Optimal. Leaf size=185 \[ \frac{b c^3 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )}{2 d}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac{c^3 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac{a+b \tanh ^{-1}(c x)}{3 d x^3}-\frac{2 b c^3 \log \left (1-c^2 x^2\right )}{3 d}+\frac{b c^2}{2 d x}+\frac{4 b c^3 \log (x)}{3 d}-\frac{b c^3 \tanh ^{-1}(c x)}{2 d}-\frac{b c}{6 d x^2} \]
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Rubi [A] time = 0.345908, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {5934, 5916, 266, 44, 325, 206, 36, 29, 31, 5932, 2447} \[ \frac{b c^3 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )}{2 d}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac{c^3 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac{a+b \tanh ^{-1}(c x)}{3 d x^3}-\frac{2 b c^3 \log \left (1-c^2 x^2\right )}{3 d}+\frac{b c^2}{2 d x}+\frac{4 b c^3 \log (x)}{3 d}-\frac{b c^3 \tanh ^{-1}(c x)}{2 d}-\frac{b c}{6 d x^2} \]
Antiderivative was successfully verified.
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Rule 5934
Rule 5916
Rule 266
Rule 44
Rule 325
Rule 206
Rule 36
Rule 29
Rule 31
Rule 5932
Rule 2447
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{x^4 (d+c d x)} \, dx &=-\left (c \int \frac{a+b \tanh ^{-1}(c x)}{x^3 (d+c d x)} \, dx\right )+\frac{\int \frac{a+b \tanh ^{-1}(c x)}{x^4} \, dx}{d}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{3 d x^3}+c^2 \int \frac{a+b \tanh ^{-1}(c x)}{x^2 (d+c d x)} \, dx-\frac{c \int \frac{a+b \tanh ^{-1}(c x)}{x^3} \, dx}{d}+\frac{(b c) \int \frac{1}{x^3 \left (1-c^2 x^2\right )} \, dx}{3 d}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-c^3 \int \frac{a+b \tanh ^{-1}(c x)}{x (d+c d x)} \, dx+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )}{6 d}+\frac{c^2 \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d}-\frac{\left (b c^2\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx}{2 d}\\ &=\frac{b c^2}{2 d x}-\frac{a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac{c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{(b c) \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 )}{6 d}+\frac{\left (b c^3\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx}{d}-\frac{\left (b c^4\right ) \int \frac{1}{1-c^2 x^2} \, dx}{2 d}+\frac{\left (b c^4\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac{b c}{6 d x^2}+\frac{b c^2}{2 d x}-\frac{b c^3 \tanh ^{-1}(c x)}{2 d}-\frac{a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac{b c^3 \log (x)}{3 d}-\frac{b c^3 \log \left (1-c^2 x^2\right )}{6 d}-\frac{c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{b c^3 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{2 d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{b c}{6 d x^2}+\frac{b c^2}{2 d x}-\frac{b c^3 \tanh ^{-1}(c x)}{2 d}-\frac{a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac{b c^3 \log (x)}{3 d}-\frac{b c^3 \log \left (1-c^2 x^2\right )}{6 d}-\frac{c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{b c^3 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{2 d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 d}+\frac{\left (b c^5\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{b c}{6 d x^2}+\frac{b c^2}{2 d x}-\frac{b c^3 \tanh ^{-1}(c x)}{2 d}-\frac{a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac{4 b c^3 \log (x)}{3 d}-\frac{2 b c^3 \log \left (1-c^2 x^2\right )}{3 d}-\frac{c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{b c^3 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.42305, size = 172, normalized size = 0.93 \[ \frac{3 b c^3 x^3 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )-6 a c^2 x^2-6 a c^3 x^3 \log (x)+6 a c^3 x^3 \log (c x+1)+3 a c x-2 a+b c^3 x^3+3 b c^2 x^2+8 b c^3 x^3 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )-b \tanh ^{-1}(c x) \left (3 c^3 x^3+6 c^2 x^2+6 c^3 x^3 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-3 c x+2\right )-b c x}{6 d x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.059, size = 328, normalized size = 1.8 \begin{align*} -{\frac{a}{3\,d{x}^{3}}}-{\frac{a{c}^{2}}{dx}}+{\frac{ac}{2\,d{x}^{2}}}-{\frac{{c}^{3}a\ln \left ( cx \right ) }{d}}+{\frac{{c}^{3}a\ln \left ( cx+1 \right ) }{d}}-{\frac{b{\it Artanh} \left ( cx \right ) }{3\,d{x}^{3}}}-{\frac{{c}^{2}b{\it Artanh} \left ( cx \right ) }{dx}}+{\frac{bc{\it Artanh} \left ( cx \right ) }{2\,d{x}^{2}}}-{\frac{{c}^{3}b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) }{d}}+{\frac{{c}^{3}b{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{d}}-{\frac{5\,{c}^{3}b\ln \left ( cx-1 \right ) }{12\,d}}-{\frac{bc}{6\,d{x}^{2}}}+{\frac{{c}^{2}b}{2\,dx}}+{\frac{4\,{c}^{3}b\ln \left ( cx \right ) }{3\,d}}-{\frac{11\,{c}^{3}b\ln \left ( cx+1 \right ) }{12\,d}}+{\frac{{c}^{3}b{\it dilog} \left ( cx \right ) }{2\,d}}+{\frac{{c}^{3}b{\it dilog} \left ( cx+1 \right ) }{2\,d}}+{\frac{{c}^{3}b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2\,d}}-{\frac{{c}^{3}b}{2\,d}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{{c}^{3}b\ln \left ( cx+1 \right ) }{2\,d}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }-{\frac{{c}^{3}b}{2\,d}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{{c}^{3}b \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{4\,d}} \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}{6} \,{\left (\frac{6 \, c^{3} \log \left (c x + 1\right )}{d} - \frac{6 \, c^{3} \log \left (x\right )}{d} - \frac{6 \, c^{2} x^{2} - 3 \, c x + 2}{d x^{3}}\right )} a + \frac{1}{2} \, b \int \frac{\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{c d x^{5} + d x^{4}}\,{d 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{b \operatorname{artanh}\left (c x\right ) + a}{c d x^{5} + d 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*} \frac{\int \frac{a}{c x^{5} + x^{4}}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{c x^{5} + x^{4}}\, dx}{d} \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{b \operatorname{artanh}\left (c x\right ) + a}{{\left (c d x + d\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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