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 = 0.01, 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 {Li}_2\left (-\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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 6.42, size = 284, normalized size = 9.79 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\frac {\text {polylog}\left [2,\frac {a+b x}{a}\right ]}{b},\text {Abs}\left [\frac {a}{b}+x\right ]<1\text {\&\&}\text {Abs}\left [\frac {b}{a+b x}\right ]<1\right \},\left \{\frac {I \text {Pi} \text {Log}\left [\frac {a}{b}+x\right ]-\text {polylog}\left [2,\frac {a+b x}{a}\right ]+\text {Log}\left [\frac {a}{b}\right ] \text {Log}\left [\frac {a}{b}+x\right ]}{b},\text {Abs}\left [\frac {a}{b}+x\right ]<1\right \},\left \{\frac {-I \text {Pi} \text {Log}\left [\frac {b}{a+b x}\right ]-\text {Log}\left [\frac {a}{b}\right ] \text {Log}\left [\frac {b}{a+b x}\right ]-\text {polylog}\left [2,\frac {a+b x}{a}\right ]}{b},\text {Abs}\left [\frac {b}{a+b x}\right ]<1\right \}\right \},-\frac {\text {Log}\left [\frac {a}{b}\right ] \text {meijerg}\left [\left \{\left \{\right \},\left \{1,1\right \}\right \},\left \{\left \{0,0\right \},\left \{\right \}\right \},\frac {a}{b}+x\right ]}{b}-\frac {\text {polylog}\left [2,\frac {b \left (\frac {a}{b}+x\right )}{a}\right ]}{b}+\frac {\text {Log}\left [\frac {a}{b}\right ] \text {meijerg}\left [\left \{\left \{1,1\right \},\left \{\right \}\right \},\left \{\left \{\right \},\left \{0,0\right \}\right \},\frac {a}{b}+x\right ]}{b}-\frac {I \text {Pi} \text {meijerg}\left [\left \{\left \{\right \},\left \{1,1\right \}\right \},\left \{\left \{0,0\right \},\left \{\right \}\right \},\frac {a}{b}+x\right ]}{b}+\frac {I \text {Pi} \text {meijerg}\left [\left \{\left \{1,1\right \},\left \{\right \}\right \},\left \{\left \{\right \},\left \{0,0\right \}\right \},\frac {a}{b}+x\right ]}{b}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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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 = 0.25, 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 = 0.65, 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 [A]
time = 3.79, 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] N/A
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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