Optimal. Leaf size=81 \[ -\frac {\text {Li}_2\left (-\frac {a+b x+1}{-a-b x+1}\right )}{b}+\frac {(a+b x) \tanh ^{-1}(a+b x)^2}{b}+\frac {\tanh ^{-1}(a+b x)^2}{b}-\frac {2 \log \left (\frac {2}{-a-b x+1}\right ) \tanh ^{-1}(a+b x)}{b} \]
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Rubi [A] time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6103, 5910, 5984, 5918, 2402, 2315} \[ -\frac {\text {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )}{b}+\frac {(a+b x) \tanh ^{-1}(a+b x)^2}{b}+\frac {\tanh ^{-1}(a+b x)^2}{b}-\frac {2 \log \left (\frac {2}{-a-b x+1}\right ) \tanh ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 5910
Rule 5918
Rule 5984
Rule 6103
Rubi steps
\begin {align*} \int \tanh ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \tanh ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac {2 \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\tanh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac {2 \operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)}{1-x} \, dx,x,a+b x\right )}{b}\\ &=\frac {\tanh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac {2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b}+\frac {2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\tanh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac {2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a-b x}\right )}{b}\\ &=\frac {\tanh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac {2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b}-\frac {\text {Li}_2\left (1-\frac {2}{1-a-b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 55, normalized size = 0.68 \[ \frac {\text {Li}_2\left (-e^{-2 \tanh ^{-1}(a+b x)}\right )+\tanh ^{-1}(a+b x) \left ((a+b x-1) \tanh ^{-1}(a+b x)-2 \log \left (e^{-2 \tanh ^{-1}(a+b x)}+1\right )\right )}{b} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {artanh}\left (b x + a\right )^{2}, x\right ) \]
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 \[ \int \operatorname {artanh}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 103, normalized size = 1.27 \[ x \arctanh \left (b x +a \right )^{2}+\frac {\arctanh \left (b x +a \right )^{2} a}{b}+\frac {\arctanh \left (b x +a \right )^{2}}{b}-\frac {2 \arctanh \left (b x +a \right ) \ln \left (1+\frac {\left (b x +a +1\right )^{2}}{1-\left (b x +a \right )^{2}}\right )}{b}-\frac {\polylog \left (2, -\frac {\left (b x +a +1\right )^{2}}{1-\left (b x +a \right )^{2}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 139, normalized size = 1.72 \[ -\frac {1}{4} \, b^{2} {\left (\frac {{\left (a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a - 1\right )} \log \left (b x + a - 1\right )^{2}}{b^{3}} + \frac {4 \, {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )}}{b^{3}}\right )} + b {\left (\frac {{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2}} - \frac {{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{2}}\right )} \operatorname {artanh}\left (b x + a\right ) + x \operatorname {artanh}\left (b x + a\right )^{2} \]
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 \[ \int {\mathrm {atanh}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int \operatorname {atanh}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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