Optimal. Leaf size=74 \[ \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c} \]
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Rubi [A]
time = 0.07, 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 = {6021, 6131,
6055, 2449, 2352} \begin {gather*} 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 6021
Rule 6055
Rule 6131
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 ) \text {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 = 0.10, size = 82, normalized size = 1.11 \begin {gather*} \frac {b^2 (-1+c x) \tanh ^{-1}(c x)^2+2 b \tanh ^{-1}(c x) \left (a c x-b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+a \left (a c x+b \log \left (1-c^2 x^2\right )\right )+b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 118, normalized size = 1.59
method | result | size |
derivativedivides | \(\frac {c x \,a^{2}+b^{2} c x \arctanh \left (c x \right )^{2}+b^{2} \arctanh \left (c x \right )^{2}-2 \arctanh \left (c x \right ) \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right ) b^{2}-\polylog \left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right ) b^{2}+2 a b c x \arctanh \left (c x \right )+a b \ln \left (-c^{2} x^{2}+1\right )}{c}\) | \(118\) |
default | \(\frac {c x \,a^{2}+b^{2} c x \arctanh \left (c x \right )^{2}+b^{2} \arctanh \left (c x \right )^{2}-2 \arctanh \left (c x \right ) \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right ) b^{2}-\polylog \left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right ) b^{2}+2 a b c x \arctanh \left (c x \right )+a b \ln \left (-c^{2} x^{2}+1\right )}{c}\) | \(118\) |
risch | \(-\frac {a^{2}}{c}-\frac {b^{2}}{c}+a^{2} x -\frac {b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{c}-\frac {b^{2} \ln \left (c x -1\right )}{c}+\frac {b^{2} \ln \left (c x +1\right )^{2} x}{4}+\frac {b^{2} \ln \left (c x +1\right )^{2}}{4 c}-\frac {2 a b}{c}+\frac {\ln \left (-c x +1\right )^{2} x \,b^{2}}{4}-\frac {\ln \left (-c x +1\right )^{2} b^{2}}{4 c}+\frac {\ln \left (-c x +1\right ) b^{2}}{c}+b a \ln \left (c x +1\right ) x +\frac {b a \ln \left (c x +1\right )}{c}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right ) x}{2}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right )}{2 c}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{c}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{c}-\ln \left (-c x +1\right ) x a b +\frac {\ln \left (-c x +1\right ) a b}{c}\) | \(264\) |
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 \begin {gather*} \text {Failed to integrate} \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 \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}\, 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 {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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