Optimal. Leaf size=260 \[ -\frac{2 b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2}-\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c^3 d^2}-\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{c^3 d^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2 (c x+1)}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2 (c x+1)}+\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^2}-\frac{2 b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2}+\frac{2 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{b^2}{2 c^3 d^2 (c x+1)}+\frac{b^2 \tanh ^{-1}(c x)}{2 c^3 d^2} \]
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Rubi [A] time = 0.475892, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {5940, 5910, 5984, 5918, 2402, 2315, 5928, 5926, 627, 44, 207, 5948, 6056, 6610} \[ -\frac{2 b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2}-\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c^3 d^2}-\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{c^3 d^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2 (c x+1)}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2 (c x+1)}+\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^2}-\frac{2 b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2}+\frac{2 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{b^2}{2 c^3 d^2 (c x+1)}+\frac{b^2 \tanh ^{-1}(c x)}{2 c^3 d^2} \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5910
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 5928
Rule 5926
Rule 627
Rule 44
Rule 207
Rule 5948
Rule 6056
Rule 6610
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{(d+c d x)^2} \, dx &=\int \left (\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)^2}-\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}\right ) \, dx\\ &=\frac{\int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^2 d^2}+\frac{\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{c^2 d^2}-\frac{2 \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{c^2 d^2}\\ &=\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2 (1+c x)}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^2}+\frac{(2 b) \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}{c^2 d^2}-\frac{(4 b) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^2 d^2}-\frac{(2 b) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d^2}\\ &=\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2 (1+c x)}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}+\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^2 d^2}-\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^2 d^2}-\frac{(2 b) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c^2 d^2}+\frac{\left (2 b^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^2 d^2}\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2 (1+c x)}+\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^2}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2 (1+c x)}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^3 d^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}+\frac{b^2 \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^2 d^2}+\frac{\left (2 b^2\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2 d^2}\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2 (1+c x)}+\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^2}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2 (1+c x)}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^3 d^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )}{c^3 d^2}+\frac{b^2 \int \frac{1}{(1-c x) (1+c x)^2} \, dx}{c^2 d^2}\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2 (1+c x)}+\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^2}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2 (1+c x)}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^3 d^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3 d^2}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}+\frac{b^2 \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^2 d^2}\\ &=-\frac{b^2}{2 c^3 d^2 (1+c x)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2 (1+c x)}+\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^2}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2 (1+c x)}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^3 d^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3 d^2}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{b^2 \int \frac{1}{-1+c^2 x^2} \, dx}{2 c^2 d^2}\\ &=-\frac{b^2}{2 c^3 d^2 (1+c x)}+\frac{b^2 \tanh ^{-1}(c x)}{2 c^3 d^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2 (1+c x)}+\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^2}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2 (1+c x)}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^3 d^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3 d^2}-\frac{2 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^3 d^2}\\ \end{align*}
Mathematica [A] time = 0.918927, size = 295, normalized size = 1.13 \[ \frac{2 a b \left (-4 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+2 \log \left (1-c^2 x^2\right )+\sinh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \left (2 c x+4 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+\sinh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (2 \tanh ^{-1}(c x)\right )\right )\right )+b^2 \left (\left (4-8 \tanh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-4 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+4 c x \tanh ^{-1}(c x)^2-4 \tanh ^{-1}(c x)^2+8 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-8 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+2 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (2 \tanh ^{-1}(c x)\right )-2 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )-2 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (2 \tanh ^{-1}(c x)\right )\right )+4 a^2 c x-\frac{4 a^2}{c x+1}-8 a^2 \log (c x+1)}{4 c^3 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.507, size = 5542, normalized size = 21.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -a^{2}{\left (\frac{1}{c^{4} d^{2} x + c^{3} d^{2}} - \frac{x}{c^{2} d^{2}} + \frac{2 \, \log \left (c x + 1\right )}{c^{3} d^{2}}\right )} + \frac{{\left (b^{2} c^{2} x^{2} + b^{2} c x - b^{2} - 2 \,{\left (b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{4 \,{\left (c^{4} d^{2} x + c^{3} d^{2}\right )}} - \int -\frac{{\left (b^{2} c^{3} x^{3} - b^{2} c^{2} x^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b c^{3} x^{3} - a b c^{2} x^{2}\right )} \log \left (c x + 1\right ) - 2 \,{\left ({\left (2 \, a b c^{3} + b^{2} c^{3}\right )} x^{3} - 2 \,{\left (a b c^{2} - b^{2} c^{2}\right )} x^{2} - b^{2} +{\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c^{2} x^{2} - 4 \, b^{2} c x - 2 \, b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \,{\left (c^{5} d^{2} x^{3} + c^{4} d^{2} x^{2} - c^{3} d^{2} x - c^{2} d^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{2} \operatorname{artanh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{artanh}\left (c x\right ) + a^{2} x^{2}}{c^{2} d^{2} x^{2} + 2 \, c d^{2} x + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{2} x^{2}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac{b^{2} x^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac{2 a b x^{2} \operatorname{atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c d x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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