Optimal. Leaf size=244 \[ -\frac{4}{3} b^2 c^3 d^2 \text{PolyLog}(2,-c x)+\frac{4}{3} b^2 c^3 d^2 \text{PolyLog}(2,c x)+\frac{4}{3} b^2 c^3 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )+\frac{8}{3} a b c^3 d^2 \log (x)-\frac{2 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{8}{3} b c^3 d^2 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-b^2 c^3 d^2 \log \left (1-c^2 x^2\right )-\frac{b^2 c^2 d^2}{3 x}+2 b^2 c^3 d^2 \log (x)+\frac{1}{3} b^2 c^3 d^2 \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.256134, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {37, 5938, 5916, 325, 206, 266, 36, 29, 31, 5912, 5918, 2402, 2315} \[ -\frac{4}{3} b^2 c^3 d^2 \text{PolyLog}(2,-c x)+\frac{4}{3} b^2 c^3 d^2 \text{PolyLog}(2,c x)+\frac{4}{3} b^2 c^3 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )+\frac{8}{3} a b c^3 d^2 \log (x)-\frac{2 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{8}{3} b c^3 d^2 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-b^2 c^3 d^2 \log \left (1-c^2 x^2\right )-\frac{b^2 c^2 d^2}{3 x}+2 b^2 c^3 d^2 \log (x)+\frac{1}{3} b^2 c^3 d^2 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 37
Rule 5938
Rule 5916
Rule 325
Rule 206
Rule 266
Rule 36
Rule 29
Rule 31
Rule 5912
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{(d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-(2 b c) \int \left (-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{4 c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x}+\frac{4 c^3 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 (-1+c x)}\right ) \, dx\\ &=-\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (2 b c d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (2 b c^2 d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac{1}{3} \left (8 b c^3 d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx-\frac{1}{3} \left (8 b c^4 d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{-1+c x} \, dx\\ &=-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{2 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{8}{3} a b c^3 d^2 \log (x)+\frac{8}{3} b c^3 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(-c x)+\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(c x)+\frac{1}{3} \left (b^2 c^2 d^2\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (2 b^2 c^3 d^2\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx-\frac{1}{3} \left (8 b^2 c^4 d^2\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac{b^2 c^2 d^2}{3 x}-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{2 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{8}{3} a b c^3 d^2 \log (x)+\frac{8}{3} b c^3 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(-c x)+\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(c x)+\left (b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac{1}{3} \left (8 b^2 c^3 d^2\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^4 d^2\right ) \int \frac{1}{1-c^2 x^2} \, dx\\ &=-\frac{b^2 c^2 d^2}{3 x}+\frac{1}{3} b^2 c^3 d^2 \tanh ^{-1}(c x)-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{2 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{8}{3} a b c^3 d^2 \log (x)+\frac{8}{3} b c^3 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(-c x)+\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(c x)+\frac{4}{3} b^2 c^3 d^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )+\left (b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\left (b^2 c^5 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b^2 c^2 d^2}{3 x}+\frac{1}{3} b^2 c^3 d^2 \tanh ^{-1}(c x)-\frac{b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{2 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{8}{3} a b c^3 d^2 \log (x)+2 b^2 c^3 d^2 \log (x)+\frac{8}{3} b c^3 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-b^2 c^3 d^2 \log \left (1-c^2 x^2\right )-\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(-c x)+\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(c x)+\frac{4}{3} b^2 c^3 d^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )\\ \end{align*}
Mathematica [A] time = 0.639116, size = 270, normalized size = 1.11 \[ -\frac{d^2 \left (4 b^2 c^3 x^3 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+3 a^2 c^2 x^2+3 a^2 c x+a^2+6 a b c^2 x^2-8 a b c^3 x^3 \log (c x)+3 a b c^3 x^3 \log (1-c x)-3 a b c^3 x^3 \log (c x+1)+4 a b c^3 x^3 \log \left (1-c^2 x^2\right )+b \tanh ^{-1}(c x) \left (a \left (6 c^2 x^2+6 c x+2\right )+b c x \left (-c^2 x^2+6 c x+1\right )-8 b c^3 x^3 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )+a b c x+b^2 c^2 x^2-6 b^2 c^3 x^3 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+b^2 \left (-7 c^3 x^3+3 c^2 x^2+3 c x+1\right ) \tanh ^{-1}(c x)^2\right )}{3 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.076, size = 550, normalized size = 2.3 \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 [B] time = 3.07804, size = 749, normalized size = 3.07 \begin{align*} -\frac{4}{3} \,{\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} c^{3} d^{2} - \frac{4}{3} \,{\left (\log \left (c x\right ) \log \left (-c x + 1\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} b^{2} c^{3} d^{2} + \frac{4}{3} \,{\left (\log \left (c x + 1\right ) \log \left (-c x\right ) +{\rm Li}_2\left (c x + 1\right )\right )} b^{2} c^{3} d^{2} - \frac{5}{6} \, b^{2} c^{3} d^{2} \log \left (c x + 1\right ) - \frac{7}{6} \, b^{2} c^{3} d^{2} \log \left (c x - 1\right ) + 2 \, b^{2} c^{3} d^{2} \log \left (x\right ) -{\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^{2} d^{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 c d^{2} - \frac{1}{3} \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac{1}{x^{2}}\right )} c + \frac{2 \, \operatorname{artanh}\left (c x\right )}{x^{3}}\right )} a b d^{2} - \frac{a^{2} c^{2} d^{2}}{x} - \frac{a^{2} c d^{2}}{x^{2}} - \frac{a^{2} d^{2}}{3 \, x^{3}} - \frac{4 \, b^{2} c^{2} d^{2} x^{2} +{\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} -{\left (7 \, b^{2} c^{3} d^{2} x^{3} - 3 \, b^{2} c^{2} d^{2} x^{2} - 3 \, b^{2} c d^{2} x - b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \,{\left (6 \, b^{2} c^{2} d^{2} x^{2} + b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \,{\left (6 \, b^{2} c^{2} d^{2} x^{2} + b^{2} c d^{2} x +{\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \, x^{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 (\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^{4}}, 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^{4}}\, dx + \int \frac{2 a^{2} c}{x^{3}}\, dx + \int \frac{a^{2} c^{2}}{x^{2}}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{2 b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{b^{2} c^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{4 a b c \operatorname{atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{2 a b c^{2} \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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