Optimal. Leaf size=275 \[ -\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)^2}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (c x+1)^3}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (c x+1)^3}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (c x+1)^3}-\frac {85 b^3}{576 c (c x+1)}-\frac {19 b^3}{576 c (c x+1)^2}-\frac {b^3}{108 c (c x+1)^3}+\frac {85 b^3 \tanh ^{-1}(c x)}{576 c} \]
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Rubi [A] time = 0.61, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 42, 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} \[ -\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)^2}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (c x+1)^3}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (c x+1)^3}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (c x+1)^3}-\frac {85 b^3}{576 c (c x+1)}-\frac {19 b^3}{576 c (c x+1)^2}-\frac {b^3}{108 c (c x+1)^3}+\frac {85 b^3 \tanh ^{-1}(c x)}{576 c} \]
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)^4} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+b \int \left (\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 (1+c x)^4}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 (1+c x)^3}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 (1+c x)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{8} b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx-\frac {1}{8} b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{-1+c^2 x^2} \, dx+\frac {1}{4} b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^3} \, dx+\frac {1}{2} b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^4} \, dx\\ &=-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{4} b^2 \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} b^2 \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 {1}{3} b^2 \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^4}+\frac {a+b \tanh ^{-1}(c x)}{4 (1+c x)^3}+\frac {a+b \tanh ^{-1}(c x)}{8 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{8 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{24} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{24} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac {1}{16} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{16} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac {1}{12} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx+\frac {1}{8} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx+\frac {1}{8} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{8} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac {1}{6} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^4} \, dx\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{24} b^3 \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{24} b^3 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx+\frac {1}{18} b^3 \int \frac {1}{(1+c x)^3 \left (1-c^2 x^2\right )} \, dx+\frac {1}{16} b^3 \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{16} b^3 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx+\frac {1}{8} b^3 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{24} b^3 \int \frac {1}{(1-c x) (1+c x)^3} \, dx+\frac {1}{24} b^3 \int \frac {1}{(1-c x) (1+c x)^2} \, dx+\frac {1}{18} b^3 \int \frac {1}{(1-c x) (1+c x)^4} \, dx+\frac {1}{16} b^3 \int \frac {1}{(1-c x) (1+c x)^3} \, dx+\frac {1}{16} b^3 \int \frac {1}{(1-c x) (1+c x)^2} \, dx+\frac {1}{8} b^3 \int \frac {1}{(1-c x) (1+c x)^2} \, dx\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{24} b^3 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{24} b^3 \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}{18} b^3 \int \left (\frac {1}{2 (1+c x)^4}+\frac {1}{4 (1+c x)^3}+\frac {1}{8 (1+c x)^2}-\frac {1}{8 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{16} b^3 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{16} b^3 \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} b^3 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b^3}{108 c (1+c x)^3}-\frac {19 b^3}{576 c (1+c x)^2}-\frac {85 b^3}{576 c (1+c x)}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}-\frac {1}{144} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{96} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{64} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{48} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{32} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{16} b^3 \int \frac {1}{-1+c^2 x^2} \, dx\\ &=-\frac {b^3}{108 c (1+c x)^3}-\frac {19 b^3}{576 c (1+c x)^2}-\frac {85 b^3}{576 c (1+c x)}+\frac {85 b^3 \tanh ^{-1}(c x)}{576 c}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 279, normalized size = 1.01 \[ -\frac {24 b \tanh ^{-1}(c x) \left (144 a^2+12 a b \left (3 c^2 x^2+9 c x+10\right )+b^2 \left (33 c^2 x^2+81 c x+56\right )\right )+6 b \left (72 a^2+132 a b+85 b^2\right ) (c x+1)^2+6 b \left (72 a^2+60 a b+19 b^2\right ) (c x+1)+3 b \left (72 a^2+132 a b+85 b^2\right ) (c x+1)^3 \log (1-c x)-3 b \left (72 a^2+132 a b+85 b^2\right ) (c x+1)^3 \log (c x+1)+32 \left (36 a^3+18 a^2 b+6 a b^2+b^3\right )-36 b^2 (c x-1) \tanh ^{-1}(c x)^2 \left (12 a \left (c^2 x^2+4 c x+7\right )+b \left (11 c^2 x^2+32 c x+29\right )\right )-144 b^3 \left (c^3 x^3+3 c^2 x^2+3 c x-7\right ) \tanh ^{-1}(c x)^3}{3456 c (c x+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 345, normalized size = 1.25 \[ -\frac {6 \, {\left (72 \, a^{2} b + 132 \, a b^{2} + 85 \, b^{3}\right )} c^{2} x^{2} - 18 \, {\left (b^{3} c^{3} x^{3} + 3 \, b^{3} c^{2} x^{2} + 3 \, b^{3} c x - 7 \, b^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3} + 1152 \, a^{3} + 1440 \, a^{2} b + 1344 \, a b^{2} + 656 \, b^{3} + 162 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c x - 9 \, {\left ({\left (12 \, a b^{2} + 11 \, b^{3}\right )} c^{3} x^{3} + 3 \, {\left (12 \, a b^{2} + 7 \, b^{3}\right )} c^{2} x^{2} - 84 \, a b^{2} - 29 \, b^{3} + 3 \, {\left (12 \, a b^{2} - b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 3 \, {\left ({\left (72 \, a^{2} b + 132 \, a b^{2} + 85 \, b^{3}\right )} c^{3} x^{3} + 3 \, {\left (72 \, a^{2} b + 84 \, a b^{2} + 41 \, b^{3}\right )} c^{2} x^{2} - 504 \, a^{2} b - 348 \, a b^{2} - 139 \, b^{3} + 3 \, {\left (72 \, a^{2} b - 12 \, a b^{2} - 23 \, b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{3456 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 555, normalized size = 2.02 \[ \frac {1}{6912} \, {\left (\frac {36 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {3 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + b^{3}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3}}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {18 \, {\left (\frac {36 \, {\left (c x + 1\right )}^{2} a b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {36 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} + 12 \, a b^{2} + \frac {18 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {9 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + 2 \, b^{3}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {6 \, {\left (\frac {216 \, {\left (c x + 1\right )}^{2} a^{2} b}{{\left (c x - 1\right )}^{2}} - \frac {216 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} + 72 \, a^{2} b + \frac {216 \, {\left (c x + 1\right )}^{2} a b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {108 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} + 24 \, a b^{2} + \frac {108 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {27 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + 4 \, b^{3}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {{\left (\frac {864 \, {\left (c x + 1\right )}^{2} a^{3}}{{\left (c x - 1\right )}^{2}} - \frac {864 \, {\left (c x + 1\right )} a^{3}}{c x - 1} + 288 \, a^{3} + \frac {1296 \, {\left (c x + 1\right )}^{2} a^{2} b}{{\left (c x - 1\right )}^{2}} - \frac {648 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} + 144 \, a^{2} b + \frac {1296 \, {\left (c x + 1\right )}^{2} a b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {324 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} + 48 \, a b^{2} + \frac {648 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {81 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + 8 \, b^{3}\right )} {\left (c x - 1\right )}^{3}}{{\left (c x + 1\right )}^{3} c^{2}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.89, size = 3637, normalized size = 13.23 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 1085, normalized size = 3.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.49, size = 1304, normalized size = 4.74 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{\left (c x + 1\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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