Optimal. Leaf size=191 \[ -\frac {3 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}-\frac {3 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(c x+1)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c} \]
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Rubi [A] time = 0.30, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5928, 5910, 5984, 5918, 2402, 2315, 1586, 5948, 6058, 6610} \[ -\frac {3 b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}-\frac {3 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}+\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c}-\frac {3 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(c x+1)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c} \]
Antiderivative was successfully verified.
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Rule 1586
Rule 2315
Rule 2402
Rule 5910
Rule 5918
Rule 5928
Rule 5948
Rule 5984
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int (1+c x) \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {1}{2} (3 b) \int \left (-\left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2 (1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}\right ) \, dx\\ &=\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}+\frac {1}{2} (3 b) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx-(3 b) \int \frac {(1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-(3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx-\left (3 b^2 c\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}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (6 b^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\left (3 b^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx+\left (3 b^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c}-\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}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 334, normalized size = 1.75 \[ \frac {2 a^3 c^2 x^2+4 a^3 c x+6 a^2 b c^2 x^2 \tanh ^{-1}(c x)+6 a^2 b c x+9 a^2 b \log (1-c x)+3 a^2 b \log (c x+1)+12 a^2 b c x \tanh ^{-1}(c x)+6 a b^2 \log \left (1-c^2 x^2\right )+6 a b^2 c^2 x^2 \tanh ^{-1}(c x)^2+6 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right ) \left (2 a+2 b \tanh ^{-1}(c x)+b\right )-18 a b^2 \tanh ^{-1}(c x)^2+12 a b^2 c x \tanh ^{-1}(c x)^2+12 a b^2 c x \tanh ^{-1}(c x)-24 a b^2 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+2 b^3 c^2 x^2 \tanh ^{-1}(c x)^3+6 b^3 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )-6 b^3 \tanh ^{-1}(c x)^3+4 b^3 c x \tanh ^{-1}(c x)^3-6 b^3 \tanh ^{-1}(c x)^2+6 b^3 c x \tanh ^{-1}(c x)^2-12 b^3 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-12 b^3 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )}{4 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{3} c x + {\left (b^{3} c x + b^{3}\right )} \operatorname {artanh}\left (c x\right )^{3} + a^{3} + 3 \, {\left (a b^{2} c x + a b^{2}\right )} \operatorname {artanh}\left (c x\right )^{2} + 3 \, {\left (a^{2} b c x + a^{2} b\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
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 \[ \int {\left (c x + 1\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.80, size = 6440, normalized size = 33.72 \[ \text {output too large to display} \]
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 \[ \frac {1}{2} \, a^{3} c x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a^{2} b c + a^{3} x + \frac {3 \, {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a^{2} b}{2 \, c} - \frac {{\left (b^{3} c^{2} x^{2} + 2 \, b^{3} c x - 3 \, b^{3}\right )} \log \left (-c x + 1\right )^{3} - 3 \, {\left (2 \, a b^{2} c^{2} x^{2} + 2 \, {\left (2 \, a b^{2} c + b^{3} c\right )} x + {\left (b^{3} c^{2} x^{2} + 2 \, b^{3} c x + b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{16 \, c} - \int -\frac {{\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \left (c x + 1\right )^{3} + 6 \, {\left (a b^{2} c^{2} x^{2} - a b^{2}\right )} \log \left (c x + 1\right )^{2} - 3 \, {\left (2 \, a b^{2} c^{2} x^{2} + {\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \left (c x + 1\right )^{2} + 2 \, {\left (2 \, a b^{2} c + b^{3} c\right )} x + {\left (2 \, b^{3} c x - 4 \, a b^{2} + b^{3} + {\left (4 \, a b^{2} c^{2} + b^{3} c^{2}\right )} x^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, {\left (c x - 1\right )}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,\left (c\,x+1\right ) \,d x \]
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 \[ \int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3} \left (c x + 1\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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