Optimal. Leaf size=306 \[ -\frac {6 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {11 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+3 a b^2 x+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {6 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {11 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{c}+3 b^3 x \tanh ^{-1}(c x)-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+\frac {b^3 x}{4} \]
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Rubi [A] time = 0.66, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5928, 5910, 5984, 5918, 2402, 2315, 5916, 5980, 260, 5948, 321, 206, 1586, 6058, 6610} \[ -\frac {6 b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}-\frac {11 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}+\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{c}+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {11 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+3 a b^2 x+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {6 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}+3 b^3 x \tanh ^{-1}(c x)-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+\frac {b^3 x}{4} \]
Antiderivative was successfully verified.
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Rule 206
Rule 260
Rule 321
Rule 1586
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5928
Rule 5948
Rule 5980
Rule 5984
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {1}{4} (3 b) \int \left (-7 \left (a+b \tanh ^{-1}(c x)\right )^2-4 c x \left (a+b \tanh ^{-1}(c x)\right )^2-c^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {8 (1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}\right ) \, dx\\ &=\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}+\frac {1}{4} (21 b) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx-(6 b) \int \frac {(1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx+(3 b c) \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\frac {1}{4} \left (3 b c^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-(6 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx-\frac {1}{2} \left (21 b^2 c\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (3 b^2 c^2\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{2} \left (b^2 c^3\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {21 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {6 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 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx-\frac {1}{2} \left (21 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (12 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 {1}{2} \left (b^2 c\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{2} \left (b^2 c\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=3 a b^2 x+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {21 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}-\frac {1}{2} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (3 b^3\right ) \int \tanh ^{-1}(c x) \, dx+\left (6 b^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx+\frac {1}{2} \left (21 b^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\frac {1}{4} \left (b^3 c^2\right ) \int \frac {x^2}{1-c^2 x^2} \, dx\\ &=3 a b^2 x+\frac {b^3 x}{4}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 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 )}{c}-\frac {1}{4} b^3 \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{2} b^3 \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\frac {\left (21 b^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{2 c}-\left (3 b^3 c\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=3 a b^2 x+\frac {b^3 x}{4}-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {21 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 c}-\frac {6 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 )}{c}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{2 c}\\ &=3 a b^2 x+\frac {b^3 x}{4}-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {11 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c}-\frac {6 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 )}{c}\\ \end {align*}
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Mathematica [B] time = 1.47, size = 644, normalized size = 2.10 \[ \frac {2 a^3 c^4 x^4+8 a^3 c^3 x^3+12 a^3 c^2 x^2+8 a^3 c x+6 a^2 b c^4 x^4 \tanh ^{-1}(c x)+2 a^2 b c^3 x^3+24 a^2 b c^3 x^3 \tanh ^{-1}(c x)+12 a^2 b c^2 x^2+36 a^2 b c^2 x^2 \tanh ^{-1}(c x)+42 a^2 b c x+45 a^2 b \log (1-c x)+3 a^2 b \log (c x+1)+24 a^2 b c x \tanh ^{-1}(c x)+6 a b^2 c^4 x^4 \tanh ^{-1}(c x)^2+24 a b^2 c^3 x^3 \tanh ^{-1}(c x)^2+4 a b^2 c^3 x^3 \tanh ^{-1}(c x)+2 a b^2 c^2 x^2+44 a b^2 \log \left (1-c^2 x^2\right )+36 a b^2 c^2 x^2 \tanh ^{-1}(c x)^2+24 a b^2 c^2 x^2 \tanh ^{-1}(c x)+4 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right ) \left (12 a+12 b \tanh ^{-1}(c x)+11 b\right )+24 a b^2 c x+24 a b^2 c x \tanh ^{-1}(c x)^2+84 a b^2 c x \tanh ^{-1}(c x)-90 a b^2 \tanh ^{-1}(c x)^2-24 a b^2 \tanh ^{-1}(c x)-96 a b^2 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-2 a b^2+2 b^3 c^4 x^4 \tanh ^{-1}(c x)^3+8 b^3 c^3 x^3 \tanh ^{-1}(c x)^3+2 b^3 c^3 x^3 \tanh ^{-1}(c x)^2+12 b^3 \log \left (1-c^2 x^2\right )+12 b^3 c^2 x^2 \tanh ^{-1}(c x)^3+12 b^3 c^2 x^2 \tanh ^{-1}(c x)^2+2 b^3 c^2 x^2 \tanh ^{-1}(c x)+24 b^3 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )+2 b^3 c x+8 b^3 c x \tanh ^{-1}(c x)^3+42 b^3 c x \tanh ^{-1}(c x)^2+24 b^3 c x \tanh ^{-1}(c x)-30 b^3 \tanh ^{-1}(c x)^3-56 b^3 \tanh ^{-1}(c x)^2-2 b^3 \tanh ^{-1}(c x)-48 b^3 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-88 b^3 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )}{8 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{3} c^{3} x^{3} + 3 \, a^{3} c^{2} x^{2} + 3 \, a^{3} c x + {\left (b^{3} c^{3} x^{3} + 3 \, b^{3} c^{2} x^{2} + 3 \, b^{3} c x + b^{3}\right )} \operatorname {artanh}\left (c x\right )^{3} + a^{3} + 3 \, {\left (a b^{2} c^{3} x^{3} + 3 \, a b^{2} c^{2} x^{2} + 3 \, a b^{2} c x + a b^{2}\right )} \operatorname {artanh}\left (c x\right )^{2} + 3 \, {\left (a^{2} b c^{3} x^{3} + 3 \, a^{2} b c^{2} x^{2} + 3 \, a^{2} b c x + a^{2} b\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
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 \[ \int {\left (c x + 1\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.59, size = 963, normalized size = 3.15 \[ \text {result too large to display} \]
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}{4} \, a^{3} c^{3} x^{4} + a^{3} c^{2} x^{3} + \frac {1}{8} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} a^{2} b c^{3} + \frac {3}{2} \, {\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^{2} b c^{2} + \frac {3}{2} \, a^{3} c x^{2} + \frac {9}{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^{4} x^{4} + 4 \, b^{3} c^{3} x^{3} + 6 \, b^{3} c^{2} x^{2} + 4 \, b^{3} c x - 15 \, b^{3}\right )} \log \left (-c x + 1\right )^{3} - {\left (6 \, a b^{2} c^{4} x^{4} + 2 \, {\left (12 \, a b^{2} c^{3} + b^{3} c^{3}\right )} x^{3} + 12 \, {\left (3 \, a b^{2} c^{2} + b^{3} c^{2}\right )} x^{2} + 6 \, {\left (4 \, a b^{2} c + 7 \, b^{3} c\right )} x + 3 \, {\left (b^{3} c^{4} x^{4} + 4 \, b^{3} c^{3} x^{3} + 6 \, b^{3} c^{2} x^{2} + 4 \, b^{3} c x + b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{32 \, c} - \int -\frac {2 \, {\left (b^{3} c^{4} x^{4} + 2 \, b^{3} c^{3} x^{3} - 2 \, b^{3} c x - b^{3}\right )} \log \left (c x + 1\right )^{3} + 12 \, {\left (a b^{2} c^{4} x^{4} + 2 \, a b^{2} c^{3} x^{3} - 2 \, a b^{2} c x - a b^{2}\right )} \log \left (c x + 1\right )^{2} - {\left (6 \, a b^{2} c^{4} x^{4} + 2 \, {\left (12 \, a b^{2} c^{3} + b^{3} c^{3}\right )} x^{3} + 12 \, {\left (3 \, a b^{2} c^{2} + b^{3} c^{2}\right )} x^{2} + 6 \, {\left (b^{3} c^{4} x^{4} + 2 \, b^{3} c^{3} x^{3} - 2 \, b^{3} c x - b^{3}\right )} \log \left (c x + 1\right )^{2} + 6 \, {\left (4 \, a b^{2} c + 7 \, b^{3} c\right )} x + 3 \, {\left (6 \, b^{3} c^{2} x^{2} + {\left (8 \, a b^{2} c^{4} + b^{3} c^{4}\right )} x^{4} + 4 \, {\left (4 \, a b^{2} c^{3} + b^{3} c^{3}\right )} x^{3} - 8 \, a b^{2} + b^{3} - 4 \, {\left (4 \, a b^{2} c - b^{3} c\right )} x\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{16 \, {\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 = -0.00 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,{\left (c\,x+1\right )}^3 \,d x \]
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 \[ \int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3} \left (c x + 1\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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