Optimal. Leaf size=130 \[ \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}-\frac {2 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^3}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^3}+\frac {b^2 x}{3 c^2} \]
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Rubi [A] time = 0.20, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5916, 5980, 321, 206, 5984, 5918, 2402, 2315} \[ -\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^3}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}-\frac {2 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {b^2 x}{3 c^2}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 2315
Rule 2402
Rule 5916
Rule 5918
Rule 5980
Rule 5984
Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{3} (2 b c) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(2 b) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c}-\frac {(2 b) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c}\\ &=\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{3} b^2 \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c^2}\\ &=\frac {b^2 x}{3 c^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \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 )}{3 c^3}-\frac {b^2 \int \frac {1}{1-c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c^2}\\ &=\frac {b^2 x}{3 c^2}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^3}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \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 )}{3 c^3}-\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 )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^3}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \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 )}{3 c^3}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^3}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 122, normalized size = 0.94 \[ \frac {a^2 c^3 x^3+a b c^2 x^2+a b \log \left (c^2 x^2-1\right )+b \tanh ^{-1}(c x) \left (2 a c^3 x^3+b c^2 x^2-2 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-b\right )+b^2 \left (c^3 x^3-1\right ) \tanh ^{-1}(c x)^2+b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+b^2 c x}{3 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{2} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname {artanh}\left (c x\right ) + a^{2} x^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 270, normalized size = 2.08 \[ \frac {x^{3} a^{2}}{3}+\frac {b^{2} x^{3} \arctanh \left (c x \right )^{2}}{3}+\frac {b^{2} \arctanh \left (c x \right ) x^{2}}{3 c}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{3 c^{3}}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{3 c^{3}}+\frac {b^{2} x}{3 c^{2}}+\frac {b^{2} \ln \left (c x -1\right )}{6 c^{3}}-\frac {b^{2} \ln \left (c x +1\right )}{6 c^{3}}+\frac {b^{2} \ln \left (c x -1\right )^{2}}{12 c^{3}}-\frac {b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{3 c^{3}}-\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{6 c^{3}}-\frac {b^{2} \ln \left (c x +1\right )^{2}}{12 c^{3}}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{6 c^{3}}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{6 c^{3}}+\frac {2 a b \,x^{3} \arctanh \left (c x \right )}{3}+\frac {a b \,x^{2}}{3 c}+\frac {a b \ln \left (c x -1\right )}{3 c^{3}}+\frac {a b \ln \left (c x +1\right )}{3 c^{3}} \]
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 \[ \frac {1}{3} \, a^{2} x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b - \frac {1}{216} \, {\left (2 \, c^{4} {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{6}} - \frac {3 \, \log \left (c x + 1\right )}{c^{7}} + \frac {3 \, \log \left (c x - 1\right )}{c^{7}}\right )} - 3 \, c^{3} {\left (\frac {x^{2}}{c^{4}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )} - 648 \, c^{3} \int \frac {x^{3} \log \left (c x + 1\right )}{9 \, {\left (c^{4} x^{2} - c^{2}\right )}}\,{d x} + 9 \, c^{2} {\left (\frac {2 \, x}{c^{4}} - \frac {\log \left (c x + 1\right )}{c^{5}} + \frac {\log \left (c x - 1\right )}{c^{5}}\right )} - 324 \, c \int \frac {x \log \left (c x + 1\right )}{9 \, {\left (c^{4} x^{2} - c^{2}\right )}}\,{d x} - \frac {6 \, {\left (3 \, c^{3} x^{3} \log \left (c x + 1\right )^{2} + {\left (2 \, c^{3} x^{3} - 3 \, c^{2} x^{2} + 6 \, c x - 6 \, {\left (c^{3} x^{3} + 1\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )\right )}}{c^{3}} - \frac {2 \, {\left (c x - 1\right )}^{3} {\left (9 \, \log \left (-c x + 1\right )^{2} - 6 \, \log \left (-c x + 1\right ) + 2\right )} + 27 \, {\left (c x - 1\right )}^{2} {\left (2 \, \log \left (-c x + 1\right )^{2} - 2 \, \log \left (-c x + 1\right ) + 1\right )} + 54 \, {\left (c x - 1\right )} {\left (\log \left (-c x + 1\right )^{2} - 2 \, \log \left (-c x + 1\right ) + 2\right )}}{c^{3}} + \frac {18 \, \log \left (9 \, c^{4} x^{2} - 9 \, c^{2}\right )}{c^{3}} - 324 \, \int \frac {\log \left (c x + 1\right )}{9 \, {\left (c^{4} x^{2} - c^{2}\right )}}\,{d x}\right )} b^{2} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2 \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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