Optimal. Leaf size=94 \[ \frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d}+\frac {a x}{c d}-\frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 c^2 d}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^2 d}+\frac {b x \tanh ^{-1}(c x)}{c d} \]
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Rubi [A] time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5930, 5910, 260, 5918, 2402, 2315} \[ -\frac {b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 c^2 d}+\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d}+\frac {a x}{c d}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^2 d}+\frac {b x \tanh ^{-1}(c x)}{c d} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2402
Rule 5910
Rule 5918
Rule 5930
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{d+c d x} \, dx &=-\frac {\int \frac {a+b \tanh ^{-1}(c x)}{d+c d x} \, dx}{c}+\frac {\int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c d}\\ &=\frac {a x}{c d}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^2 d}+\frac {b \int \tanh ^{-1}(c x) \, dx}{c d}-\frac {b \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c d}\\ &=\frac {a x}{c d}+\frac {b x \tanh ^{-1}(c x)}{c d}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^2 d}-\frac {b \int \frac {x}{1-c^2 x^2} \, dx}{d}-\frac {b \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{c^2 d}\\ &=\frac {a x}{c d}+\frac {b x \tanh ^{-1}(c x)}{c d}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^2 d}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^2 d}-\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c^2 d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 75, normalized size = 0.80 \[ \frac {2 a c x-2 a \log (c x+1)+b \log \left (1-c^2 x^2\right )-b \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+2 b \tanh ^{-1}(c x) \left (c x+\log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )}{2 c^2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \operatorname {artanh}\left (c x\right ) + a x}{c d x + d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x}{c d x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 157, normalized size = 1.67 \[ \frac {a x}{c d}-\frac {a \ln \left (c x +1\right )}{c^{2} d}-\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{c^{2} d}+\frac {b x \arctanh \left (c x \right )}{c d}+\frac {b \ln \left (c x +1\right )^{2}}{4 c^{2} d}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 c^{2} d}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 c^{2} d}+\frac {b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 c^{2} d}+\frac {b \ln \left (\left (c x -1\right ) \left (c x +1\right )\right )}{2 c^{2} d} \]
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 \[ \frac {1}{4} \, {\left (c^{2} {\left (\frac {2 \, x}{c^{3} d} - \frac {\log \left (c x + 1\right )}{c^{4} d} + \frac {\log \left (c x - 1\right )}{c^{4} d}\right )} + 2 \, c^{2} \int \frac {x^{2} \log \left (c x + 1\right )}{c^{3} d x^{2} - c d}\,{d x} - 4 \, c \int \frac {x \log \left (c x + 1\right )}{c^{3} d x^{2} - c d}\,{d x} - \frac {2 \, {\left (c x - \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{c^{2} d} + \frac {\log \left (c^{3} d x^{2} - c d\right )}{c^{2} d} - 2 \, \int \frac {\log \left (c x + 1\right )}{c^{3} d x^{2} - c d}\,{d x}\right )} b + a {\left (\frac {x}{c d} - \frac {\log \left (c x + 1\right )}{c^{2} d}\right )} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{d+c\,d\,x} \,d x \]
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 \[ \frac {\int \frac {a x}{c x + 1}\, dx + \int \frac {b x \operatorname {atanh}{\left (c x \right )}}{c x + 1}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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