Optimal. Leaf size=123 \[ -\frac {3 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c^2} \]
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Rubi [A] time = 0.25, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5916, 5980, 5910, 5984, 5918, 2402, 2315, 5948} \[ -\frac {3 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c^2}-\frac {3 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5948
Rule 5980
Rule 5984
Rubi steps
\begin {align*} \int x \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {1}{2} (3 b c) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {(3 b) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{2 c}-\frac {(3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{2 c}\\ &=\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\left (3 b^2\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}+\frac {\left (3 b^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^2}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 161, normalized size = 1.31 \[ \frac {a \left (2 a^2 c^2 x^2+6 a b c x+3 a b \log (1-c x)-3 a b \log (c x+1)+6 b^2 \log \left (1-c^2 x^2\right )\right )+6 b^2 (c x-1) \tanh ^{-1}(c x)^2 (a c x+a+b)+6 b \tanh ^{-1}(c x) \left (a c x (a c x+2 b)-2 b^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+2 b^3 \left (c^2 x^2-1\right ) \tanh ^{-1}(c x)^3+6 b^3 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )}{4 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x \operatorname {artanh}\left (c x\right )^{3} + 3 \, a b^{2} x \operatorname {artanh}\left (c x\right )^{2} + 3 \, a^{2} b x \operatorname {artanh}\left (c x\right ) + a^{3} x, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.64, size = 6097, normalized size = 49.57 \[ \text {output too large to display} \]
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 {3}{2} \, a b^{2} x^{2} \operatorname {artanh}\left (c x\right )^{2} + \frac {1}{2} \, a^{3} x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\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 )}\right )} a^{2} b + \frac {3}{8} \, {\left (4 \, c {\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 )} \operatorname {artanh}\left (c x\right ) - \frac {2 \, {\left (\log \left (c x - 1\right ) - 2\right )} \log \left (c x + 1\right ) - \log \left (c x + 1\right )^{2} - \log \left (c x - 1\right )^{2} - 4 \, \log \left (c x - 1\right )}{c^{2}}\right )} a b^{2} - \frac {1}{64} \, {\left (3 \, c^{3} {\left (\frac {x^{2}}{c^{3}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{5}}\right )} + 21 \, c^{2} {\left (\frac {2 \, x}{c^{3}} - \frac {\log \left (c x + 1\right )}{c^{4}} + \frac {\log \left (c x - 1\right )}{c^{4}}\right )} - 576 \, c \int \frac {x \log \left (c x + 1\right )}{4 \, {\left (c^{3} x^{2} - c\right )}}\,{d x} - \frac {2 \, {\left (12 \, c x \log \left (c x + 1\right )^{2} + 2 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right )^{3} - 3 \, {\left (c^{2} x^{2} - 2 \, c x - 2 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + 1\right )} \log \left (-c x + 1\right )^{2} + 3 \, {\left (c^{2} x^{2} - 2 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right )^{2} + 6 \, c x - 8 \, {\left (c x + 1\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )\right )}}{c^{2}} + \frac {{\left (4 \, \log \left (-c x + 1\right )^{3} - 6 \, \log \left (-c x + 1\right )^{2} + 6 \, \log \left (-c x + 1\right ) - 3\right )} {\left (c x - 1\right )}^{2} + 8 \, {\left (\log \left (-c x + 1\right )^{3} - 3 \, \log \left (-c x + 1\right )^{2} + 6 \, \log \left (-c x + 1\right ) - 6\right )} {\left (c x - 1\right )}}{c^{2}} + \frac {18 \, \log \left (4 \, c^{3} x^{2} - 4 \, c\right )}{c^{2}} - 192 \, \int \frac {\log \left (c x + 1\right )}{4 \, {\left (c^{3} x^{2} - c\right )}}\,{d x}\right )} b^{3} \]
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 \[ \int x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3 \,d x \]
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 \[ \int x \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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