Optimal. Leaf size=244 \[ \frac{1}{4} b c^4 e \text{PolyLog}(2,-c x)-\frac{1}{4} b c^4 e \text{PolyLog}(2,c x)-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x^4}+\frac{1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac{1}{12} c^4 e (3 a-4 b) \log (c x+1)+\frac{a c^2 e}{4 x^2}-\frac{1}{2} a c^4 e \log (x)-\frac{b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x}-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{12 x^3}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac{b c^2 e \tanh ^{-1}(c x)}{4 x^2}+\frac{5 b c^3 e}{12 x}-\frac{1}{4} b c^4 e \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.260507, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {5916, 325, 206, 6085, 1802, 6044, 5912} \[ \frac{1}{4} b c^4 e \text{PolyLog}(2,-c x)-\frac{1}{4} b c^4 e \text{PolyLog}(2,c x)-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x^4}+\frac{1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac{1}{12} c^4 e (3 a-4 b) \log (c x+1)+\frac{a c^2 e}{4 x^2}-\frac{1}{2} a c^4 e \log (x)-\frac{b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x}-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{12 x^3}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac{b c^2 e \tanh ^{-1}(c x)}{4 x^2}+\frac{5 b c^3 e}{12 x}-\frac{1}{4} b c^4 e \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5916
Rule 325
Rule 206
Rule 6085
Rule 1802
Rule 6044
Rule 5912
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx &=-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac{b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\left (2 c^2 e\right ) \int \left (\frac{3 a+b c x+3 b c^3 x^3}{12 x^3 \left (-1+c^2 x^2\right )}-\frac{b \left (1+c^2 x^2\right ) \tanh ^{-1}(c x)}{4 x^3}\right ) \, dx\\ &=-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac{b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac{1}{6} \left (c^2 e\right ) \int \frac{3 a+b c x+3 b c^3 x^3}{x^3 \left (-1+c^2 x^2\right )} \, dx-\frac{1}{2} \left (b c^2 e\right ) \int \frac{\left (1+c^2 x^2\right ) \tanh ^{-1}(c x)}{x^3} \, dx\\ &=-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac{b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac{1}{6} \left (c^2 e\right ) \int \left (-\frac{3 a}{x^3}-\frac{b c}{x^2}-\frac{3 a c^2}{x}+\frac{(3 a+4 b) c^3}{2 (-1+c x)}+\frac{(3 a-4 b) c^3}{2 (1+c x)}\right ) \, dx-\frac{1}{2} \left (b c^2 e\right ) \int \left (\frac{\tanh ^{-1}(c x)}{x^3}+\frac{c^2 \tanh ^{-1}(c x)}{x}\right ) \, dx\\ &=\frac{a c^2 e}{4 x^2}+\frac{b c^3 e}{6 x}-\frac{1}{2} a c^4 e \log (x)+\frac{1}{12} (3 a+4 b) c^4 e \log (1-c x)+\frac{1}{12} (3 a-4 b) c^4 e \log (1+c x)-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac{b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}-\frac{1}{2} \left (b c^2 e\right ) \int \frac{\tanh ^{-1}(c x)}{x^3} \, dx-\frac{1}{2} \left (b c^4 e\right ) \int \frac{\tanh ^{-1}(c x)}{x} \, dx\\ &=\frac{a c^2 e}{4 x^2}+\frac{b c^3 e}{6 x}+\frac{b c^2 e \tanh ^{-1}(c x)}{4 x^2}-\frac{1}{2} a c^4 e \log (x)+\frac{1}{12} (3 a+4 b) c^4 e \log (1-c x)+\frac{1}{12} (3 a-4 b) c^4 e \log (1+c x)-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac{b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac{1}{4} b c^4 e \text{Li}_2(-c x)-\frac{1}{4} b c^4 e \text{Li}_2(c x)-\frac{1}{4} \left (b c^3 e\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=\frac{a c^2 e}{4 x^2}+\frac{5 b c^3 e}{12 x}+\frac{b c^2 e \tanh ^{-1}(c x)}{4 x^2}-\frac{1}{2} a c^4 e \log (x)+\frac{1}{12} (3 a+4 b) c^4 e \log (1-c x)+\frac{1}{12} (3 a-4 b) c^4 e \log (1+c x)-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac{b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac{1}{4} b c^4 e \text{Li}_2(-c x)-\frac{1}{4} b c^4 e \text{Li}_2(c x)-\frac{1}{4} \left (b c^5 e\right ) \int \frac{1}{1-c^2 x^2} \, dx\\ &=\frac{a c^2 e}{4 x^2}+\frac{5 b c^3 e}{12 x}-\frac{1}{4} b c^4 e \tanh ^{-1}(c x)+\frac{b c^2 e \tanh ^{-1}(c x)}{4 x^2}-\frac{1}{2} a c^4 e \log (x)+\frac{1}{12} (3 a+4 b) c^4 e \log (1-c x)+\frac{1}{12} (3 a-4 b) c^4 e \log (1+c x)-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac{b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac{1}{4} b c^4 e \text{Li}_2(-c x)-\frac{1}{4} b c^4 e \text{Li}_2(c x)\\ \end{align*}
Mathematica [A] time = 0.15077, size = 299, normalized size = 1.23 \[ -\frac{1}{4} b c^4 e (\text{PolyLog}(2,c x)-\text{PolyLog}(2,-c x))+\frac{e \log \left (1-c^2 x^2\right ) \left (-3 a-3 b c^3 x^3+3 b c^4 x^4 \tanh ^{-1}(c x)-b c x-3 b \tanh ^{-1}(c x)\right )}{12 x^4}+\frac{1}{12} \log (1-c x) \left (3 a c^4 e+4 b c^4 e\right )+\frac{1}{12} \log (c x+1) \left (3 a c^4 e-4 b c^4 e\right )+\frac{a c^2 e}{4 x^2}-\frac{1}{2} a c^4 e \log (x)-\frac{a d}{4 x^4}+b c^4 d \left (\frac{1}{4} \left (-\frac{1}{3 c^3 x^3}-\frac{1}{c x}-\frac{1}{2} \log (1-c x)+\frac{1}{2} \log (c x+1)\right )-\frac{\tanh ^{-1}(c x)}{4 c^4 x^4}\right )-\frac{1}{2} b c^4 e \left (\frac{1}{2} \left (-\frac{1}{c x}-\frac{1}{2} \log (1-c x)+\frac{1}{2} \log (c x+1)\right )-\frac{\tanh ^{-1}(c x)}{2 c^2 x^2}\right )+\frac{b c^3 e}{6 x} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 14.305, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Artanh} \left ( cx \right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{5}}}\, dx \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}{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 + \frac{1}{4} \,{\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^{2} - \frac{\log \left (-c^{2} x^{2} + 1\right )}{x^{4}}\right )} a e + \frac{1}{8} \, b e{\left (\frac{\log \left (-c x + 1\right )^{2}}{x^{4}} - 4 \, \int -\frac{2 \,{\left (c x - 1\right )} \log \left (c x + 1\right )^{2} - c x \log \left (-c x + 1\right )}{2 \,{\left (c x^{6} - x^{5}\right )}}\,{d x}\right )} - \frac{a d}{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{b d \operatorname{artanh}\left (c x\right ) + a d +{\left (b e \operatorname{artanh}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\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*} \int \frac{\left (a + b \operatorname{atanh}{\left (c x \right )}\right ) \left (d + e \log{\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{5}}\, dx \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 (b \operatorname{artanh}\left (c x\right ) + a\right )}{\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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