Optimal. Leaf size=313 \[ -b c^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c^2 d^2 \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-2 b^2 c^2 d^2 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{1}{2} b^2 c^2 d^2 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^2 d^2 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+\frac{5}{2} c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2+2 c^2 d^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+4 b c^2 d^2 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{1}{2} b^2 c^2 d^2 \log \left (1-c^2 x^2\right )+b^2 c^2 d^2 \log (x) \]
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Rubi [A] time = 0.673329, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {5940, 5916, 5982, 266, 36, 29, 31, 5948, 5988, 5932, 2447, 5914, 6052, 6058, 6610} \[ -b c^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c^2 d^2 \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-2 b^2 c^2 d^2 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{1}{2} b^2 c^2 d^2 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^2 d^2 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+\frac{5}{2} c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2+2 c^2 d^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+4 b c^2 d^2 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{1}{2} b^2 c^2 d^2 \log \left (1-c^2 x^2\right )+b^2 c^2 d^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5916
Rule 5982
Rule 266
Rule 36
Rule 29
Rule 31
Rule 5948
Rule 5988
Rule 5932
Rule 2447
Rule 5914
Rule 6052
Rule 6058
Rule 6610
Rubi steps
\begin{align*} \int \frac{(d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx &=\int \left (\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3}+\frac{2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2}+\frac{c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d^2 \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx+\left (2 c d^2\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\left (c^2 d^2\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx\\ &=-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+\left (b c d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (4 b c^2 d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx-\left (4 b c^3 d^2\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\\ &=2 c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+\left (b c d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (4 b c^2 d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (b c^3 d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx+\left (2 b c^3 d^2\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^3 d^2\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\\ &=-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{5}{2} c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+4 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\left (b^2 c^2 d^2\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx+\left (b^2 c^3 d^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c^3 d^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (4 b^2 c^3 d^2\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{5}{2} c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+4 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-2 b^2 c^2 d^2 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} \left (b^2 c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{5}{2} c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+4 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-2 b^2 c^2 d^2 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} \left (b^2 c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} \left (b^2 c^4 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{5}{2} c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+b^2 c^2 d^2 \log (x)-\frac{1}{2} b^2 c^2 d^2 \log \left (1-c^2 x^2\right )+4 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-2 b^2 c^2 d^2 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )\\ \end{align*}
Mathematica [C] time = 0.749529, size = 370, normalized size = 1.18 \[ -\frac{d^2 \left (2 a b c^2 x^2 (\text{PolyLog}(2,-c x)-\text{PolyLog}(2,c x))-2 b^2 c^2 x^2 \left (\tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )-\frac{2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+\tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+\frac{i \pi ^3}{24}\right )+4 b^2 c x \left (c x \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \left ((1-c x) \tanh ^{-1}(c x)-2 c x \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )\right )-2 a^2 c^2 x^2 \log (x)+4 a^2 c x+a^2+4 a b c x \left (c x \left (\log \left (1-c^2 x^2\right )-2 \log (c x)\right )+2 \tanh ^{-1}(c x)\right )+a b \left (c x (c x \log (1-c x)-c x \log (c x+1)+2)+2 \tanh ^{-1}(c x)\right )+b^2 \left (-2 c^2 x^2 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+\left (1-c^2 x^2\right ) \tanh ^{-1}(c x)^2+2 c x \tanh ^{-1}(c x)\right )\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.217, size = 1167, normalized size = 3.7 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} c^{2} d^{2} \log \left (x\right ) - 2 \,{\left (c{\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac{2 \, \operatorname{artanh}\left (c x\right )}{x}\right )} a b c d^{2} + \frac{1}{2} \,{\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 b d^{2} - \frac{2 \, a^{2} c d^{2}}{x} - \frac{a^{2} d^{2}}{2 \, x^{2}} - \frac{{\left (4 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2}}{8 \, x^{2}} - \int -\frac{{\left (b^{2} c^{3} d^{2} x^{3} + b^{2} c^{2} d^{2} x^{2} - b^{2} c d^{2} x - b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b c^{3} d^{2} x^{3} - a b c^{2} d^{2} x^{2}\right )} \log \left (c x + 1\right ) -{\left (4 \, a b c^{3} d^{2} x^{3} - b^{2} c d^{2} x - 4 \,{\left (a b c^{2} d^{2} + b^{2} c^{2} d^{2}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{2} x^{3} + b^{2} c^{2} d^{2} x^{2} - b^{2} c d^{2} x - b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \,{\left (c x^{4} - x^{3}\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^{2} d^{2} x^{2} + 2 \, a^{2} c d^{2} x + a^{2} d^{2} +{\left (b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d^{2} x^{2} + 2 \, a b c d^{2} x + a b d^{2}\right )} \operatorname{artanh}\left (c x\right )}{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*} d^{2} \left (\int \frac{a^{2}}{x^{3}}\, dx + \int \frac{2 a^{2} c}{x^{2}}\, dx + \int \frac{a^{2} c^{2}}{x}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{2 b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{b^{2} c^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac{4 a b c \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{2 a b c^{2} \operatorname{atanh}{\left (c x \right )}}{x}\, 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 )}^{2}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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