Optimal. Leaf size=185 \[ -\frac{1}{2} b d^4 \text{PolyLog}(2,-c x)+\frac{1}{2} b d^4 \text{PolyLog}(2,c x)+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 x+a d^4 \log (x)+\frac{1}{12} b c^3 d^4 x^3+\frac{2}{3} b c^2 d^4 x^2+\frac{8}{3} b d^4 \log \left (1-c^2 x^2\right )+\frac{13}{4} b c d^4 x-\frac{13}{4} b d^4 \tanh ^{-1}(c x)+4 b c d^4 x \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.197096, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5940, 5910, 260, 5912, 5916, 321, 206, 266, 43, 302} \[ -\frac{1}{2} b d^4 \text{PolyLog}(2,-c x)+\frac{1}{2} b d^4 \text{PolyLog}(2,c x)+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 x+a d^4 \log (x)+\frac{1}{12} b c^3 d^4 x^3+\frac{2}{3} b c^2 d^4 x^2+\frac{8}{3} b d^4 \log \left (1-c^2 x^2\right )+\frac{13}{4} b c d^4 x-\frac{13}{4} b d^4 \tanh ^{-1}(c x)+4 b c d^4 x \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5910
Rule 260
Rule 5912
Rule 5916
Rule 321
Rule 206
Rule 266
Rule 43
Rule 302
Rubi steps
\begin{align*} \int \frac{(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x} \, dx &=\int \left (4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+6 c^2 d^4 x \left (a+b \tanh ^{-1}(c x)\right )+4 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )\right ) \, dx\\ &=d^4 \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx+\left (4 c d^4\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (6 c^2 d^4\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (4 c^3 d^4\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (c^4 d^4\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=4 a c d^4 x+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+a d^4 \log (x)-\frac{1}{2} b d^4 \text{Li}_2(-c x)+\frac{1}{2} b d^4 \text{Li}_2(c x)+\left (4 b c d^4\right ) \int \tanh ^{-1}(c x) \, dx-\left (3 b c^3 d^4\right ) \int \frac{x^2}{1-c^2 x^2} \, dx-\frac{1}{3} \left (4 b c^4 d^4\right ) \int \frac{x^3}{1-c^2 x^2} \, dx-\frac{1}{4} \left (b c^5 d^4\right ) \int \frac{x^4}{1-c^2 x^2} \, dx\\ &=4 a c d^4 x+3 b c d^4 x+4 b c d^4 x \tanh ^{-1}(c x)+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+a d^4 \log (x)-\frac{1}{2} b d^4 \text{Li}_2(-c x)+\frac{1}{2} b d^4 \text{Li}_2(c x)-\left (3 b c d^4\right ) \int \frac{1}{1-c^2 x^2} \, dx-\left (4 b c^2 d^4\right ) \int \frac{x}{1-c^2 x^2} \, dx-\frac{1}{3} \left (2 b c^4 d^4\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )-\frac{1}{4} \left (b c^5 d^4\right ) \int \left (-\frac{1}{c^4}-\frac{x^2}{c^2}+\frac{1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=4 a c d^4 x+\frac{13}{4} b c d^4 x+\frac{1}{12} b c^3 d^4 x^3-3 b d^4 \tanh ^{-1}(c x)+4 b c d^4 x \tanh ^{-1}(c x)+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+a d^4 \log (x)+2 b d^4 \log \left (1-c^2 x^2\right )-\frac{1}{2} b d^4 \text{Li}_2(-c x)+\frac{1}{2} b d^4 \text{Li}_2(c x)-\frac{1}{4} \left (b c d^4\right ) \int \frac{1}{1-c^2 x^2} \, dx-\frac{1}{3} \left (2 b c^4 d^4\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=4 a c d^4 x+\frac{13}{4} b c d^4 x+\frac{2}{3} b c^2 d^4 x^2+\frac{1}{12} b c^3 d^4 x^3-\frac{13}{4} b d^4 \tanh ^{-1}(c x)+4 b c d^4 x \tanh ^{-1}(c x)+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+a d^4 \log (x)+\frac{8}{3} b d^4 \log \left (1-c^2 x^2\right )-\frac{1}{2} b d^4 \text{Li}_2(-c x)+\frac{1}{2} b d^4 \text{Li}_2(c x)\\ \end{align*}
Mathematica [A] time = 0.170134, size = 179, normalized size = 0.97 \[ \frac{1}{24} d^4 \left (-12 b \text{PolyLog}(2,-c x)+12 b \text{PolyLog}(2,c x)+6 a c^4 x^4+32 a c^3 x^3+72 a c^2 x^2+96 a c x+24 a \log (x)+2 b c^3 x^3+16 b c^2 x^2+48 b \log \left (1-c^2 x^2\right )+16 b \log \left (c^2 x^2-1\right )+6 b c^4 x^4 \tanh ^{-1}(c x)+32 b c^3 x^3 \tanh ^{-1}(c x)+72 b c^2 x^2 \tanh ^{-1}(c x)+78 b c x+39 b \log (1-c x)-39 b \log (c x+1)+96 b c x \tanh ^{-1}(c x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.044, size = 222, normalized size = 1.2 \begin{align*}{\frac{{d}^{4}a{c}^{4}{x}^{4}}{4}}+{\frac{4\,{d}^{4}a{c}^{3}{x}^{3}}{3}}+3\,{d}^{4}a{c}^{2}{x}^{2}+4\,ac{d}^{4}x+{d}^{4}a\ln \left ( cx \right ) +{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ){c}^{4}{x}^{4}}{4}}+{\frac{4\,{d}^{4}b{\it Artanh} \left ( cx \right ){c}^{3}{x}^{3}}{3}}+3\,{d}^{4}b{\it Artanh} \left ( cx \right ){c}^{2}{x}^{2}+4\,bc{d}^{4}x{\it Artanh} \left ( cx \right ) +{d}^{4}b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) -{\frac{{d}^{4}b{\it dilog} \left ( cx \right ) }{2}}-{\frac{{d}^{4}b{\it dilog} \left ( cx+1 \right ) }{2}}-{\frac{{d}^{4}b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2}}+{\frac{b{c}^{3}{d}^{4}{x}^{3}}{12}}+{\frac{2\,b{c}^{2}{d}^{4}{x}^{2}}{3}}+{\frac{13\,bc{d}^{4}x}{4}}+{\frac{103\,{d}^{4}b\ln \left ( cx-1 \right ) }{24}}+{\frac{25\,{d}^{4}b\ln \left ( cx+1 \right ) }{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46915, size = 373, normalized size = 2.02 \begin{align*} \frac{1}{4} \, a c^{4} d^{4} x^{4} + \frac{4}{3} \, a c^{3} d^{4} x^{3} + \frac{1}{12} \, b c^{3} d^{4} x^{3} + 3 \, a c^{2} d^{4} x^{2} + \frac{2}{3} \, b c^{2} d^{4} x^{2} + 4 \, a c d^{4} x + \frac{13}{4} \, b c d^{4} x + 2 \,{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{4} - \frac{1}{2} \,{\left (\log \left (c x\right ) \log \left (-c x + 1\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} b d^{4} + \frac{1}{2} \,{\left (\log \left (c x + 1\right ) \log \left (-c x\right ) +{\rm Li}_2\left (c x + 1\right )\right )} b d^{4} - \frac{23}{24} \, b d^{4} \log \left (c x + 1\right ) + \frac{55}{24} \, b d^{4} \log \left (c x - 1\right ) + a d^{4} \log \left (x\right ) + \frac{1}{24} \,{\left (3 \, b c^{4} d^{4} x^{4} + 16 \, b c^{3} d^{4} x^{3} + 36 \, b c^{2} d^{4} x^{2}\right )} \log \left (c x + 1\right ) - \frac{1}{24} \,{\left (3 \, b c^{4} d^{4} x^{4} + 16 \, b c^{3} d^{4} x^{3} + 36 \, b c^{2} d^{4} x^{2}\right )} \log \left (-c x + 1\right ) \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^{4} d^{4} x^{4} + 4 \, a c^{3} d^{4} x^{3} + 6 \, a c^{2} d^{4} x^{2} + 4 \, a c d^{4} x + a d^{4} +{\left (b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 4 \, b c d^{4} x + b d^{4}\right )} \operatorname{artanh}\left (c x\right )}{x}, 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^{4} \left (\int 4 a c\, dx + \int \frac{a}{x}\, dx + \int 6 a c^{2} x\, dx + \int 4 a c^{3} x^{2}\, dx + \int a c^{4} x^{3}\, dx + \int 4 b c \operatorname{atanh}{\left (c x \right )}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{x}\, dx + \int 6 b c^{2} x \operatorname{atanh}{\left (c x \right )}\, dx + \int 4 b c^{3} x^{2} \operatorname{atanh}{\left (c x \right )}\, dx + \int b c^{4} x^{3} \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 )}^{4}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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