Optimal. Leaf size=175 \[ \frac {1}{3} b c d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c}-\frac {8 b d^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+2 a b d^2 x+\frac {b^2 d^2 \log \left (1-c^2 x^2\right )}{c}-\frac {4 b^2 d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c}-\frac {b^2 d^2 \tanh ^{-1}(c x)}{3 c}+2 b^2 d^2 x \tanh ^{-1}(c x)+\frac {1}{3} b^2 d^2 x \]
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Rubi [A] time = 0.16, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {5928, 5910, 260, 5916, 321, 206, 1586, 5918, 2402, 2315} \[ -\frac {4 b^2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c}+\frac {1}{3} b c d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c}-\frac {8 b d^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+2 a b d^2 x+\frac {b^2 d^2 \log \left (1-c^2 x^2\right )}{c}-\frac {b^2 d^2 \tanh ^{-1}(c x)}{3 c}+2 b^2 d^2 x \tanh ^{-1}(c x)+\frac {1}{3} b^2 d^2 x \]
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
Rubi steps
\begin {align*} \int (d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c}-\frac {(2 b) \int \left (-3 d^3 \left (a+b \tanh ^{-1}(c x)\right )-c d^3 x \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 \left (d^3+c d^3 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}\right ) \, dx}{3 d}\\ &=\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c}-\frac {(8 b) \int \frac {\left (d^3+c d^3 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d}+\left (2 b d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\frac {1}{3} \left (2 b c d^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=2 a b d^2 x+\frac {1}{3} b c d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c}-\frac {(8 b) \int \frac {a+b \tanh ^{-1}(c x)}{\frac {1}{d^3}-\frac {c x}{d^3}} \, dx}{3 d}+\left (2 b^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx-\frac {1}{3} \left (b^2 c^2 d^2\right ) \int \frac {x^2}{1-c^2 x^2} \, dx\\ &=2 a b d^2 x+\frac {1}{3} b^2 d^2 x+2 b^2 d^2 x \tanh ^{-1}(c x)+\frac {1}{3} b c d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c}-\frac {8 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c}-\frac {1}{3} \left (b^2 d^2\right ) \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{3} \left (8 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b^2 c d^2\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=2 a b d^2 x+\frac {1}{3} b^2 d^2 x-\frac {b^2 d^2 \tanh ^{-1}(c x)}{3 c}+2 b^2 d^2 x \tanh ^{-1}(c x)+\frac {1}{3} b c d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c}-\frac {8 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c}+\frac {b^2 d^2 \log \left (1-c^2 x^2\right )}{c}-\frac {\left (8 b^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{3 c}\\ &=2 a b d^2 x+\frac {1}{3} b^2 d^2 x-\frac {b^2 d^2 \tanh ^{-1}(c x)}{3 c}+2 b^2 d^2 x \tanh ^{-1}(c x)+\frac {1}{3} b c d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c}-\frac {8 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c}+\frac {b^2 d^2 \log \left (1-c^2 x^2\right )}{c}-\frac {4 b^2 d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 227, normalized size = 1.30 \[ \frac {d^2 \left (a^2 c^3 x^3+3 a^2 c^2 x^2+3 a^2 c x+a b c^2 x^2+3 a b \log \left (1-c^2 x^2\right )+a b \log \left (c^2 x^2-1\right )+b \tanh ^{-1}(c x) \left (2 a c x \left (c^2 x^2+3 c x+3\right )+b \left (c^2 x^2+6 c x-1\right )-8 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+6 a b c x+3 a b \log (1-c x)-3 a b \log (c x+1)+3 b^2 \log \left (1-c^2 x^2\right )+b^2 \left (c^3 x^3+3 c^2 x^2+3 c x-7\right ) \tanh ^{-1}(c x)^2+4 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+b^2 c x\right )}{3 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} c^{2} d^{2} x^{2} + 2 \, a^{2} c d^{2} x + a^{2} d^{2} + {\left (b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d^{2} x^{2} + 2 \, a b c d^{2} x + a b d^{2}\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
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 \[ \int {\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 372, normalized size = 2.13 \[ \frac {b^{2} d^{2} x}{3}+2 c \,d^{2} a b \arctanh \left (c x \right ) x^{2}+\frac {2 c^{2} d^{2} a b \arctanh \left (c x \right ) x^{3}}{3}-\frac {4 d^{2} b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{3 c}+2 b^{2} d^{2} x \arctanh \left (c x \right )+d^{2} b^{2} \arctanh \left (c x \right )^{2} x +\frac {d^{2} b^{2} \arctanh \left (c x \right )^{2}}{3 c}+c \,d^{2} a^{2} x^{2}+\frac {c^{2} d^{2} a^{2} x^{3}}{3}+\frac {5 d^{2} b^{2} \ln \left (c x +1\right )}{6 c}+\frac {7 d^{2} b^{2} \ln \left (c x -1\right )}{6 c}+\frac {2 d^{2} b^{2} \ln \left (c x -1\right )^{2}}{3 c}-\frac {d^{2} b^{2}}{3 c}+\frac {d^{2} a^{2}}{3 c}+x \,a^{2} d^{2}+2 a b \,d^{2} x +\frac {c \,d^{2} b^{2} \arctanh \left (c x \right ) x^{2}}{3}+2 d^{2} a b \arctanh \left (c x \right ) x +\frac {8 d^{2} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{3 c}+\frac {2 d^{2} a b \arctanh \left (c x \right )}{3 c}+\frac {c^{2} d^{2} b^{2} \arctanh \left (c x \right )^{2} x^{3}}{3}+c \,d^{2} b^{2} \arctanh \left (c x \right )^{2} x^{2}+\frac {8 d^{2} a b \ln \left (c x -1\right )}{3 c}-\frac {4 d^{2} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{3 c}+\frac {c \,d^{2} a b \,x^{2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 464, normalized size = 2.65 \[ \frac {1}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b c^{2} d^{2} + a^{2} c d^{2} x^{2} + {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b c d^{2} + a^{2} d^{2} x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d^{2}}{c} + \frac {4 \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d^{2}}{3 \, c} + \frac {5 \, b^{2} d^{2} \log \left (c x + 1\right )}{6 \, c} + \frac {7 \, b^{2} d^{2} \log \left (c x - 1\right )}{6 \, c} + \frac {4 \, b^{2} c d^{2} x + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x - 7 \, b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \, {\left (b^{2} c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (b^{2} c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^2 \,d x \]
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 \[ d^{2} \left (\int a^{2}\, dx + \int b^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a^{2} c x\, dx + \int a^{2} c^{2} x^{2}\, dx + \int 2 b^{2} c x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 4 a b c x \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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