Integrand size = 27, antiderivative size = 339 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx=\frac {a c^2 e}{4 x^2}+\frac {5 b c^3 e}{12 x}+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2-\frac {1}{4} b c^4 e \text {arctanh}(c x)-\frac {1}{4} b c^4 e \text {arctanh}(c x)^2-\frac {1}{2} a c^4 e \log (x)+\frac {1}{2} b c^4 e \text {arctanh}(c x) \log \left (\frac {2}{1-c x}\right )+\frac {1}{12} (3 a+4 b) c^4 e \log (1-c x)+\frac {1}{12} (3 a-4 b) c^4 e \log (1+c x)-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac {1}{4} b c^4 \text {arctanh}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {1}{2} b c^4 e \coth ^{-1}(c x) \log \left (2-\frac {2}{1+c x}\right )+\frac {1}{4} b c^4 e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{4} b c^4 e \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right ) \]
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Time = 0.55 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {6038, 331, 212, 6233, 6857, 1816, 6130, 6136, 6080, 2497, 6131, 6055, 2449, 2352} \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx=\frac {1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac {1}{12} c^4 e (3 a-4 b) \log (c x+1)-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x^4}-\frac {1}{2} a c^4 e \log (x)+\frac {a c^2 e}{4 x^2}-\frac {1}{4} b c^4 e \text {arctanh}(c x)^2-\frac {1}{4} b c^4 e \text {arctanh}(c x)+\frac {1}{2} b c^4 e \text {arctanh}(c x) \log \left (\frac {2}{1-c x}\right )+\frac {1}{4} b c^4 \text {arctanh}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac {1}{4} b c^4 e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{4} b c^4 e \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2-\frac {1}{2} b c^4 e \log \left (2-\frac {2}{c x+1}\right ) \coth ^{-1}(c x)+\frac {5 b c^3 e}{12 x}-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{12 x^3}+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}-\frac {b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x} \]
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Rule 212
Rule 331
Rule 1816
Rule 2352
Rule 2449
Rule 2497
Rule 6038
Rule 6055
Rule 6080
Rule 6130
Rule 6131
Rule 6136
Rule 6233
Rule 6857
Rubi steps \begin{align*} \text {integral}& = -\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 {\left (a+b \coth ^{-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 \text {arctanh}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\left (2 c^2 e\right ) \int \left (\frac {3 a+b c x+3 b c^3 x^3+3 b \coth ^{-1}(c x)}{12 x^3 \left (-1+c^2 x^2\right )}-\frac {b c^4 x \text {arctanh}(c x)}{4 \left (-1+c^2 x^2\right )}\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 {\left (a+b \coth ^{-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 \text {arctanh}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{6} \left (c^2 e\right ) \int \frac {3 a+b c x+3 b c^3 x^3+3 b \coth ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (b c^6 e\right ) \int \frac {x \text {arctanh}(c x)}{-1+c^2 x^2} \, dx \\ & = -\frac {1}{4} b c^4 e \text {arctanh}(c x)^2-\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 {\left (a+b \coth ^{-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 \text {arctanh}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{6} \left (c^2 e\right ) \int \left (\frac {3 a+b c x+3 b c^3 x^3}{x^3 \left (-1+c^2 x^2\right )}+\frac {3 b \coth ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{2} \left (b c^5 e\right ) \int \frac {\text {arctanh}(c x)}{1-c x} \, dx \\ & = -\frac {1}{4} b c^4 e \text {arctanh}(c x)^2+\frac {1}{2} b c^4 e \text {arctanh}(c x) \log \left (\frac {2}{1-c x}\right )-\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 {\left (a+b \coth ^{-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 \text {arctanh}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\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 {\coth ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (b c^5 e\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx \\ & = -\frac {1}{4} b c^4 e \text {arctanh}(c x)^2+\frac {1}{2} b c^4 e \text {arctanh}(c x) \log \left (\frac {2}{1-c x}\right )-\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 {\left (a+b \coth ^{-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 \text {arctanh}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\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 \frac {\coth ^{-1}(c x)}{x^3} \, dx+\frac {1}{2} \left (b c^4 e\right ) \int \frac {\coth ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx+\frac {1}{2} \left (b c^4 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right ) \\ & = \frac {a c^2 e}{4 x^2}+\frac {b c^3 e}{6 x}+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2-\frac {1}{4} b c^4 e \text {arctanh}(c x)^2-\frac {1}{2} a c^4 e \log (x)+\frac {1}{2} b c^4 e \text {arctanh}(c x) \log \left (\frac {2}{1-c x}\right )+\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 {\left (a+b \coth ^{-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 \text {arctanh}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{4} b c^4 e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-\frac {1}{4} \left (b c^3 e\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx-\frac {1}{2} \left (b c^4 e\right ) \int \frac {\coth ^{-1}(c x)}{x (1+c x)} \, dx \\ & = \frac {a c^2 e}{4 x^2}+\frac {5 b c^3 e}{12 x}+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2-\frac {1}{4} b c^4 e \text {arctanh}(c x)^2-\frac {1}{2} a c^4 e \log (x)+\frac {1}{2} b c^4 e \text {arctanh}(c x) \log \left (\frac {2}{1-c x}\right )+\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 {\left (a+b \coth ^{-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 \text {arctanh}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {1}{2} b c^4 e \coth ^{-1}(c x) \log \left (2-\frac {2}{1+c x}\right )+\frac {1}{4} b c^4 e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-\frac {1}{4} \left (b c^5 e\right ) \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{2} \left (b c^5 e\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx \\ & = \frac {a c^2 e}{4 x^2}+\frac {5 b c^3 e}{12 x}+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2-\frac {1}{4} b c^4 e \text {arctanh}(c x)-\frac {1}{4} b c^4 e \text {arctanh}(c x)^2-\frac {1}{2} a c^4 e \log (x)+\frac {1}{2} b c^4 e \text {arctanh}(c x) \log \left (\frac {2}{1-c x}\right )+\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 {\left (a+b \coth ^{-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 \text {arctanh}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {1}{2} b c^4 e \coth ^{-1}(c x) \log \left (2-\frac {2}{1+c x}\right )+\frac {1}{4} b c^4 e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{4} b c^4 e \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx=-\frac {a d}{4 x^4}+\frac {a c^2 e}{4 x^2}+\frac {b c^3 e}{6 x}-\frac {1}{2} a c^4 e \log (x)+\frac {1}{12} \left (3 a c^4 e+4 b c^4 e\right ) \log (1-c x)-\frac {1}{2} b c^4 e \left (-\frac {\coth ^{-1}(c x)}{2 c^2 x^2}+\frac {1}{2} \left (-\frac {1}{c x}-\frac {1}{2} \log (1-c x)+\frac {1}{2} \log (1+c x)\right )\right )+b c^4 d \left (-\frac {\coth ^{-1}(c x)}{4 c^4 x^4}+\frac {1}{4} \left (-\frac {1}{3 c^3 x^3}-\frac {1}{c x}-\frac {1}{2} \log (1-c x)+\frac {1}{2} \log (1+c x)\right )\right )+\frac {1}{12} \left (3 a c^4 e-4 b c^4 e\right ) \log (1+c x)+\frac {e \left (-3 a-b c x-3 b c^3 x^3-3 b \coth ^{-1}(c x)+3 b c^4 x^4 \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{12 x^4}-\frac {1}{4} b c^4 e \left (\operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right ) \]
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\[\int \frac {\left (a +b \,\operatorname {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{5}}d x\]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{5}} \,d x } \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{5}}\, dx \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{5}} \,d x } \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx=\int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^5} \,d x \]
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