Optimal. Leaf size=130 \[ \frac {\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{f}-\frac {b \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {b \text {Li}_2\left (1-\frac {2}{c+d x+1}\right )}{2 f} \]
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Rubi [A] time = 0.14, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6111, 5920, 2402, 2315, 2447} \[ -\frac {b \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac {b \text {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{2 f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{f} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 2447
Rule 5920
Rule 6111
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c+d x)}{e+f x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{\frac {d e-c f}{d}+\frac {f x}{d}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{f}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )}{\left (\frac {f}{d}+\frac {d e-c f}{d}\right ) (1+x)}\right )}{1-x^2} \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}-\frac {b \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac {b \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c+d x}\right )}{f}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \text {Li}_2\left (1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 126, normalized size = 0.97 \[ \frac {2 a \log (e+f x)-b \text {Li}_2\left (\frac {f (c+d x-1)}{(c-1) f-d e}\right )+b \text {Li}_2\left (\frac {f (c+d x+1)}{-d e+c f+f}\right )-b \log (-c-d x+1) \log \left (\frac {d (e+f x)}{-c f+d e+f}\right )+b \log (c+d x+1) \log \left (\frac {d (e+f x)}{d e-(c+1) f}\right )}{2 f} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (d x + c\right ) + a}{f x + e}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (d x + c\right ) + a}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 202, normalized size = 1.55 \[ \frac {a \ln \left (\left (d x +c \right ) f -c f +d e \right )}{f}+\frac {b \ln \left (\left (d x +c \right ) f -c f +d e \right ) \arctanh \left (d x +c \right )}{f}+\frac {b \ln \left (\left (d x +c \right ) f -c f +d e \right ) \ln \left (\frac {\left (d x +c \right ) f -f}{c f -d e -f}\right )}{2 f}+\frac {b \dilog \left (\frac {\left (d x +c \right ) f -f}{c f -d e -f}\right )}{2 f}-\frac {b \ln \left (\left (d x +c \right ) f -c f +d e \right ) \ln \left (\frac {\left (d x +c \right ) f +f}{c f -d e +f}\right )}{2 f}-\frac {b \dilog \left (\frac {\left (d x +c \right ) f +f}{c f -d e +f}\right )}{2 f} \]
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 \[ \frac {1}{2} \, b \int \frac {\log \left (d x + c + 1\right ) - \log \left (-d x - c + 1\right )}{f x + e}\,{d x} + \frac {a \log \left (f x + e\right )}{f} \]
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 \[ \int \frac {a+b\,\mathrm {atanh}\left (c+d\,x\right )}{e+f\,x} \,d x \]
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 \[ \int \frac {a + b \operatorname {atanh}{\left (c + d x \right )}}{e + f x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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