Optimal. Leaf size=15 \[ \frac {\text {Li}_2\left (-\frac {1}{a+b x}\right )}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2447} \[ \frac {\text {PolyLog}\left (2,-\frac {1}{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2447
Rubi steps
\begin {align*} \int \frac {\log \left (1+\frac {1}{a+b x}\right )}{a+b x} \, dx &=\frac {\text {Li}_2\left (-\frac {1}{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 140, normalized size = 9.33 \[ -\frac {\text {Li}_2(-a-b x)}{b}+\frac {\log ^2\left (\frac {a b-(a+1) b}{b (a+b x)}\right )}{2 b}+\frac {\log \left (\frac {b (-a-b x-1)}{(-a-1) b+a b}\right ) \log \left (\frac {a b-(a+1) b}{b (a+b x)}\right )}{b}-\frac {\log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (\frac {a b-(a+1) b}{b (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 22, normalized size = 1.47 \[ \frac {{\rm Li}_2\left (-\frac {b x + a + 1}{b x + a} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 39.25, size = 320, normalized size = 21.33 \[ \frac {1}{2} \, {\left ({\left (a + 1\right )} b - a b\right )}^{2} {\left (\frac {\log \left (\frac {{\left | b x + a + 1 \right |}}{{\left | b x + a \right |}}\right )}{b^{4}} - \frac {\log \left ({\left | \frac {b x + a + 1}{b x + a} - 1 \right |}\right )}{b^{4}} - \frac {1}{b^{4} {\left (\frac {b x + a + 1}{b x + a} - 1\right )}} - \frac {\log \left (\frac {1}{a - \frac {{\left (\frac {{\left (a - \frac {{\left (\frac {{\left (b x + a + 1\right )} a}{b x + a} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a} - b} + 1\right )} a}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} a}{b x + a} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a} - b}} - a - 1\right )} b}{\frac {{\left (a - \frac {{\left (\frac {{\left (b x + a + 1\right )} a}{b x + a} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a} - b} + 1\right )} b}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} a}{b x + a} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a} - b}} - b}} + 1\right )}{b^{4} {\left (\frac {b x + a + 1}{b x + a} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 15, normalized size = 1.00 \[ \frac {\dilog \left (1+\frac {1}{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.21, size = 61, normalized size = 4.07 \[ \frac {2 \, \log \left (b x + a + 1\right ) \log \left (b x + a\right ) - \log \left (b x + a\right )^{2}}{2 \, b} - \frac {\log \left (b x + a + 1\right ) \log \left (b x + a\right ) + {\rm Li}_2\left (-b x - a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.03, size = 15, normalized size = 1.00 \[ \frac {\mathrm {polylog}\left (2,-\frac {1}{a+b\,x}\right )}{b} \]
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 \[ \int \frac {\log {\left (1 + \frac {1}{a + b x} \right )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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