Optimal. Leaf size=312 \[ -\frac{8 b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{15 c^3}+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{a b d^2 x}{c^2}+\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}-\frac{16 b d^2 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{15 c^3}+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}+\frac{19 b^2 d^2 x}{30 c^2}+\frac{b^2 d^2 x \tanh ^{-1}(c x)}{c^2}-\frac{19 b^2 d^2 \tanh ^{-1}(c x)}{30 c^3}+\frac{b^2 d^2 x^2}{6 c}+\frac{1}{30} b^2 d^2 x^3 \]
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Rubi [A] time = 0.883728, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 36, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {5940, 5916, 5980, 321, 206, 5984, 5918, 2402, 2315, 266, 43, 5910, 260, 5948, 302} \[ -\frac{8 b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{15 c^3}+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{a b d^2 x}{c^2}+\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}-\frac{16 b d^2 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{15 c^3}+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}+\frac{19 b^2 d^2 x}{30 c^2}+\frac{b^2 d^2 x \tanh ^{-1}(c x)}{c^2}-\frac{19 b^2 d^2 \tanh ^{-1}(c x)}{30 c^3}+\frac{b^2 d^2 x^2}{6 c}+\frac{1}{30} b^2 d^2 x^3 \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5916
Rule 5980
Rule 321
Rule 206
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 266
Rule 43
Rule 5910
Rule 260
Rule 5948
Rule 302
Rubi steps
\begin{align*} \int x^2 (d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+2 c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (2 c d^2\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^2 d^2\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{3} \left (2 b c d^2\right ) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (b c^2 d^2\right ) \int \frac{x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{1}{5} \left (2 b c^3 d^2\right ) \int \frac{x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\left (b d^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (b d^2\right ) \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac{\left (2 b d^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c}-\frac{\left (2 b d^2\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c}+\frac{1}{5} \left (2 b c d^2\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{5} \left (2 b c d^2\right ) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac{1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{3} \left (b^2 d^2\right ) \int \frac{x^2}{1-c^2 x^2} \, dx-\frac{\left (2 b d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c^2}+\frac{\left (b d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2}-\frac{\left (b d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^2}+\frac{\left (2 b d^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c}-\frac{\left (2 b d^2\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c}-\frac{1}{3} \left (b^2 c d^2\right ) \int \frac{x^3}{1-c^2 x^2} \, dx-\frac{1}{10} \left (b^2 c^2 d^2\right ) \int \frac{x^4}{1-c^2 x^2} \, dx\\ &=\frac{a b d^2 x}{c^2}+\frac{b^2 d^2 x}{3 c^2}+\frac{8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{2 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{3 c^3}-\frac{1}{5} \left (b^2 d^2\right ) \int \frac{x^2}{1-c^2 x^2} \, dx-\frac{\left (2 b d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^2}-\frac{\left (b^2 d^2\right ) \int \frac{1}{1-c^2 x^2} \, dx}{3 c^2}+\frac{\left (2 b^2 d^2\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c^2}+\frac{\left (b^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx}{c^2}-\frac{1}{6} \left (b^2 c d^2\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )-\frac{1}{10} \left (b^2 c^2 d^2\right ) \int \left (-\frac{1}{c^4}-\frac{x^2}{c^2}+\frac{1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac{a b d^2 x}{c^2}+\frac{19 b^2 d^2 x}{30 c^2}+\frac{1}{30} b^2 d^2 x^3-\frac{b^2 d^2 \tanh ^{-1}(c x)}{3 c^3}+\frac{b^2 d^2 x \tanh ^{-1}(c x)}{c^2}+\frac{8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{16 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{15 c^3}-\frac{\left (2 b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )}{3 c^3}-\frac{\left (b^2 d^2\right ) \int \frac{1}{1-c^2 x^2} \, dx}{10 c^2}-\frac{\left (b^2 d^2\right ) \int \frac{1}{1-c^2 x^2} \, dx}{5 c^2}+\frac{\left (2 b^2 d^2\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c^2}-\frac{\left (b^2 d^2\right ) \int \frac{x}{1-c^2 x^2} \, dx}{c}-\frac{1}{6} \left (b^2 c d^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{a b d^2 x}{c^2}+\frac{19 b^2 d^2 x}{30 c^2}+\frac{b^2 d^2 x^2}{6 c}+\frac{1}{30} b^2 d^2 x^3-\frac{19 b^2 d^2 \tanh ^{-1}(c x)}{30 c^3}+\frac{b^2 d^2 x \tanh ^{-1}(c x)}{c^2}+\frac{8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{16 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{15 c^3}+\frac{2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}-\frac{b^2 d^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{3 c^3}-\frac{\left (2 b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )}{5 c^3}\\ &=\frac{a b d^2 x}{c^2}+\frac{19 b^2 d^2 x}{30 c^2}+\frac{b^2 d^2 x^2}{6 c}+\frac{1}{30} b^2 d^2 x^3-\frac{19 b^2 d^2 \tanh ^{-1}(c x)}{30 c^3}+\frac{b^2 d^2 x \tanh ^{-1}(c x)}{c^2}+\frac{8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{16 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{15 c^3}+\frac{2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}-\frac{8 b^2 d^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{15 c^3}\\ \end{align*}
Mathematica [A] time = 0.999511, size = 297, normalized size = 0.95 \[ \frac{d^2 \left (16 b^2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+6 a^2 c^5 x^5+15 a^2 c^4 x^4+10 a^2 c^3 x^3+3 a b c^4 x^4+10 a b c^3 x^3+16 a b c^2 x^2+16 a b \log \left (c^2 x^2-1\right )+b \tanh ^{-1}(c x) \left (2 a c^3 x^3 \left (6 c^2 x^2+15 c x+10\right )+b \left (3 c^4 x^4+10 c^3 x^3+16 c^2 x^2+30 c x-19\right )-32 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+30 a b c x+15 a b \log (1-c x)-15 a b \log (c x+1)-9 a b+b^2 c^3 x^3+5 b^2 c^2 x^2+20 b^2 \log \left (1-c^2 x^2\right )+b^2 \left (6 c^5 x^5+15 c^4 x^4+10 c^3 x^3-31\right ) \tanh ^{-1}(c x)^2+19 b^2 c x-5 b^2\right )}{30 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.053, size = 521, normalized size = 1.7 \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 = 2.15045, size = 815, normalized size = 2.61 \begin{align*} \frac{1}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac{1}{2} \, a^{2} c d^{2} x^{4} + \frac{1}{10} \,{\left (4 \, x^{5} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} a b c^{2} d^{2} + \frac{1}{3} \, a^{2} d^{2} x^{3} + \frac{1}{6} \,{\left (6 \, x^{4} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \,{\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac{3 \, \log \left (c x + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} a b c d^{2} + \frac{1}{3} \,{\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 )} a b d^{2} + \frac{8 \,{\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^{2}}{15 \, c^{3}} + \frac{7 \, b^{2} d^{2} \log \left (c x + 1\right )}{20 \, c^{3}} + \frac{59 \, b^{2} d^{2} \log \left (c x - 1\right )}{60 \, c^{3}} + \frac{4 \, b^{2} c^{3} d^{2} x^{3} + 20 \, b^{2} c^{2} d^{2} x^{2} + 76 \, b^{2} c d^{2} x +{\left (6 \, b^{2} c^{5} d^{2} x^{5} + 15 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} +{\left (6 \, b^{2} c^{5} d^{2} x^{5} + 15 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} - 31 \, b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \,{\left (3 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 30 \, b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \,{\left (3 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 30 \, b^{2} c d^{2} x +{\left (6 \, b^{2} c^{5} d^{2} x^{5} + 15 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{120 \, c^{3}} \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^{2} d^{2} x^{4} + 2 \, a^{2} c d^{2} x^{3} + a^{2} d^{2} x^{2} +{\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c d^{2} x^{3} + b^{2} d^{2} x^{2}\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d^{2} x^{4} + 2 \, a b c d^{2} x^{3} + a b d^{2} x^{2}\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^{2} \left (\int a^{2} x^{2}\, dx + \int 2 a^{2} c x^{3}\, dx + \int a^{2} c^{2} x^{4}\, dx + \int b^{2} x^{2} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{2} \operatorname{atanh}{\left (c x \right )}\, dx + \int 2 b^{2} c x^{3} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{2} x^{4} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 4 a b c x^{3} \operatorname{atanh}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{4} \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 )}^{2}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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