} - \frac {i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{b} + \frac {{G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} \log {\left (\frac {a}{b} \right )}}{b} + \frac {i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{b} - \frac {\operatorname {Li}_{2}\left (\frac {b \left (\frac {a}{b} + x\right )}{a}\right )}{b} & \text {otherwise} \end {cases} \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 = -0.03 \begin {gather*} \int \frac {\ln \left (x\right )}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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