Optimal. Leaf size=112 \[ \frac {d (c x+1)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+a b d x+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+b^2 d x \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.12, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5928, 5910, 260, 1586, 5918, 2402, 2315} \[ -\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {d (c x+1)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+a b d x+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c}+b^2 d x \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 260
Rule 1586
Rule 2315
Rule 2402
Rule 5910
Rule 5918
Rule 5928
Rubi steps
\begin {align*} \int (d+c d x) \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\frac {d (1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {b \int \left (-d^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {2 \left (d^2+c d^2 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}\right ) \, dx}{d}\\ &=\frac {d (1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {(2 b) \int \frac {\left (d^2+c d^2 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d}+(b d) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=a b d x+\frac {d (1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{\frac {1}{d^2}-\frac {c x}{d^2}} \, dx}{d}+\left (b^2 d\right ) \int \tanh ^{-1}(c x) \, dx\\ &=a b d x+b^2 d x \tanh ^{-1}(c x)+\frac {d (1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}+\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c d\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=a b d x+b^2 d x \tanh ^{-1}(c x)+\frac {d (1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c}-\frac {\left (2 b^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c}\\ &=a b d x+b^2 d x \tanh ^{-1}(c x)+\frac {d (1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 156, normalized size = 1.39 \[ \frac {d \left (a^2 c^2 x^2+2 a^2 c x+2 a b \log \left (1-c^2 x^2\right )+2 a b c x+a b \log (1-c x)-a b \log (c x+1)+2 b \tanh ^{-1}(c x) \left (c x (a c x+2 a+b)-2 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+b^2 \log \left (1-c^2 x^2\right )+b^2 \left (c^2 x^2+2 c x-3\right ) \tanh ^{-1}(c x)^2+2 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{2 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} c d x + a^{2} d + {\left (b^{2} c d x + b^{2} d\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c d x + a b d\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
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 \[ \int {\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 296, normalized size = 2.64 \[ \frac {c \,a^{2} d \,x^{2}}{2}+a^{2} d x +\frac {c d \,b^{2} \arctanh \left (c x \right )^{2} x^{2}}{2}+d \,b^{2} \arctanh \left (c x \right )^{2} x +b^{2} d x \arctanh \left (c x \right )+\frac {3 d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{2 c}+\frac {d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{2 c}-\frac {b^{2} \ln \left (c x +1\right )^{2} d}{8 c}+\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{4 c}-\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{4 c}-\frac {d \,b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{c}+\frac {d \,b^{2} \ln \left (c x -1\right )}{2 c}+\frac {d \,b^{2} \ln \left (c x +1\right )}{2 c}+\frac {3 d \,b^{2} \ln \left (c x -1\right )^{2}}{8 c}-\frac {3 d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{4 c}+c d a b \arctanh \left (c x \right ) x^{2}+2 d a b \arctanh \left (c x \right ) x +a b d x +\frac {3 d a b \ln \left (c x -1\right )}{2 c}+\frac {d a b \ln \left (c x +1\right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 290, normalized size = 2.59 \[ \frac {1}{2} \, a^{2} c d x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b c d + a^{2} d x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d}{c} + \frac {{\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d}{c} + \frac {b^{2} d \log \left (c x + 1\right )}{2 \, c} + \frac {b^{2} d \log \left (c x - 1\right )}{2 \, c} + \frac {4 \, b^{2} c d x \log \left (c x + 1\right ) + {\left (b^{2} c^{2} d x^{2} + 2 \, b^{2} c d x + b^{2} d\right )} \log \left (c x + 1\right )^{2} + {\left (b^{2} c^{2} d x^{2} + 2 \, b^{2} c d x - 3 \, b^{2} d\right )} \log \left (-c x + 1\right )^{2} - 2 \, {\left (2 \, b^{2} c d x + {\left (b^{2} c^{2} d x^{2} + 2 \, b^{2} c d x + b^{2} d\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, c} \]
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 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right ) \,d x \]
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 \[ d \left (\int a^{2}\, dx + \int b^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b \operatorname {atanh}{\left (c x \right )}\, dx + \int a^{2} c x\, dx + \int b^{2} c x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b c x \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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