Optimal. Leaf size=188 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (c x+1)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}+\frac{b^2}{2 c^2 d^2 (c x+1)}-\frac{b^2 \tanh ^{-1}(c x)}{2 c^2 d^2} \]
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Rubi [A] time = 0.342551, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5940, 5928, 5926, 627, 44, 207, 5948, 5918, 6056, 6610} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (c x+1)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}+\frac{b^2}{2 c^2 d^2 (c x+1)}-\frac{b^2 \tanh ^{-1}(c x)}{2 c^2 d^2} \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5928
Rule 5926
Rule 627
Rule 44
Rule 207
Rule 5948
Rule 5918
Rule 6056
Rule 6610
Rubi steps
\begin{align*} \int \frac{x \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 d^2 (1+c x)^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c d^2 (1+c x)}\right ) \, dx\\ &=-\frac{\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{c d^2}+\frac{\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{c d^2}\\ &=\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^2 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 d^2}+\frac{(2 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 d^2}\\ &=\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^2 d^2}-\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c d^2}+\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c d^2}-\frac{b^2 \int \frac{\text{Li}_2\left (1-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c d^2}\\ &=\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^2 d^2}-\frac{b^2 \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c d^2}\\ &=\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^2 d^2}-\frac{b^2 \int \frac{1}{(1-c x) (1+c x)^2} \, dx}{c d^2}\\ &=\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^2 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 d^2}\\ &=\frac{b^2}{2 c^2 d^2 (1+c x)}+\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^2 d^2}+\frac{b^2 \int \frac{1}{-1+c^2 x^2} \, dx}{2 c d^2}\\ &=\frac{b^2}{2 c^2 d^2 (1+c x)}-\frac{b^2 \tanh ^{-1}(c x)}{2 c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^2 d^2}\\ \end{align*}
Mathematica [A] time = 0.625016, size = 233, normalized size = 1.24 \[ \frac{2 a b \left (2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \left (-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )\right )\right )+b^2 \left (4 \tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+2 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-4 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-2 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )-2 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )\right )+\frac{4 a^2}{c x+1}+4 a^2 \log (c x+1)}{4 c^2 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.277, size = 1030, normalized size = 5.5 \begin{align*} \text{result 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^{3} d^{2} x + c^{2} d^{2}} + \frac{\log \left (c x + 1\right )}{c^{2} d^{2}}\right )} + \frac{{\left (b^{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^{3} d^{2} x + c^{2} d^{2}\right )}} - \int -\frac{{\left (b^{2} c^{2} x^{2} - b^{2} c x\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b c^{2} x^{2} - a b c x\right )} \log \left (c x + 1\right ) - 2 \,{\left (2 \, a b c^{2} x^{2} + b^{2} -{\left (2 \, a b c - b^{2} c\right )} x +{\left (2 \, b^{2} c^{2} x^{2} + b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \,{\left (c^{4} d^{2} x^{3} + c^{3} d^{2} x^{2} - c^{2} d^{2} x - c 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 \operatorname{artanh}\left (c x\right )^{2} + 2 \, a b x \operatorname{artanh}\left (c x\right ) + a^{2} x}{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}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac{b^{2} x \operatorname{atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac{2 a b x \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}{{\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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