Optimal. Leaf size=196 \[ -\frac{b^2 d \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{3 c^2}-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}-\frac{2 b d \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^2}+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{a b d x}{c}+\frac{b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac{b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac{b^2 d x}{3 c}+\frac{b^2 d x \tanh ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.393827, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5940, 5916, 5980, 5910, 260, 5948, 321, 206, 5984, 5918, 2402, 2315} \[ -\frac{b^2 d \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{3 c^2}-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}-\frac{2 b d \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^2}+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{a b d x}{c}+\frac{b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac{b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac{b^2 d x}{3 c}+\frac{b^2 d x \tanh ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5916
Rule 5980
Rule 5910
Rule 260
Rule 5948
Rule 321
Rule 206
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int x (d+c d x) \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d x \left (a+b \tanh ^{-1}(c x)\right )^2+c d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+(c d) \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-(b c d) \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{1}{3} \left (2 b c^2 d\right ) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} (2 b d) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{3} (2 b d) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac{(b d) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}-\frac{(b d) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c}\\ &=\frac{a b d x}{c}+\frac{1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{(2 b d) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c}+\frac{\left (b^2 d\right ) \int \tanh ^{-1}(c x) \, dx}{c}-\frac{1}{3} \left (b^2 c d\right ) \int \frac{x^2}{1-c^2 x^2} \, dx\\ &=\frac{a b d x}{c}+\frac{b^2 d x}{3 c}+\frac{b^2 d x \tanh ^{-1}(c x)}{c}+\frac{1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{3 c^2}-\left (b^2 d\right ) \int \frac{x}{1-c^2 x^2} \, dx-\frac{\left (b^2 d\right ) \int \frac{1}{1-c^2 x^2} \, dx}{3 c}+\frac{\left (2 b^2 d\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c}\\ &=\frac{a b d x}{c}+\frac{b^2 d x}{3 c}-\frac{b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac{b^2 d x \tanh ^{-1}(c x)}{c}+\frac{1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{3 c^2}+\frac{b^2 d \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 )}{3 c^2}\\ &=\frac{a b d x}{c}+\frac{b^2 d x}{3 c}-\frac{b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac{b^2 d x \tanh ^{-1}(c x)}{c}+\frac{1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{3 c^2}+\frac{b^2 d \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 )}{3 c^2}\\ \end{align*}
Mathematica [A] time = 0.479758, size = 201, normalized size = 1.03 \[ \frac{d \left (2 b^2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+2 a^2 c^3 x^3+3 a^2 c^2 x^2+2 a b c^2 x^2+2 a b \log \left (c^2 x^2-1\right )+2 b \tanh ^{-1}(c x) \left (a c^2 x^2 (2 c x+3)+b \left (c^2 x^2+3 c x-1\right )-2 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+6 a b c x+3 a b \log (1-c x)-3 a b \log (c x+1)+3 b^2 \log \left (1-c^2 x^2\right )+b^2 \left (2 c^3 x^3+3 c^2 x^2-5\right ) \tanh ^{-1}(c x)^2+2 b^2 c x\right )}{6 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.048, size = 341, normalized size = 1.7 \begin{align*}{\frac{c{a}^{2}d{x}^{3}}{3}}+{\frac{{a}^{2}d{x}^{2}}{2}}+{\frac{cd{b}^{2} \left ({\it Artanh} \left ( cx \right ) \right ) ^{2}{x}^{3}}{3}}+{\frac{d{b}^{2} \left ({\it Artanh} \left ( cx \right ) \right ) ^{2}{x}^{2}}{2}}+{\frac{d{b}^{2}{\it Artanh} \left ( cx \right ){x}^{2}}{3}}+{\frac{{b}^{2}dx{\it Artanh} \left ( cx \right ) }{c}}+{\frac{5\,d{b}^{2}{\it Artanh} \left ( cx \right ) \ln \left ( cx-1 \right ) }{6\,{c}^{2}}}-{\frac{d{b}^{2}{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{6\,{c}^{2}}}+{\frac{5\,d{b}^{2} \left ( \ln \left ( cx-1 \right ) \right ) ^{2}}{24\,{c}^{2}}}-{\frac{d{b}^{2}}{3\,{c}^{2}}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{5\,d{b}^{2}\ln \left ( cx-1 \right ) }{12\,{c}^{2}}\ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{d{b}^{2}}{12\,{c}^{2}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{d{b}^{2}\ln \left ( cx+1 \right ) }{12\,{c}^{2}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }+{\frac{d{b}^{2} \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{24\,{c}^{2}}}+{\frac{{b}^{2}dx}{3\,c}}+{\frac{2\,d{b}^{2}\ln \left ( cx-1 \right ) }{3\,{c}^{2}}}+{\frac{d{b}^{2}\ln \left ( cx+1 \right ) }{3\,{c}^{2}}}+{\frac{2\,cdab{\it Artanh} \left ( cx \right ){x}^{3}}{3}}+dab{\it Artanh} \left ( cx \right ){x}^{2}+{\frac{dab{x}^{2}}{3}}+{\frac{abdx}{c}}+{\frac{5\,dab\ln \left ( cx-1 \right ) }{6\,{c}^{2}}}-{\frac{dab\ln \left ( cx+1 \right ) }{6\,{c}^{2}}} \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*} \text{result too large to display} \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} c d x^{2} + a^{2} d x +{\left (b^{2} c d x^{2} + b^{2} d x\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c d x^{2} + a b d x\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*} d \left (\int a^{2} x\, dx + \int a^{2} c x^{2}\, dx + \int b^{2} x \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x \operatorname{atanh}{\left (c x \right )}\, dx + \int b^{2} c x^{2} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b c x^{2} \operatorname{atanh}{\left (c x \right )}\, dx\right ) \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 (c d x + d\right )}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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