Optimal. Leaf size=112 \[ a b d x+b^2 d x \tanh ^{-1}(c x)+\frac {d (1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c}-\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c} \]
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Rubi [A]
time = 0.09, 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 = {6065, 6021,
266, 1600, 6055, 2449, 2352} \begin {gather*} \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) \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 1600
Rule 2352
Rule 2449
Rule 6021
Rule 6055
Rule 6065
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 ) \text {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 = 0.18, size = 156, normalized size = 1.39 \begin {gather*} \frac {d \left (2 a^2 c x+2 a b c x+a^2 c^2 x^2+b^2 \left (-3+2 c x+c^2 x^2\right ) \tanh ^{-1}(c x)^2+2 b \tanh ^{-1}(c x) \left (c x (2 a+b+a c x)-2 b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+a b \log (1-c x)-a b \log (1+c x)+2 a b \log \left (1-c^2 x^2\right )+b^2 \log \left (1-c^2 x^2\right )+2 b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(272\) vs.
\(2(108)=216\).
time = 0.20, size = 273, normalized size = 2.44
method | result | size |
derivativedivides | \(\frac {d \,a^{2} \left (\frac {1}{2} c^{2} x^{2}+c x \right )+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{2} x^{2}}{2}+d \,b^{2} \arctanh \left (c x \right )^{2} c x +d \,b^{2} \arctanh \left (c x \right ) c x +\frac {3 d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{2}+\frac {d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}+\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{4}-d \,b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )-\frac {d \,b^{2} \ln \left (c x +1\right )^{2}}{8}+\frac {d \,b^{2} \ln \left (c x -1\right )}{2}+\frac {d \,b^{2} \ln \left (c x +1\right )}{2}-\frac {3 d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}+\frac {3 d \,b^{2} \ln \left (c x -1\right )^{2}}{8}+d a b \arctanh \left (c x \right ) c^{2} x^{2}+2 d a b \arctanh \left (c x \right ) c x +d a b c x +\frac {3 d a b \ln \left (c x -1\right )}{2}+\frac {d a b \ln \left (c x +1\right )}{2}}{c}\) | \(273\) |
default | \(\frac {d \,a^{2} \left (\frac {1}{2} c^{2} x^{2}+c x \right )+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{2} x^{2}}{2}+d \,b^{2} \arctanh \left (c x \right )^{2} c x +d \,b^{2} \arctanh \left (c x \right ) c x +\frac {3 d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{2}+\frac {d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}+\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{4}-d \,b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )-\frac {d \,b^{2} \ln \left (c x +1\right )^{2}}{8}+\frac {d \,b^{2} \ln \left (c x -1\right )}{2}+\frac {d \,b^{2} \ln \left (c x +1\right )}{2}-\frac {3 d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}+\frac {3 d \,b^{2} \ln \left (c x -1\right )^{2}}{8}+d a b \arctanh \left (c x \right ) c^{2} x^{2}+2 d a b \arctanh \left (c x \right ) c x +d a b c x +\frac {3 d a b \ln \left (c x -1\right )}{2}+\frac {d a b \ln \left (c x +1\right )}{2}}{c}\) | \(273\) |
risch | \(\frac {d \,b^{2} \left (c^{2} x^{2}+2 c x +1\right ) \ln \left (c x +1\right )^{2}}{8 c}+\left (-\frac {d \,b^{2} x \left (c x +2\right ) \ln \left (-c x +1\right )}{4}+\frac {d b \left (2 a \,c^{2} x^{2}+4 c x a +2 b c x +3 b \ln \left (-c x +1\right )\right )}{4 c}\right ) \ln \left (c x +1\right )+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d}{c}-\frac {b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right ) d}{c}+\frac {b \ln \left (-c x -1\right ) a d}{2 c}-\frac {3 \ln \left (-c x +1\right )^{2} b^{2} d}{8 c}+\frac {7 \ln \left (-c x +1\right ) b^{2} d}{8 c}-\frac {3 \ln \left (-c x +1\right ) x \,b^{2} d}{4}+\frac {\ln \left (-c x +1\right )^{2} x \,b^{2} d}{4}+a^{2} d x +\frac {3 \ln \left (-c x +1\right ) a b d}{2 c}-\ln \left (-c x +1\right ) x a b d +\frac {d c \,x^{2} a^{2}}{2}-\frac {b^{2} \left (-c x +1\right ) \ln \left (-c x +1\right ) d}{2 c}-\frac {3 d \,a^{2}}{2 c}+a b d x +\frac {d \,b^{2} \ln \left (-c x +1\right ) \left (-c x +1\right )^{2}}{8 c}-\frac {d c a b \ln \left (-c x +1\right ) x^{2}}{2}+\frac {b^{2} \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right ) d}{c}-\frac {d a b}{c}+\frac {d c \,b^{2} \ln \left (-c x +1\right )^{2} x^{2}}{8}-\frac {d c \,b^{2} \ln \left (-c x +1\right ) x^{2}}{8}+\frac {d \,b^{2} \ln \left (-c x -1\right )}{2 c}\) | \(415\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 290 vs.
\(2 (105) = 210\).
time = 0.41, size = 290, normalized size = 2.59 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} 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 ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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