Optimal. Leaf size=84 \[ \frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{5 b d \log (1-c x)}{12 c^2}-\frac{b d \log (c x+1)}{12 c^2}+\frac{b d x}{2 c}+\frac{1}{6} b d x^2 \]
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Rubi [A] time = 0.074793, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {43, 5936, 12, 801, 633, 31} \[ \frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{5 b d \log (1-c x)}{12 c^2}-\frac{b d \log (c x+1)}{12 c^2}+\frac{b d x}{2 c}+\frac{1}{6} b d x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 5936
Rule 12
Rule 801
Rule 633
Rule 31
Rubi steps
\begin{align*} \int x (d+c d x) \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac{d x^2 (3+2 c x)}{6-6 c^2 x^2} \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )-(b c d) \int \frac{x^2 (3+2 c x)}{6-6 c^2 x^2} \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )-(b c d) \int \left (-\frac{1}{2 c^2}-\frac{x}{3 c}+\frac{3+2 c x}{c^2 \left (6-6 c^2 x^2\right )}\right ) \, dx\\ &=\frac{b d x}{2 c}+\frac{1}{6} b d x^2+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )-\frac{(b d) \int \frac{3+2 c x}{6-6 c^2 x^2} \, dx}{c}\\ &=\frac{b d x}{2 c}+\frac{1}{6} b d x^2+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} (b d) \int \frac{1}{-6 c-6 c^2 x} \, dx-\frac{1}{2} (5 b d) \int \frac{1}{6 c-6 c^2 x} \, dx\\ &=\frac{b d x}{2 c}+\frac{1}{6} b d x^2+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{5 b d \log (1-c x)}{12 c^2}-\frac{b d \log (1+c x)}{12 c^2}\\ \end{align*}
Mathematica [A] time = 0.0554376, size = 79, normalized size = 0.94 \[ \frac{d \left (4 a c^3 x^3+6 a c^2 x^2+2 b c^2 x^2+2 b c^2 x^2 (2 c x+3) \tanh ^{-1}(c x)+6 b c x+5 b \log (1-c x)-b \log (c x+1)\right )}{12 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 81, normalized size = 1. \begin{align*}{\frac{cda{x}^{3}}{3}}+{\frac{da{x}^{2}}{2}}+{\frac{cdb{\it Artanh} \left ( cx \right ){x}^{3}}{3}}+{\frac{db{\it Artanh} \left ( cx \right ){x}^{2}}{2}}+{\frac{bd{x}^{2}}{6}}+{\frac{bdx}{2\,c}}+{\frac{5\,db\ln \left ( cx-1 \right ) }{12\,{c}^{2}}}-{\frac{db\ln \left ( cx+1 \right ) }{12\,{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97236, size = 134, normalized size = 1.6 \begin{align*} \frac{1}{3} \, a c d 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 d + \frac{1}{2} \, a d x^{2} + \frac{1}{4} \,{\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 d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96771, size = 219, normalized size = 2.61 \begin{align*} \frac{4 \, a c^{3} d x^{3} + 2 \,{\left (3 \, a + b\right )} c^{2} d x^{2} + 6 \, b c d x - b d \log \left (c x + 1\right ) + 5 \, b d \log \left (c x - 1\right ) +{\left (2 \, b c^{3} d x^{3} + 3 \, b c^{2} d x^{2}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{12 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.61303, size = 100, normalized size = 1.19 \begin{align*} \begin{cases} \frac{a c d x^{3}}{3} + \frac{a d x^{2}}{2} + \frac{b c d x^{3} \operatorname{atanh}{\left (c x \right )}}{3} + \frac{b d x^{2} \operatorname{atanh}{\left (c x \right )}}{2} + \frac{b d x^{2}}{6} + \frac{b d x}{2 c} + \frac{b d \log{\left (x - \frac{1}{c} \right )}}{3 c^{2}} - \frac{b d \operatorname{atanh}{\left (c x \right )}}{6 c^{2}} & \text{for}\: c \neq 0 \\\frac{a d x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22319, size = 120, normalized size = 1.43 \begin{align*} \frac{1}{3} \, a c d x^{3} + \frac{1}{6} \,{\left (3 \, a d + b d\right )} x^{2} + \frac{b d x}{2 \, c} + \frac{1}{12} \,{\left (2 \, b c d x^{3} + 3 \, b d x^{2}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) - \frac{b d \log \left (c x + 1\right )}{12 \, c^{2}} + \frac{5 \, b d \log \left (c x - 1\right )}{12 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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