Optimal. Leaf size=160 \[ -\frac{b^2 d \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c}-\frac{\left (\frac{e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac{2 b d \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac{a b e x}{c}+\frac{b^2 e \log \left (1-c^2 x^2\right )}{2 c^2}+\frac{b^2 e x \tanh ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.329908, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5928, 5910, 260, 6048, 5948, 5984, 5918, 2402, 2315} \[ -\frac{b^2 d \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c}-\frac{\left (\frac{e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac{2 b d \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac{a b e x}{c}+\frac{b^2 e \log \left (1-c^2 x^2\right )}{2 c^2}+\frac{b^2 e x \tanh ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 5928
Rule 5910
Rule 260
Rule 6048
Rule 5948
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int (d+e x) \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac{(b c) \int \left (-\frac{e^2 \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac{\left (c^2 d^2+e^2+2 c^2 d e x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 \left (1-c^2 x^2\right )}\right ) \, dx}{e}\\ &=\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac{b \int \frac{\left (c^2 d^2+e^2+2 c^2 d e x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c e}+\frac{(b e) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}\\ &=\frac{a b e x}{c}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac{b \int \left (\frac{c^2 d^2 \left (1+\frac{e^2}{c^2 d^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}+\frac{2 c^2 d e x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}\right ) \, dx}{c e}+\frac{\left (b^2 e\right ) \int \tanh ^{-1}(c x) \, dx}{c}\\ &=\frac{a b e x}{c}+\frac{b^2 e x \tanh ^{-1}(c x)}{c}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-(2 b c d) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (b^2 e\right ) \int \frac{x}{1-c^2 x^2} \, dx-\frac{\left (b \left (c^2 d^2+e^2\right )\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c e}\\ &=\frac{a b e x}{c}+\frac{b^2 e x \tanh ^{-1}(c x)}{c}+\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac{\left (d^2+\frac{e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac{b^2 e \log \left (1-c^2 x^2\right )}{2 c^2}-(2 b d) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx\\ &=\frac{a b e x}{c}+\frac{b^2 e x \tanh ^{-1}(c x)}{c}+\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac{\left (d^2+\frac{e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac{2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c}+\frac{b^2 e \log \left (1-c^2 x^2\right )}{2 c^2}+\left (2 b^2 d\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac{a b e x}{c}+\frac{b^2 e x \tanh ^{-1}(c x)}{c}+\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac{\left (d^2+\frac{e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac{2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c}+\frac{b^2 e \log \left (1-c^2 x^2\right )}{2 c^2}-\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}\\ &=\frac{a b e x}{c}+\frac{b^2 e x \tanh ^{-1}(c x)}{c}+\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac{\left (d^2+\frac{e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac{2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c}+\frac{b^2 e \log \left (1-c^2 x^2\right )}{2 c^2}-\frac{b^2 d \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.423255, size = 174, normalized size = 1.09 \[ \frac{2 b^2 c d \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+2 a^2 c^2 d x+a^2 c^2 e x^2+2 a b c d \log \left (1-c^2 x^2\right )+2 b c \tanh ^{-1}(c x) \left (a c x (2 d+e x)-2 b d \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+b e x\right )+2 a b c e x+a b e \log (1-c x)-a b e \log (c x+1)+b^2 e \log \left (1-c^2 x^2\right )+b^2 (c x-1) \tanh ^{-1}(c x)^2 (2 c d+c e x+e)}{2 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.052, size = 462, normalized size = 2.9 \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 [B] time = 1.76164, size = 423, normalized size = 2.64 \begin{align*} \frac{1}{2} \, a^{2} e 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 e + 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} e \log \left (c x + 1\right )}{2 \, c^{2}} + \frac{b^{2} e \log \left (c x - 1\right )}{2 \, c^{2}} + \frac{4 \, b^{2} c e x \log \left (c x + 1\right ) +{\left (b^{2} c^{2} e x^{2} + 2 \, b^{2} c^{2} d x +{\left (2 \, c d - e\right )} b^{2}\right )} \log \left (c x + 1\right )^{2} +{\left (b^{2} c^{2} e x^{2} + 2 \, b^{2} c^{2} d x -{\left (2 \, c d + e\right )} b^{2}\right )} \log \left (-c x + 1\right )^{2} - 2 \,{\left (2 \, b^{2} c e x +{\left (b^{2} c^{2} e x^{2} + 2 \, b^{2} c^{2} d x +{\left (2 \, c d - e\right )} b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, c^{2}} \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 (a^{2} e x + a^{2} d +{\left (b^{2} e x + b^{2} d\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b e x + a b d\right )} \operatorname{artanh}\left (c x\right ), 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*} \int \left (a + b \operatorname{atanh}{\left (c x \right )}\right )^{2} \left (d + e x\right )\, dx \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{\left (e x + d\right )}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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