Optimal. Leaf size=206 \[ \frac {1}{6} b c^2 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+b c d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {4 b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {7}{2} a b d^3 x+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{6 c}-\frac {2 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {1}{12} b^2 c d^3 x^2-\frac {b^2 d^3 \tanh ^{-1}(c x)}{c}+\frac {7}{2} b^2 d^3 x \tanh ^{-1}(c x)+b^2 d^3 x \]
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Rubi [A] time = 0.21, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {5928, 5910, 260, 5916, 321, 206, 266, 43, 1586, 5918, 2402, 2315} \[ -\frac {2 b^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {1}{6} b c^2 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+b c d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {4 b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {7}{2} a b d^3 x+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{6 c}+\frac {1}{12} b^2 c d^3 x^2-\frac {b^2 d^3 \tanh ^{-1}(c x)}{c}+\frac {7}{2} b^2 d^3 x \tanh ^{-1}(c x)+b^2 d^3 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 206
Rule 260
Rule 266
Rule 321
Rule 1586
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5928
Rubi steps
\begin {align*} \int (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {b \int \left (-7 d^4 \left (a+b \tanh ^{-1}(c x)\right )-4 c d^4 x \left (a+b \tanh ^{-1}(c x)\right )-c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {8 \left (d^4+c d^4 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}\right ) \, dx}{2 d}\\ &=\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {(4 b) \int \frac {\left (d^4+c d^4 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d}+\frac {1}{2} \left (7 b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (2 b c d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\frac {1}{2} \left (b c^2 d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=\frac {7}{2} a b d^3 x+b c d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} b c^2 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {(4 b) \int \frac {a+b \tanh ^{-1}(c x)}{\frac {1}{d^4}-\frac {c x}{d^4}} \, dx}{d}+\frac {1}{2} \left (7 b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx-\left (b^2 c^2 d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {1}{6} \left (b^2 c^3 d^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx\\ &=\frac {7}{2} a b d^3 x+b^2 d^3 x+\frac {7}{2} b^2 d^3 x \tanh ^{-1}(c x)+b c d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} b c^2 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {4 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\left (b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx+\left (4 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\frac {1}{2} \left (7 b^2 c d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx-\frac {1}{12} \left (b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac {7}{2} a b d^3 x+b^2 d^3 x-\frac {b^2 d^3 \tanh ^{-1}(c x)}{c}+\frac {7}{2} b^2 d^3 x \tanh ^{-1}(c x)+b c d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} b c^2 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {4 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}+\frac {7 b^2 d^3 \log \left (1-c^2 x^2\right )}{4 c}-\frac {\left (4 b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c}-\frac {1}{12} \left (b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {7}{2} a b d^3 x+b^2 d^3 x+\frac {1}{12} b^2 c d^3 x^2-\frac {b^2 d^3 \tanh ^{-1}(c x)}{c}+\frac {7}{2} b^2 d^3 x \tanh ^{-1}(c x)+b c d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} b c^2 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {4 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{6 c}-\frac {2 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 293, normalized size = 1.42 \[ \frac {d^3 \left (3 a^2 c^4 x^4+12 a^2 c^3 x^3+18 a^2 c^2 x^2+12 a^2 c x+2 a b c^3 x^3+12 a b c^2 x^2+12 a b \log \left (1-c^2 x^2\right )+12 a b \log \left (c^2 x^2-1\right )+2 b \tanh ^{-1}(c x) \left (3 a c x \left (c^3 x^3+4 c^2 x^2+6 c x+4\right )+b \left (c^3 x^3+6 c^2 x^2+21 c x-6\right )-24 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+42 a b c x+21 a b \log (1-c x)-21 a b \log (c x+1)+b^2 c^2 x^2+22 b^2 \log \left (1-c^2 x^2\right )+3 b^2 \left (c^4 x^4+4 c^3 x^3+6 c^2 x^2+4 c x-15\right ) \tanh ^{-1}(c x)^2+24 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+12 b^2 c x-b^2\right )}{12 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} c^{3} d^{3} x^{3} + 3 \, a^{2} c^{2} d^{3} x^{2} + 3 \, a^{2} c d^{3} x + a^{2} d^{3} + {\left (b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} + 3 \, b^{2} c d^{3} x + b^{2} d^{3}\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c^{3} d^{3} x^{3} + 3 \, a b c^{2} d^{3} x^{2} + 3 \, a b c d^{3} x + a b d^{3}\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
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 \[ \int {\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 462, normalized size = 2.24 \[ b^{2} d^{3} x +\frac {c^{3} d^{3} a b \arctanh \left (c x \right ) x^{4}}{2}+d^{3} b^{2} \arctanh \left (c x \right )^{2} x +\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{4 c}-\frac {2 d^{3} b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{c}+\frac {7 b^{2} d^{3} x \arctanh \left (c x \right )}{2}+\frac {3 c \,d^{3} a^{2} x^{2}}{2}+c^{2} d^{3} a^{2} x^{3}+\frac {d^{3} b^{2} \ln \left (c x -1\right )^{2}}{c}+\frac {c^{3} d^{3} a^{2} x^{4}}{4}+\frac {7 d^{3} b^{2} \ln \left (c x -1\right )}{3 c}+\frac {4 d^{3} b^{2} \ln \left (c x +1\right )}{3 c}+3 c \,d^{3} a b \arctanh \left (c x \right ) x^{2}+2 c^{2} d^{3} a b \arctanh \left (c x \right ) x^{3}-\frac {13 d^{3} b^{2}}{12 c}+\frac {d^{3} a^{2}}{4 c}+\frac {4 d^{3} a b \ln \left (c x -1\right )}{c}+\frac {4 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{c}+\frac {d^{3} a b \arctanh \left (c x \right )}{2 c}+c^{2} d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{3}+2 d^{3} a b \arctanh \left (c x \right ) x +\frac {c^{2} d^{3} b^{2} \arctanh \left (c x \right ) x^{3}}{6}+c \,d^{3} b^{2} \arctanh \left (c x \right ) x^{2}+\frac {c^{3} d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{4}}{4}+\frac {3 c \,d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{2}}{2}+a^{2} x \,d^{3}+\frac {c^{2} d^{3} a b \,x^{3}}{6}+c \,d^{3} a b \,x^{2}-\frac {2 d^{3} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{c}+\frac {7 a b \,d^{3} x}{2}+\frac {b^{2} c \,d^{3} x^{2}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 627, normalized size = 3.04 \[ \frac {1}{4} \, a^{2} c^{3} d^{3} x^{4} + a^{2} c^{2} d^{3} x^{3} + \frac {1}{12} \, {\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 b c^{3} d^{3} + {\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 b c^{2} d^{3} + \frac {3}{2} \, a^{2} c d^{3} x^{2} + \frac {3}{2} \, {\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 b c d^{3} + a^{2} d^{3} x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d^{3}}{c} + \frac {2 \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d^{3}}{c} + \frac {4 \, b^{2} d^{3} \log \left (c x + 1\right )}{3 \, c} + \frac {7 \, b^{2} d^{3} \log \left (c x - 1\right )}{3 \, c} + \frac {4 \, b^{2} c^{2} d^{3} x^{2} + 48 \, b^{2} c d^{3} x + 3 \, {\left (b^{2} c^{4} d^{3} x^{4} + 4 \, b^{2} c^{3} d^{3} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c d^{3} x + b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (b^{2} c^{4} d^{3} x^{4} + 4 \, b^{2} c^{3} d^{3} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c d^{3} x - 15 \, b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (b^{2} c^{3} d^{3} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 21 \, b^{2} c d^{3} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (2 \, b^{2} c^{3} d^{3} x^{3} + 12 \, b^{2} c^{2} d^{3} x^{2} + 42 \, b^{2} c d^{3} x + 3 \, {\left (b^{2} c^{4} d^{3} x^{4} + 4 \, b^{2} c^{3} d^{3} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c d^{3} x + b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{48 \, c} \]
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 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3 \,d x \]
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 \[ d^{3} \left (\int a^{2}\, dx + \int b^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b \operatorname {atanh}{\left (c x \right )}\, dx + \int 3 a^{2} c x\, dx + \int 3 a^{2} c^{2} x^{2}\, dx + \int a^{2} c^{3} x^{3}\, dx + \int 3 b^{2} c x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x \operatorname {atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{3} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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