Optimal. Leaf size=355 \[ -b d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b d^3 \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{10}{3} b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )+\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a b c d^3 x+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{20}{3} b d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )+\frac{1}{3} b^2 c d^3 x-\frac{1}{3} b^2 d^3 \tanh ^{-1}(c x)+3 b^2 c d^3 x \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.812787, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 16, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {5940, 5910, 5984, 5918, 2402, 2315, 5914, 6052, 5948, 6058, 6610, 5916, 5980, 260, 321, 206} \[ -b d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b d^3 \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{10}{3} b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )+\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a b c d^3 x+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{20}{3} b d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )+\frac{1}{3} b^2 c d^3 x-\frac{1}{3} b^2 d^3 \tanh ^{-1}(c x)+3 b^2 c d^3 x \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5910
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 5914
Rule 6052
Rule 5948
Rule 6058
Rule 6610
Rule 5916
Rule 5980
Rule 260
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx &=\int \left (3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx+\left (3 c d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (3 c^2 d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^3 d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-\left (4 b c d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (6 b c^2 d^3\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (3 b c^3 d^3\right ) \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^4 d^3\right ) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+\left (2 b c d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b c d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx+\left (3 b c d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (3 b c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx-\left (6 b c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\frac{1}{3} \left (2 b c^2 d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{3} \left (2 b c^2 d^3\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=3 a b c d^3 x+\frac{1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-6 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-\frac{1}{3} \left (2 b c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (b^2 c d^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c d^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx+\left (3 b^2 c d^3\right ) \int \tanh ^{-1}(c x) \, dx+\left (6 b^2 c d^3\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\frac{1}{3} \left (b^2 c^3 d^3\right ) \int \frac{x^2}{1-c^2 x^2} \, dx\\ &=3 a b c d^3 x+\frac{1}{3} b^2 c d^3 x+3 b^2 c d^3 x \tanh ^{-1}(c x)+\frac{1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-\frac{20}{3} b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} b^2 d^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 d^3 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )-\left (6 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )-\frac{1}{3} \left (b^2 c d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx+\frac{1}{3} \left (2 b^2 c d^3\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (3 b^2 c^2 d^3\right ) \int \frac{x}{1-c^2 x^2} \, dx\\ &=3 a b c d^3 x+\frac{1}{3} b^2 c d^3 x-\frac{1}{3} b^2 d^3 \tanh ^{-1}(c x)+3 b^2 c d^3 x \tanh ^{-1}(c x)+\frac{1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-\frac{20}{3} b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )+\frac{3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-3 b^2 d^3 \text{Li}_2\left (1-\frac{2}{1-c x}\right )-b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} b^2 d^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 d^3 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )-\frac{1}{3} \left (2 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )\\ &=3 a b c d^3 x+\frac{1}{3} b^2 c d^3 x-\frac{1}{3} b^2 d^3 \tanh ^{-1}(c x)+3 b^2 c d^3 x \tanh ^{-1}(c x)+\frac{1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-\frac{20}{3} b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )+\frac{3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-\frac{10}{3} b^2 d^3 \text{Li}_2\left (1-\frac{2}{1-c x}\right )-b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} b^2 d^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 d^3 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )\\ \end{align*}
Mathematica [C] time = 0.827737, size = 448, normalized size = 1.26 \[ \frac{1}{24} d^3 \left (-24 a b \text{PolyLog}(2,-c x)+24 a b \text{PolyLog}(2,c x)+8 b^2 \left (3 \tanh ^{-1}(c x)+10\right ) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 \tanh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )+12 b^2 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-12 b^2 \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )+8 a^2 c^3 x^3+36 a^2 c^2 x^2+72 a^2 c x+24 a^2 \log (c x)+8 a b c^2 x^2+72 a b \log \left (1-c^2 x^2\right )+8 a b \log \left (c^2 x^2-1\right )+16 a b c^3 x^3 \tanh ^{-1}(c x)+72 a b c^2 x^2 \tanh ^{-1}(c x)+72 a b c x+36 a b \log (1-c x)-36 a b \log (c x+1)+144 a b c x \tanh ^{-1}(c x)+36 b^2 \log \left (1-c^2 x^2\right )+8 b^2 c^3 x^3 \tanh ^{-1}(c x)^2+36 b^2 c^2 x^2 \tanh ^{-1}(c x)^2+8 b^2 c^2 x^2 \tanh ^{-1}(c x)+8 b^2 c x+72 b^2 c x \tanh ^{-1}(c x)^2+72 b^2 c x \tanh ^{-1}(c x)-16 b^2 \tanh ^{-1}(c x)^3-116 b^2 \tanh ^{-1}(c x)^2-8 b^2 \tanh ^{-1}(c x)-24 b^2 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-160 b^2 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+24 b^2 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+i \pi ^3 b^2\right ) \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.046, size = 1186, normalized size = 3.3 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{2} c^{3} d^{3} x^{3} + \frac{3}{2} \, a^{2} c^{2} d^{3} x^{2} + 3 \, a^{2} c d^{3} x + 3 \,{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d^{3} + a^{2} d^{3} \log \left (x\right ) + \frac{1}{24} \,{\left (2 \, b^{2} c^{3} d^{3} x^{3} + 9 \, b^{2} c^{2} d^{3} x^{2} + 18 \, b^{2} c d^{3} x\right )} \log \left (-c x + 1\right )^{2} - \int -\frac{3 \,{\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 12 \,{\left (a b c^{4} d^{3} x^{4} + 2 \, a b c^{3} d^{3} x^{3} - 3 \, a b c^{2} d^{3} x^{2} + a b c d^{3} x - a b d^{3}\right )} \log \left (c x + 1\right ) -{\left (12 \, a b c d^{3} x - 12 \, a b d^{3} + 2 \,{\left (6 \, a b c^{4} d^{3} + b^{2} c^{4} d^{3}\right )} x^{4} + 3 \,{\left (8 \, a b c^{3} d^{3} + 3 \, b^{2} c^{3} d^{3}\right )} x^{3} - 18 \,{\left (2 \, a b c^{2} d^{3} - b^{2} c^{2} d^{3}\right )} x^{2} + 6 \,{\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \,{\left (c x^{2} - x\right )}}\,{d x} \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 (\frac{a^{2} c^{3} d^{3} x^{3} + 3 \, a^{2} c^{2} d^{3} x^{2} + 3 \, a^{2} c d^{3} x + a^{2} d^{3} +{\left (b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} + 3 \, b^{2} c d^{3} x + b^{2} d^{3}\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c^{3} d^{3} x^{3} + 3 \, a b c^{2} d^{3} x^{2} + 3 \, a b c d^{3} x + a b d^{3}\right )} \operatorname{artanh}\left (c x\right )}{x}, 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^{3} \left (\int 3 a^{2} c\, dx + \int \frac{a^{2}}{x}\, dx + \int 3 a^{2} c^{2} x\, dx + \int a^{2} c^{3} x^{2}\, dx + \int 3 b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x}\, dx + \int 6 a b c \operatorname{atanh}{\left (c x \right )}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{2} x \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{2} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname{atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} 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 \frac{{\left (c d x + d\right )}^{3}{\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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