Optimal. Leaf size=208 \[ -\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (c x+1)}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (c x+1)^2}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (c x+1)^2}-\frac {21 b^3}{64 c (c x+1)}-\frac {3 b^3}{64 c (c x+1)^2}+\frac {21 b^3 \tanh ^{-1}(c x)}{64 c} \]
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Rubi [A] time = 0.37, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5928, 5926, 627, 44, 207, 5948} \[ -\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (c x+1)}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (c x+1)^2}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (c x+1)^2}-\frac {21 b^3}{64 c (c x+1)}-\frac {3 b^3}{64 c (c x+1)^2}+\frac {21 b^3 \tanh ^{-1}(c x)}{64 c} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 627
Rule 5926
Rule 5928
Rule 5948
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{(1+c x)^3} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{2} (3 b) \int \left (\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 (1+c x)^3}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 (1+c x)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{8} (3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx-\frac {1}{8} (3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{-1+c^2 x^2} \, dx+\frac {1}{4} (3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^3} \, dx\\ &=-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{4} \left (3 b^2\right ) \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{4} \left (3 b^2\right ) \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^3}+\frac {a+b \tanh ^{-1}(c x)}{4 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{4 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{16} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{16} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac {1}{8} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx+\frac {1}{8} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{8} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx\\ &=-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx+\frac {1}{8} \left (3 b^3\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{(1-c x) (1+c x)^3} \, dx+\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx+\frac {1}{8} \left (3 b^3\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx\\ &=-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{16} \left (3 b^3\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{16} \left (3 b^3\right ) \int \left (\frac {1}{2 (1+c x)^3}+\frac {1}{4 (1+c x)^2}-\frac {1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{8} \left (3 b^3\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {3 b^3}{64 c (1+c x)^2}-\frac {21 b^3}{64 c (1+c x)}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}-\frac {1}{64} \left (3 b^3\right ) \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{32} \left (3 b^3\right ) \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{-1+c^2 x^2} \, dx\\ &=-\frac {3 b^3}{64 c (1+c x)^2}-\frac {21 b^3}{64 c (1+c x)}+\frac {21 b^3 \tanh ^{-1}(c x)}{64 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 215, normalized size = 1.03 \[ \frac {-6 b \left (8 a^2+12 a b+7 b^2\right ) (c x+1)-3 b \left (8 a^2+12 a b+7 b^2\right ) (c x+1)^2 \log (1-c x)+3 b \left (8 a^2+12 a b+7 b^2\right ) (c x+1)^2 \log (c x+1)-24 b \tanh ^{-1}(c x) \left (8 a^2+4 a b (c x+2)+b^2 (3 c x+4)\right )-2 \left (32 a^3+24 a^2 b+12 a b^2+3 b^3\right )+12 b^2 (c x-1) \tanh ^{-1}(c x)^2 (4 a (c x+3)+b (3 c x+5))+16 b^3 \left (c^2 x^2+2 c x-3\right ) \tanh ^{-1}(c x)^3}{128 c (c x+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 250, normalized size = 1.20 \[ \frac {2 \, {\left (b^{3} c^{2} x^{2} + 2 \, b^{3} c x - 3 \, b^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3} - 64 \, a^{3} - 96 \, a^{2} b - 96 \, a b^{2} - 48 \, b^{3} - 6 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c x + 3 \, {\left ({\left (4 \, a b^{2} + 3 \, b^{3}\right )} c^{2} x^{2} - 12 \, a b^{2} - 5 \, b^{3} + 2 \, {\left (4 \, a b^{2} + b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} + 3 \, {\left ({\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c^{2} x^{2} - 24 \, a^{2} b - 20 \, a b^{2} - 9 \, b^{3} + 2 \, {\left (8 \, a^{2} b + 4 \, a b^{2} + b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{128 \, {\left (c^{3} x^{2} + 2 \, c^{2} x + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 362, normalized size = 1.74 \[ \frac {1}{256} \, {\left (\frac {4 \, {\left (\frac {2 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - b^{3}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3}}{{\left (c x + 1\right )}^{2} c^{2}} + \frac {6 \, {\left (\frac {8 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} - 4 \, a b^{2} + \frac {4 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - b^{3}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{2} c^{2}} + \frac {6 \, {\left (\frac {16 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} - 8 \, a^{2} b + \frac {16 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} - 4 \, a b^{2} + \frac {8 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - b^{3}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{2} c^{2}} + \frac {{\left (\frac {64 \, {\left (c x + 1\right )} a^{3}}{c x - 1} - 32 \, a^{3} + \frac {96 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} - 24 \, a^{2} b + \frac {96 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} - 12 \, a b^{2} + \frac {48 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - 3 \, b^{3}\right )} {\left (c x - 1\right )}^{2}}{{\left (c x + 1\right )}^{2} c^{2}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.83, size = 2752, normalized size = 13.23 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 796, normalized size = 3.83 \[ -\frac {b^{3} \operatorname {artanh}\left (c x\right )^{3}}{2 \, {\left (c^{3} x^{2} + 2 \, c^{2} x + c\right )}} - \frac {3}{16} \, {\left (c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} x^{2} + 2 \, c^{3} x + c^{2}} - \frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} + \frac {8 \, \operatorname {artanh}\left (c x\right )}{c^{3} x^{2} + 2 \, c^{2} x + c}\right )} a^{2} b - \frac {3}{32} \, {\left (4 \, c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} x^{2} + 2 \, c^{3} x + c^{2}} - \frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left ({\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 6 \, c x - {\left (3 \, c^{2} x^{2} + 6 \, c x + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 3\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 8\right )} c^{2}}{c^{5} x^{2} + 2 \, c^{4} x + c^{3}}\right )} a b^{2} - \frac {1}{128} \, {\left (24 \, c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} x^{2} + 2 \, c^{3} x + c^{2}} - \frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} \operatorname {artanh}\left (c x\right )^{2} - {\left (\frac {{\left (2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{3} - 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{3} - 3 \, {\left (3 \, c^{2} x^{2} + 6 \, c x + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 3\right )} \log \left (c x + 1\right )^{2} - 9 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} - 42 \, c x + 3 \, {\left (7 \, c^{2} x^{2} + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 14 \, c x + 6 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 7\right )} \log \left (c x + 1\right ) - 21 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) - 48\right )} c^{2}}{c^{6} x^{2} + 2 \, c^{5} x + c^{4}} - \frac {12 \, {\left ({\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 6 \, c x - {\left (3 \, c^{2} x^{2} + 6 \, c x + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 3\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 8\right )} c \operatorname {artanh}\left (c x\right )}{c^{5} x^{2} + 2 \, c^{4} x + c^{3}}\right )} c\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {artanh}\left (c x\right )^{2}}{2 \, {\left (c^{3} x^{2} + 2 \, c^{2} x + c\right )}} - \frac {a^{3}}{2 \, {\left (c^{3} x^{2} + 2 \, c^{2} x + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.46, size = 930, normalized size = 4.47 \[ \frac {102\,b^3\,\ln \left (1-c\,x\right )-102\,b^3\,\ln \left (c\,x+1\right )-96\,a\,b^2-96\,a^2\,b-15\,b^3\,{\ln \left (c\,x+1\right )}^2-6\,b^3\,{\ln \left (c\,x+1\right )}^3-15\,b^3\,{\ln \left (1-c\,x\right )}^2+6\,b^3\,{\ln \left (1-c\,x\right )}^3+150\,b^3\,\mathrm {atanh}\left (c\,x\right )-64\,a^3-48\,b^3+144\,a\,b^2\,\mathrm {atanh}\left (c\,x\right )+48\,a^2\,b\,\mathrm {atanh}\left (c\,x\right )+30\,b^3\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-132\,a\,b^2\,\ln \left (c\,x+1\right )-96\,a^2\,b\,\ln \left (c\,x+1\right )+132\,a\,b^2\,\ln \left (1-c\,x\right )+96\,a^2\,b\,\ln \left (1-c\,x\right )-18\,b^3\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2+18\,b^3\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )-36\,a\,b^2\,{\ln \left (c\,x+1\right )}^2-36\,a\,b^2\,{\ln \left (1-c\,x\right )}^2-42\,b^3\,c\,x-144\,b^3\,c\,x\,\ln \left (c\,x+1\right )+144\,b^3\,c\,x\,\ln \left (1-c\,x\right )+9\,b^3\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2+2\,b^3\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^3+9\,b^3\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2-2\,b^3\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^3+150\,b^3\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )-72\,a\,b^2\,c\,x-48\,a^2\,b\,c\,x+6\,b^3\,c\,x\,{\ln \left (c\,x+1\right )}^2+4\,b^3\,c\,x\,{\ln \left (c\,x+1\right )}^3+6\,b^3\,c\,x\,{\ln \left (1-c\,x\right )}^2-4\,b^3\,c\,x\,{\ln \left (1-c\,x\right )}^3+72\,a\,b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+300\,b^3\,c\,x\,\mathrm {atanh}\left (c\,x\right )-54\,b^3\,c^2\,x^2\,\ln \left (c\,x+1\right )+54\,b^3\,c^2\,x^2\,\ln \left (1-c\,x\right )-12\,b^3\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-36\,a\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )+36\,a\,b^2\,c^2\,x^2\,\ln \left (1-c\,x\right )+6\,b^3\,c^2\,x^2\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2-6\,b^3\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )-120\,a\,b^2\,c\,x\,\ln \left (c\,x+1\right )+120\,a\,b^2\,c\,x\,\ln \left (1-c\,x\right )+12\,b^3\,c\,x\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2-12\,b^3\,c\,x\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )+12\,a\,b^2\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2+12\,a\,b^2\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2+144\,a\,b^2\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+48\,a^2\,b\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+24\,a\,b^2\,c\,x\,{\ln \left (c\,x+1\right )}^2+24\,a\,b^2\,c\,x\,{\ln \left (1-c\,x\right )}^2-18\,b^3\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+288\,a\,b^2\,c\,x\,\mathrm {atanh}\left (c\,x\right )+96\,a^2\,b\,c\,x\,\mathrm {atanh}\left (c\,x\right )-24\,a\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-48\,a\,b^2\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{128\,c\,{\left (c\,x+1\right )}^2} \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{\left (c x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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