Optimal. Leaf size=151 \[ -b^2 c^2 d \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{3}{2} c^2 d \left (a+b \tanh ^{-1}(c x)\right )^2+2 b c^2 d \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac{b c d \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{1}{2} b^2 c^2 d \log \left (1-c^2 x^2\right )+b^2 c^2 d \log (x) \]
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Rubi [A] time = 0.365271, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {5940, 5916, 5982, 266, 36, 29, 31, 5948, 5988, 5932, 2447} \[ -b^2 c^2 d \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{3}{2} c^2 d \left (a+b \tanh ^{-1}(c x)\right )^2+2 b c^2 d \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac{b c d \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{1}{2} b^2 c^2 d \log \left (1-c^2 x^2\right )+b^2 c^2 d \log (x) \]
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
Rubi steps
\begin{align*} \int \frac{(d+c d x) \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx &=\int \left (\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3}+\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2}\right ) \, dx\\ &=d \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx+(c d) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx\\ &=-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+(b c d) \int \frac{a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (2 b c^2 d\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx\\ &=c^2 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+(b c d) \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (2 b c^2 d\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (b c^3 d\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=-\frac{b c d \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{3}{2} c^2 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 b c^2 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )+\left (b^2 c^2 d\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx-\left (2 b^2 c^3 d\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac{b c d \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{3}{2} c^2 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 b c^2 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b^2 c^2 d \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{1}{2} \left (b^2 c^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{b c d \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{3}{2} c^2 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 b c^2 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b^2 c^2 d \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{1}{2} \left (b^2 c^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} \left (b^2 c^4 d\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{3}{2} c^2 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+b^2 c^2 d \log (x)-\frac{1}{2} b^2 c^2 d \log \left (1-c^2 x^2\right )+2 b c^2 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b^2 c^2 d \text{Li}_2\left (-1+\frac{2}{1+c x}\right )\\ \end{align*}
Mathematica [A] time = 0.281313, size = 206, normalized size = 1.36 \[ -\frac{d \left (2 b^2 c^2 x^2 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+2 a^2 c x+a^2-4 a b c^2 x^2 \log (c x)+a b c^2 x^2 \log (1-c x)-a b c^2 x^2 \log (c x+1)+2 a b c^2 x^2 \log \left (1-c^2 x^2\right )+2 b \tanh ^{-1}(c x) \left (2 a c x+a-2 b c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+b c x\right )+2 a b c x-2 b^2 c^2 x^2 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+b^2 \left (-3 c^2 x^2+2 c x+1\right ) \tanh ^{-1}(c x)^2\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.071, size = 400, normalized size = 2.7 \begin{align*} 2\,{c}^{2}d{b}^{2}{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) -2\,{\frac{cdab{\it Artanh} \left ( cx \right ) }{x}}+{c}^{2}d{b}^{2}\ln \left ( cx \right ) -{\frac{3\,{c}^{2}d{b}^{2} \left ( \ln \left ( cx-1 \right ) \right ) ^{2}}{8}}+{c}^{2}d{b}^{2}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) +{\frac{{c}^{2}d{b}^{2} \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{8}}-{c}^{2}d{b}^{2}{\it dilog} \left ( cx \right ) -{\frac{{c}^{2}d{b}^{2}\ln \left ( cx-1 \right ) }{2}}-{\frac{{c}^{2}d{b}^{2}\ln \left ( cx+1 \right ) }{2}}-{c}^{2}d{b}^{2}{\it dilog} \left ( cx+1 \right ) -{\frac{{a}^{2}cd}{x}}-{\frac{d{b}^{2} \left ({\it Artanh} \left ( cx \right ) \right ) ^{2}}{2\,{x}^{2}}}-{\frac{{a}^{2}d}{2\,{x}^{2}}}-{\frac{cd{b}^{2}{\it Artanh} \left ( cx \right ) }{x}}-{\frac{dab{\it Artanh} \left ( cx \right ) }{{x}^{2}}}+2\,{c}^{2}dab\ln \left ( cx \right ) -{\frac{{c}^{2}dab\ln \left ( cx+1 \right ) }{2}}+{\frac{3\,{c}^{2}d{b}^{2}\ln \left ( cx-1 \right ) }{4}\ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{3\,{c}^{2}dab\ln \left ( cx-1 \right ) }{2}}+{\frac{{c}^{2}d{b}^{2}}{4}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{{c}^{2}d{b}^{2}\ln \left ( cx+1 \right ) }{4}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }-{c}^{2}d{b}^{2}\ln \left ( cx \right ) \ln \left ( cx+1 \right ) -{\frac{3\,{c}^{2}d{b}^{2}{\it Artanh} \left ( cx \right ) \ln \left ( cx-1 \right ) }{2}}-{\frac{{c}^{2}d{b}^{2}{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{2}}-{\frac{cd{b}^{2} \left ({\it Artanh} \left ( cx \right ) \right ) ^{2}}{x}}-{\frac{cdab}{x}} \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*} -{\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 - \frac{1}{4} \, b^{2} c d{\left (\frac{\log \left (-c x + 1\right )^{2}}{x} + \int -\frac{{\left (c x - 1\right )} \log \left (c x + 1\right )^{2} + 2 \,{\left (c x -{\left (c x - 1\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{c x^{3} - x^{2}}\,{d x}\right )} + \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 + \frac{1}{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 )} b^{2} d - \frac{a^{2} c d}{x} - \frac{b^{2} d \operatorname{artanh}\left (c x\right )^{2}}{2 \, x^{2}} - \frac{a^{2} d}{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{a^{2} c d x + a^{2} d +{\left (b^{2} c d x + b^{2} d\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c d x + a b d\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 \left (\int \frac{a^{2}}{x^{3}}\, dx + \int \frac{a^{2} c}{x^{2}}\, 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{b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{2 a b c \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 )}{\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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