Optimal. Leaf size=51 \[ \frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 c d}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c d} \]
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Rubi [A] time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5918, 2402, 2315} \[ \frac {b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 c d}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c d} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 5918
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{d+c d x} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c d}+\frac {b \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c d}+\frac {b \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{c d}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c d}+\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 52, normalized size = 1.02 \[ \frac {2 a \log (c x+1)+b \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )-2 b \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )}{2 c d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x\right ) + a}{c d x + d}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c x\right ) + a}{c d x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 112, normalized size = 2.20 \[ \frac {a \ln \left (c x +1\right )}{c d}+\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{c d}-\frac {b \ln \left (c x +1\right )^{2}}{4 c d}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 c d}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 c d}-\frac {b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 c d} \]
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 \[ \frac {1}{2} \, {\left (2 \, c \int \frac {x \log \left (c x + 1\right )}{c^{2} d x^{2} - d}\,{d x} - \frac {\log \left (c x + 1\right ) \log \left (-c x + 1\right )}{c d}\right )} b + \frac {a \log \left (c d x + d\right )}{c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{d+c\,d\,x} \,d x \]
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 \[ \frac {\int \frac {a}{c x + 1}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c x + 1}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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