Optimal. Leaf size=209 \[ -\frac{1}{2} b c^4 d^4 \text{PolyLog}(2,-c x)+\frac{1}{2} b c^4 d^4 \text{PolyLog}(2,c x)-\frac{3 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{4 c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}+a c^4 d^4 \log (x)-\frac{2 b c^2 d^4}{3 x^2}-\frac{8}{3} b c^4 d^4 \log \left (1-c^2 x^2\right )-\frac{13 b c^3 d^4}{4 x}+\frac{16}{3} b c^4 d^4 \log (x)+\frac{13}{4} b c^4 d^4 \tanh ^{-1}(c x)-\frac{b c d^4}{12 x^3} \]
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Rubi [A] time = 0.229174, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5940, 5916, 325, 206, 266, 44, 36, 29, 31, 5912} \[ -\frac{1}{2} b c^4 d^4 \text{PolyLog}(2,-c x)+\frac{1}{2} b c^4 d^4 \text{PolyLog}(2,c x)-\frac{3 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{4 c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}+a c^4 d^4 \log (x)-\frac{2 b c^2 d^4}{3 x^2}-\frac{8}{3} b c^4 d^4 \log \left (1-c^2 x^2\right )-\frac{13 b c^3 d^4}{4 x}+\frac{16}{3} b c^4 d^4 \log (x)+\frac{13}{4} b c^4 d^4 \tanh ^{-1}(c x)-\frac{b c d^4}{12 x^3} \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5916
Rule 325
Rule 206
Rule 266
Rule 44
Rule 36
Rule 29
Rule 31
Rule 5912
Rubi steps
\begin{align*} \int \frac{(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^5} \, dx &=\int \left (\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^5}+\frac{4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}+\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^3}+\frac{4 c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac{c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^4 \int \frac{a+b \tanh ^{-1}(c x)}{x^5} \, dx+\left (4 c d^4\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^4} \, dx+\left (6 c^2 d^4\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (4 c^3 d^4\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (c^4 d^4\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx\\ &=-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{3 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{4 c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^4 d^4 \log (x)-\frac{1}{2} b c^4 d^4 \text{Li}_2(-c x)+\frac{1}{2} b c^4 d^4 \text{Li}_2(c x)+\frac{1}{4} \left (b c d^4\right ) \int \frac{1}{x^4 \left (1-c^2 x^2\right )} \, dx+\frac{1}{3} \left (4 b c^2 d^4\right ) \int \frac{1}{x^3 \left (1-c^2 x^2\right )} \, dx+\left (3 b c^3 d^4\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (4 b c^4 d^4\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{b c d^4}{12 x^3}-\frac{3 b c^3 d^4}{x}-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{3 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{4 c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^4 d^4 \log (x)-\frac{1}{2} b c^4 d^4 \text{Li}_2(-c x)+\frac{1}{2} b c^4 d^4 \text{Li}_2(c x)+\frac{1}{3} \left (2 b c^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac{1}{4} \left (b c^3 d^4\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (2 b c^4 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\left (3 b c^5 d^4\right ) \int \frac{1}{1-c^2 x^2} \, dx\\ &=-\frac{b c d^4}{12 x^3}-\frac{13 b c^3 d^4}{4 x}+3 b c^4 d^4 \tanh ^{-1}(c x)-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{3 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{4 c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^4 d^4 \log (x)-\frac{1}{2} b c^4 d^4 \text{Li}_2(-c x)+\frac{1}{2} b c^4 d^4 \text{Li}_2(c x)+\frac{1}{3} \left (2 b c^2 d^4\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{c^2}{x}-\frac{c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\left (2 b c^4 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{4} \left (b c^5 d^4\right ) \int \frac{1}{1-c^2 x^2} \, dx+\left (2 b c^6 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d^4}{12 x^3}-\frac{2 b c^2 d^4}{3 x^2}-\frac{13 b c^3 d^4}{4 x}+\frac{13}{4} b c^4 d^4 \tanh ^{-1}(c x)-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{3 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{4 c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^4 d^4 \log (x)+\frac{16}{3} b c^4 d^4 \log (x)-\frac{8}{3} b c^4 d^4 \log \left (1-c^2 x^2\right )-\frac{1}{2} b c^4 d^4 \text{Li}_2(-c x)+\frac{1}{2} b c^4 d^4 \text{Li}_2(c x)\\ \end{align*}
Mathematica [A] time = 0.170063, size = 206, normalized size = 0.99 \[ \frac{d^4 \left (-12 b c^4 x^4 \text{PolyLog}(2,-c x)+12 b c^4 x^4 \text{PolyLog}(2,c x)-96 a c^3 x^3-72 a c^2 x^2+24 a c^4 x^4 \log (x)-32 a c x-6 a-78 b c^3 x^3-16 b c^2 x^2+128 b c^4 x^4 \log (c x)-39 b c^4 x^4 \log (1-c x)+39 b c^4 x^4 \log (c x+1)-64 b c^4 x^4 \log \left (1-c^2 x^2\right )-96 b c^3 x^3 \tanh ^{-1}(c x)-72 b c^2 x^2 \tanh ^{-1}(c x)-2 b c x-32 b c x \tanh ^{-1}(c x)-6 b \tanh ^{-1}(c x)\right )}{24 x^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.049, size = 256, normalized size = 1.2 \begin{align*} -{\frac{{d}^{4}a}{4\,{x}^{4}}}-4\,{\frac{{c}^{3}{d}^{4}a}{x}}+{c}^{4}{d}^{4}a\ln \left ( cx \right ) -3\,{\frac{{c}^{2}{d}^{4}a}{{x}^{2}}}-{\frac{4\,c{d}^{4}a}{3\,{x}^{3}}}-{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ) }{4\,{x}^{4}}}-4\,{\frac{{c}^{3}{d}^{4}b{\it Artanh} \left ( cx \right ) }{x}}+{c}^{4}{d}^{4}b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) -3\,{\frac{{c}^{2}{d}^{4}b{\it Artanh} \left ( cx \right ) }{{x}^{2}}}-{\frac{4\,c{d}^{4}b{\it Artanh} \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{103\,{c}^{4}{d}^{4}b\ln \left ( cx-1 \right ) }{24}}-{\frac{c{d}^{4}b}{12\,{x}^{3}}}-{\frac{2\,{c}^{2}{d}^{4}b}{3\,{x}^{2}}}-{\frac{13\,{c}^{3}{d}^{4}b}{4\,x}}+{\frac{16\,{c}^{4}{d}^{4}b\ln \left ( cx \right ) }{3}}-{\frac{25\,{c}^{4}{d}^{4}b\ln \left ( cx+1 \right ) }{24}}-{\frac{{c}^{4}{d}^{4}b{\it dilog} \left ( cx \right ) }{2}}-{\frac{{c}^{4}{d}^{4}b{\it dilog} \left ( cx+1 \right ) }{2}}-{\frac{{c}^{4}{d}^{4}b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2}} \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*} \frac{1}{2} \, b c^{4} d^{4} \int \frac{\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{x}\,{d x} + a c^{4} d^{4} \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 )} b c^{3} d^{4} + \frac{3}{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 )} b c^{2} d^{4} - \frac{2}{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 )} b c d^{4} - \frac{4 \, a c^{3} d^{4}}{x} + \frac{1}{24} \,{\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac{2 \,{\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac{6 \, \operatorname{artanh}\left (c x\right )}{x^{4}}\right )} b d^{4} - \frac{3 \, a c^{2} d^{4}}{x^{2}} - \frac{4 \, a c d^{4}}{3 \, x^{3}} - \frac{a d^{4}}{4 \, x^{4}} \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^{5}}, 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 \frac{a}{x^{5}}\, dx + \int \frac{4 a c}{x^{4}}\, dx + \int \frac{6 a c^{2}}{x^{3}}\, dx + \int \frac{4 a c^{3}}{x^{2}}\, dx + \int \frac{a c^{4}}{x}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{x^{5}}\, dx + \int \frac{4 b c \operatorname{atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{6 b c^{2} \operatorname{atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{4 b c^{3} \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{b c^{4} \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 )}^{4}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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