Optimal. Leaf size=150 \[ -\frac{3}{2} b c d^3 \text{PolyLog}(2,-c x)+\frac{3}{2} b c d^3 \text{PolyLog}(2,c x)+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 a c^2 d^3 x+3 a c d^3 \log (x)+b c d^3 \log \left (1-c^2 x^2\right )+\frac{1}{2} b c^2 d^3 x+3 b c^2 d^3 x \tanh ^{-1}(c x)+b c d^3 \log (x)-\frac{1}{2} b c d^3 \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.157628, antiderivative size = 150, 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, 5910, 260, 5916, 266, 36, 29, 31, 5912, 321, 206} \[ -\frac{3}{2} b c d^3 \text{PolyLog}(2,-c x)+\frac{3}{2} b c d^3 \text{PolyLog}(2,c x)+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 a c^2 d^3 x+3 a c d^3 \log (x)+b c d^3 \log \left (1-c^2 x^2\right )+\frac{1}{2} b c^2 d^3 x+3 b c^2 d^3 x \tanh ^{-1}(c x)+b c d^3 \log (x)-\frac{1}{2} b c d^3 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5910
Rule 260
Rule 5916
Rule 266
Rule 36
Rule 29
Rule 31
Rule 5912
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2} \, dx &=\int \left (3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )\right ) \, dx\\ &=d^3 \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (3 c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx+\left (3 c^2 d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (c^3 d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=3 a c^2 d^3 x-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 \log (x)-\frac{3}{2} b c d^3 \text{Li}_2(-c x)+\frac{3}{2} b c d^3 \text{Li}_2(c x)+\left (b c d^3\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx+\left (3 b c^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx-\frac{1}{2} \left (b c^4 d^3\right ) \int \frac{x^2}{1-c^2 x^2} \, dx\\ &=3 a c^2 d^3 x+\frac{1}{2} b c^2 d^3 x+3 b c^2 d^3 x \tanh ^{-1}(c x)-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 \log (x)-\frac{3}{2} b c d^3 \text{Li}_2(-c x)+\frac{3}{2} b c d^3 \text{Li}_2(c x)+\frac{1}{2} \left (b c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )-\frac{1}{2} \left (b c^2 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx-\left (3 b c^3 d^3\right ) \int \frac{x}{1-c^2 x^2} \, dx\\ &=3 a c^2 d^3 x+\frac{1}{2} b c^2 d^3 x-\frac{1}{2} b c d^3 \tanh ^{-1}(c x)+3 b c^2 d^3 x \tanh ^{-1}(c x)-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 \log (x)+\frac{3}{2} b c d^3 \log \left (1-c^2 x^2\right )-\frac{3}{2} b c d^3 \text{Li}_2(-c x)+\frac{3}{2} b c d^3 \text{Li}_2(c x)+\frac{1}{2} \left (b c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} \left (b c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )\\ &=3 a c^2 d^3 x+\frac{1}{2} b c^2 d^3 x-\frac{1}{2} b c d^3 \tanh ^{-1}(c x)+3 b c^2 d^3 x \tanh ^{-1}(c x)-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 \log (x)+b c d^3 \log (x)+b c d^3 \log \left (1-c^2 x^2\right )-\frac{3}{2} b c d^3 \text{Li}_2(-c x)+\frac{3}{2} b c d^3 \text{Li}_2(c x)\\ \end{align*}
Mathematica [A] time = 0.1409, size = 149, normalized size = 0.99 \[ \frac{d^3 \left (-6 b c x \text{PolyLog}(2,-c x)+6 b c x \text{PolyLog}(2,c x)+2 a c^3 x^3+12 a c^2 x^2+12 a c x \log (x)-4 a+2 b c^2 x^2+4 b c x \log \left (1-c^2 x^2\right )+2 b c^3 x^3 \tanh ^{-1}(c x)+12 b c^2 x^2 \tanh ^{-1}(c x)+4 b c x \log (c x)+b c x \log (1-c x)-b c x \log (c x+1)-4 b \tanh ^{-1}(c x)\right )}{4 x} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.049, size = 189, normalized size = 1.3 \begin{align*}{\frac{{d}^{3}a{c}^{3}{x}^{2}}{2}}+3\,a{c}^{2}{d}^{3}x-{\frac{{d}^{3}a}{x}}+3\,c{d}^{3}a\ln \left ( cx \right ) +{\frac{{d}^{3}b{\it Artanh} \left ( cx \right ){c}^{3}{x}^{2}}{2}}+3\,b{c}^{2}{d}^{3}x{\it Artanh} \left ( cx \right ) -{\frac{{d}^{3}b{\it Artanh} \left ( cx \right ) }{x}}+3\,c{d}^{3}b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) -{\frac{3\,c{d}^{3}b{\it dilog} \left ( cx \right ) }{2}}-{\frac{3\,c{d}^{3}b{\it dilog} \left ( cx+1 \right ) }{2}}-{\frac{3\,c{d}^{3}b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2}}+{\frac{b{c}^{2}{d}^{3}x}{2}}+{\frac{5\,c{d}^{3}b\ln \left ( cx-1 \right ) }{4}}+c{d}^{3}b\ln \left ( cx \right ) +{\frac{3\,c{d}^{3}b\ln \left ( cx+1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44768, size = 309, normalized size = 2.06 \begin{align*} \frac{1}{4} \, b c^{3} d^{3} x^{2} \log \left (c x + 1\right ) - \frac{1}{4} \, b c^{3} d^{3} x^{2} \log \left (-c x + 1\right ) + \frac{1}{2} \, a c^{3} d^{3} x^{2} + 3 \, a c^{2} d^{3} x + \frac{1}{2} \, b c^{2} d^{3} x + \frac{3}{2} \,{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b c d^{3} - \frac{3}{2} \,{\left (\log \left (c x\right ) \log \left (-c x + 1\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} b c d^{3} + \frac{3}{2} \,{\left (\log \left (c x + 1\right ) \log \left (-c x\right ) +{\rm Li}_2\left (c x + 1\right )\right )} b c d^{3} - \frac{1}{4} \, b c d^{3} \log \left (c x + 1\right ) + \frac{1}{4} \, b c d^{3} \log \left (c x - 1\right ) + 3 \, a c d^{3} \log \left (x\right ) - \frac{1}{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 )} b d^{3} - \frac{a d^{3}}{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 c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{3} x^{2} + 3 \, a c d^{3} x + a d^{3} +{\left (b c^{3} d^{3} x^{3} + 3 \, b c^{2} d^{3} x^{2} + 3 \, b c d^{3} x + b d^{3}\right )} \operatorname{artanh}\left (c x\right )}{x^{2}}, 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 c^{2}\, dx + \int \frac{a}{x^{2}}\, dx + \int \frac{3 a c}{x}\, dx + \int a c^{3} x\, dx + \int 3 b c^{2} \operatorname{atanh}{\left (c x \right )}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{3 b c \operatorname{atanh}{\left (c x \right )}}{x}\, dx + \int b c^{3} x \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 )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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