Optimal. Leaf size=74 \[ x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {2 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c} \]
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Rubi [A] time = 0.10, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5910, 5984, 5918, 2402, 2315} \[ -\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {2 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 5910
Rule 5918
Rule 5984
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=x \left (a+b \tanh ^{-1}(c x)\right )^2-(2 b c) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^2-(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx\\ &=\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}+\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c}\\ &=\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 82, normalized size = 1.11 \[ \frac {a \left (a c x+b \log \left (1-c^2 x^2\right )\right )+2 b \tanh ^{-1}(c x) \left (a c x-b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+b^2 (c x-1) \tanh ^{-1}(c x)^2}{c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x\right ) + a^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 123, normalized size = 1.66 \[ x \,b^{2} \arctanh \left (c x \right )^{2}+2 x a b \arctanh \left (c x \right )+\frac {b^{2} \arctanh \left (c x \right )^{2}}{c}-\frac {2 \arctanh \left (c x \right ) \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right ) b^{2}}{c}+a^{2} x +\frac {a b \ln \left (-c^{2} x^{2}+1\right )}{c}-\frac {\polylog \left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right ) b^{2}}{c} \]
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 \[ -\frac {1}{4} \, {\left (c^{2} {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )} - 6 \, c \int \frac {x \log \left (c x + 1\right )}{c^{2} x^{2} - 1}\,{d x} - \frac {{\left (c x - 1\right )} {\left (\log \left (-c x + 1\right )^{2} - 2 \, \log \left (-c x + 1\right ) + 2\right )}}{c} - \frac {c x \log \left (c x + 1\right )^{2} + 2 \, {\left (c x - {\left (c x + 1\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{c} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c} - 2 \, \int \frac {\log \left (c x + 1\right )}{c^{2} x^{2} - 1}\,{d x}\right )} b^{2} + a^{2} x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b}{c} \]
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 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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