Optimal. Leaf size=71 \[ \frac{d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac{b d^2 (c x+1)^2}{6 c}+\frac{4 b d^2 \log (1-c x)}{3 c}+\frac{2}{3} b d^2 x \]
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Rubi [A] time = 0.0425078, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5926, 627, 43} \[ \frac{d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac{b d^2 (c x+1)^2}{6 c}+\frac{4 b d^2 \log (1-c x)}{3 c}+\frac{2}{3} b d^2 x \]
Antiderivative was successfully verified.
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Rule 5926
Rule 627
Rule 43
Rubi steps
\begin{align*} \int (d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}-\frac{b \int \frac{(d+c d x)^3}{1-c^2 x^2} \, dx}{3 d}\\ &=\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}-\frac{b \int \frac{(d+c d x)^2}{\frac{1}{d}-\frac{c x}{d}} \, dx}{3 d}\\ &=\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}-\frac{b \int \left (-2 d^3+\frac{4 d^2}{\frac{1}{d}-\frac{c x}{d}}-d^2 (d+c d x)\right ) \, dx}{3 d}\\ &=\frac{2}{3} b d^2 x+\frac{b d^2 (1+c x)^2}{6 c}+\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac{4 b d^2 \log (1-c x)}{3 c}\\ \end{align*}
Mathematica [A] time = 0.10513, size = 92, normalized size = 1.3 \[ \frac{d^2 \left (2 a c^3 x^3+6 a c^2 x^2+6 a c x+b c^2 x^2+b \log \left (1-c^2 x^2\right )+2 b c x \left (c^2 x^2+3 c x+3\right ) \tanh ^{-1}(c x)+6 b c x+6 b \log (1-c x)\right )}{6 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 121, normalized size = 1.7 \begin{align*}{\frac{{c}^{2}{x}^{3}a{d}^{2}}{3}}+c{x}^{2}a{d}^{2}+ax{d}^{2}+{\frac{{d}^{2}a}{3\,c}}+{\frac{{c}^{2}{d}^{2}b{\it Artanh} \left ( cx \right ){x}^{3}}{3}}+c{d}^{2}b{\it Artanh} \left ( cx \right ){x}^{2}+{d}^{2}b{\it Artanh} \left ( cx \right ) x+{\frac{{d}^{2}b{\it Artanh} \left ( cx \right ) }{3\,c}}+{\frac{c{d}^{2}b{x}^{2}}{6}}+b{d}^{2}x+{\frac{4\,{d}^{2}b\ln \left ( cx-1 \right ) }{3\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.959171, size = 198, normalized size = 2.79 \begin{align*} \frac{1}{3} \, a c^{2} d^{2} x^{3} + \frac{1}{6} \,{\left (2 \, x^{3} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b c^{2} d^{2} + a c d^{2} 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 )} b c d^{2} + a d^{2} x + \frac{{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{2}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03953, size = 255, normalized size = 3.59 \begin{align*} \frac{2 \, a c^{3} d^{2} x^{3} +{\left (6 \, a + b\right )} c^{2} d^{2} x^{2} + 6 \,{\left (a + b\right )} c d^{2} x + b d^{2} \log \left (c x + 1\right ) + 7 \, b d^{2} \log \left (c x - 1\right ) +{\left (b c^{3} d^{2} x^{3} + 3 \, b c^{2} d^{2} x^{2} + 3 \, b c d^{2} x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.54826, size = 131, normalized size = 1.85 \begin{align*} \begin{cases} \frac{a c^{2} d^{2} x^{3}}{3} + a c d^{2} x^{2} + a d^{2} x + \frac{b c^{2} d^{2} x^{3} \operatorname{atanh}{\left (c x \right )}}{3} + b c d^{2} x^{2} \operatorname{atanh}{\left (c x \right )} + \frac{b c d^{2} x^{2}}{6} + b d^{2} x \operatorname{atanh}{\left (c x \right )} + b d^{2} x + \frac{4 b d^{2} \log{\left (x - \frac{1}{c} \right )}}{3 c} + \frac{b d^{2} \operatorname{atanh}{\left (c x \right )}}{3 c} & \text{for}\: c \neq 0 \\a d^{2} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21788, size = 163, normalized size = 2.3 \begin{align*} \frac{1}{3} \, a c^{2} d^{2} x^{3} + \frac{1}{6} \,{\left (6 \, a c d^{2} + b c d^{2}\right )} x^{2} + \frac{b d^{2} \log \left (c x + 1\right )}{6 \, c} + \frac{7 \, b d^{2} \log \left (c x - 1\right )}{6 \, c} +{\left (a d^{2} + b d^{2}\right )} x + \frac{1}{6} \,{\left (b c^{2} d^{2} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b d^{2} x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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