Optimal. Leaf size=123 \[ -\frac{3}{2} b^3 c^2 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+3 b^2 c^2 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}-\frac{3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x} \]
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Rubi [A] time = 0.294355, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5916, 5982, 5988, 5932, 2447, 5948} \[ -\frac{3}{2} b^3 c^2 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+3 b^2 c^2 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}-\frac{3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x} \]
Antiderivative was successfully verified.
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Rule 5916
Rule 5982
Rule 5988
Rule 5932
Rule 2447
Rule 5948
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{x^3} \, dx &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\frac{1}{2} (3 b c) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\frac{1}{2} (3 b c) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac{1}{2} \left (3 b c^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=-\frac{3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac{1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\left (3 b^2 c^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx\\ &=\frac{3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac{1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\left (3 b^2 c^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx\\ &=\frac{3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac{1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-\left (3 b^3 c^3\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac{3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac{1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-\frac{3}{2} b^3 c^2 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )\\ \end{align*}
Mathematica [A] time = 0.271884, size = 192, normalized size = 1.56 \[ \frac{-6 b^3 c^2 x^2 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+a \left (-2 a^2-3 a b c^2 x^2 \log (1-c x)+3 a b c^2 x^2 \log (c x+1)-6 a b c x+12 b^2 c^2 x^2 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )\right )-6 b \tanh ^{-1}(c x) \left (a^2+2 a b c x-2 b^2 c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )+6 b^2 (c x-1) \tanh ^{-1}(c x)^2 (a c x+a+b c x)+2 b^3 \left (c^2 x^2-1\right ) \tanh ^{-1}(c x)^3}{4 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.316, size = 5098, normalized size = 41.5 \begin{align*} \text{output 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{3}{4} \,{\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac{2}{x}\right )} c - \frac{2 \, \operatorname{artanh}\left (c x\right )}{x^{2}}\right )} a^{2} b + \frac{3}{8} \,{\left ({\left (2 \,{\left (\log \left (c x - 1\right ) - 2\right )} \log \left (c x + 1\right ) - \log \left (c x + 1\right )^{2} - \log \left (c x - 1\right )^{2} - 4 \, \log \left (c x - 1\right ) + 8 \, \log \left (x\right )\right )} c^{2} + 4 \,{\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac{2}{x}\right )} c \operatorname{artanh}\left (c x\right )\right )} a b^{2} - \frac{1}{16} \, b^{3}{\left (\frac{{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )^{3} + 3 \,{\left (2 \, c x -{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{x^{2}} + 2 \, \int -\frac{{\left (c x - 1\right )} \log \left (c x + 1\right )^{3} + 3 \,{\left (2 \, c^{2} x^{2} -{\left (c x - 1\right )} \log \left (c x + 1\right )^{2} -{\left (c^{3} x^{3} - c x\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{c x^{4} - x^{3}}\,{d x}\right )} - \frac{3 \, a b^{2} \operatorname{artanh}\left (c x\right )^{2}}{2 \, x^{2}} - \frac{a^{3}}{2 \, x^{2}} \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{b^{3} \operatorname{artanh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{artanh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{artanh}\left (c x\right ) + a^{3}}{x^{3}}, 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*} \int \frac{\left (a + b \operatorname{atanh}{\left (c x \right )}\right )^{3}}{x^{3}}\, dx \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 (b \operatorname{artanh}\left (c x\right ) + a\right )}^{3}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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