Optimal. Leaf size=120 \[ -\frac {\tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\tanh ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}+\frac {\text {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{2 d}-\frac {\text {PolyLog}\left (2,1-\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{2 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6246, 6057,
2449, 2352, 2497} \begin {gather*} -\frac {\text {Li}_2\left (1-\frac {2 b (c+d x)}{(b c-a d+d) (a+b x+1)}\right )}{2 d}+\frac {\tanh ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(a+b x+1) (-a d+b c+d)}\right )}{d}+\frac {\text {Li}_2\left (1-\frac {2}{a+b x+1}\right )}{2 d}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \tanh ^{-1}(a+b x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 2497
Rule 6057
Rule 6246
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a+b x)}{c+d x} \, dx &=\frac {\text {Subst}\left (\int \frac {\tanh ^{-1}(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\tanh ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}+\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,a+b x\right )}{d}-\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2 \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{\left (\frac {d}{b}+\frac {b c-a d}{b}\right ) (1+x)}\right )}{1-x^2} \, dx,x,a+b x\right )}{d}\\ &=-\frac {\tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\tanh ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}-\frac {\text {Li}_2\left (1-\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+a+b x}\right )}{d}\\ &=-\frac {\tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\tanh ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}+\frac {\text {Li}_2\left (1-\frac {2}{1+a+b x}\right )}{2 d}-\frac {\text {Li}_2\left (1-\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 138, normalized size = 1.15 \begin {gather*} -\frac {\log (1-a-b x) \log \left (-\frac {b (c+d x)}{-b c-(1-a) d}\right )}{2 d}+\frac {\log (1+a+b x) \log \left (\frac {b (c+d x)}{b c-(1+a) d}\right )}{2 d}-\frac {\text {PolyLog}\left (2,-\frac {d (1-a-b x)}{-b c-d+a d}\right )}{2 d}+\frac {\text {PolyLog}\left (2,\frac {d (1+a+b x)}{-b c+d+a d}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 12.58, size = 184, normalized size = 1.53
method | result | size |
risch | \(-\frac {\dilog \left (\frac {d \left (-b x -a +1\right )+d a -b c -d}{d a -b c -d}\right )}{2 d}-\frac {\ln \left (-b x -a +1\right ) \ln \left (\frac {d \left (-b x -a +1\right )+d a -b c -d}{d a -b c -d}\right )}{2 d}+\frac {\dilog \left (\frac {d \left (b x +a +1\right )-d a +b c -d}{-d a +b c -d}\right )}{2 d}+\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {d \left (b x +a +1\right )-d a +b c -d}{-d a +b c -d}\right )}{2 d}\) | \(181\) |
derivativedivides | \(\frac {\frac {b \ln \left (d a -b c -d \left (b x +a \right )\right ) \arctanh \left (b x +a \right )}{d}+\frac {b \left (\frac {\left (\dilog \left (\frac {-d \left (b x +a \right )+d}{-d a +b c +d}\right )+\ln \left (d a -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )+d}{-d a +b c +d}\right )\right ) d}{2}-\frac {\left (\dilog \left (\frac {-d \left (b x +a \right )-d}{-d a +b c -d}\right )+\ln \left (d a -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )-d}{-d a +b c -d}\right )\right ) d}{2}\right )}{d^{2}}}{b}\) | \(184\) |
default | \(\frac {\frac {b \ln \left (d a -b c -d \left (b x +a \right )\right ) \arctanh \left (b x +a \right )}{d}+\frac {b \left (\frac {\left (\dilog \left (\frac {-d \left (b x +a \right )+d}{-d a +b c +d}\right )+\ln \left (d a -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )+d}{-d a +b c +d}\right )\right ) d}{2}-\frac {\left (\dilog \left (\frac {-d \left (b x +a \right )-d}{-d a +b c -d}\right )+\ln \left (d a -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )-d}{-d a +b c -d}\right )\right ) d}{2}\right )}{d^{2}}}{b}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 192, normalized size = 1.60 \begin {gather*} -\frac {1}{2} \, b {\left (\frac {\log \left (b x + a - 1\right ) \log \left (\frac {b d x + a d - d}{b c - a d + d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d - d}{b c - a d + d}\right )}{b d} - \frac {\log \left (b x + a + 1\right ) \log \left (\frac {b d x + a d + d}{b c - a d - d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d + d}{b c - a d - d}\right )}{b d}\right )} - \frac {b {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \left (d x + c\right )}{2 \, d} + \frac {\operatorname {artanh}\left (b x + a\right ) \log \left (d x + c\right )}{d} \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*} \int \frac {\operatorname {atanh}{\left (a + b x \right )}}{c + d x}\, dx \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 {\mathrm {atanh}\left (a+b\,x\right )}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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