Optimal. Leaf size=256 \[ -\frac{1}{10} b c^5 e \text{PolyLog}\left (2,\frac{1}{1-c^2 x^2}\right )-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{5 x^5}+\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}-\frac{c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}+\frac{2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}+\frac{1}{10} b c^5 \log \left (1-\frac{1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{b c^3 \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 x^2}-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 x^4}+\frac{7 b c^3 e}{60 x^2}+\frac{19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac{5}{6} b c^5 e \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.670072, antiderivative size = 250, normalized size of antiderivative = 0.98, number of steps used = 26, number of rules used = 18, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6082, 2475, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2319, 44, 5983, 5917, 266, 36, 29, 5949} \[ -\frac{1}{10} b c^5 e \text{PolyLog}\left (2,c^2 x^2\right )-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{5 x^5}+\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}-\frac{c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}+\frac{2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac{b c^5 \left (e \log \left (1-c^2 x^2\right )+d\right )^2}{20 e}-\frac{b c^3 \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 x^2}-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 x^4}+\frac{1}{5} b c^5 d \log (x)+\frac{7 b c^3 e}{60 x^2}+\frac{19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac{5}{6} b c^5 e \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6082
Rule 2475
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2316
Rule 2315
Rule 2314
Rule 31
Rule 2319
Rule 44
Rule 5983
Rule 5917
Rule 266
Rule 36
Rule 29
Rule 5949
Rubi steps
\begin{align*} \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx &=-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac{1}{5} (b c) \int \frac{d+e \log \left (1-c^2 x^2\right )}{x^5 \left (1-c^2 x^2\right )} \, dx-\frac{1}{5} \left (2 c^2 e\right ) \int \frac{a+b \coth ^{-1}(c x)}{x^4 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac{1}{10} (b c) \operatorname{Subst}\left (\int \frac{d+e \log \left (1-c^2 x\right )}{x^3 \left (1-c^2 x\right )} \, dx,x,x^2\right )-\frac{1}{5} \left (2 c^2 e\right ) \int \frac{a+b \coth ^{-1}(c x)}{x^4} \, dx-\frac{1}{5} \left (2 c^4 e\right ) \int \frac{a+b \coth ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac{b \operatorname{Subst}\left (\int \frac{d+e \log (x)}{x \left (\frac{1}{c^2}-\frac{x}{c^2}\right )^3} \, dx,x,1-c^2 x^2\right )}{10 c}-\frac{1}{15} \left (2 b c^3 e\right ) \int \frac{1}{x^3 \left (1-c^2 x^2\right )} \, dx-\frac{1}{5} \left (2 c^4 e\right ) \int \frac{a+b \coth ^{-1}(c x)}{x^2} \, dx-\frac{1}{5} \left (2 c^6 e\right ) \int \frac{a+b \coth ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac{2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac{c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac{b \operatorname{Subst}\left (\int \frac{d+e \log (x)}{\left (\frac{1}{c^2}-\frac{x}{c^2}\right )^3} \, dx,x,1-c^2 x^2\right )}{10 c}-\frac{1}{10} (b c) \operatorname{Subst}\left (\int \frac{d+e \log (x)}{x \left (\frac{1}{c^2}-\frac{x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )-\frac{1}{15} \left (b c^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )-\frac{1}{5} \left (2 b c^5 e\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx\\ &=\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac{2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac{c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac{1}{10} (b c) \operatorname{Subst}\left (\int \frac{d+e \log (x)}{\left (\frac{1}{c^2}-\frac{x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )-\frac{1}{10} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{d+e \log (x)}{x \left (\frac{1}{c^2}-\frac{x}{c^2}\right )} \, dx,x,1-c^2 x^2\right )+\frac{1}{20} (b c e) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{1}{c^2}-\frac{x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )-\frac{1}{15} \left (b c^3 e\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 )-\frac{1}{5} \left (b c^5 e\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=\frac{b c^3 e}{15 x^2}+\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac{2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac{c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}-\frac{2}{15} b c^5 e \log (x)+\frac{1}{15} b c^5 e \log \left (1-c^2 x^2\right )-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac{b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac{1}{10} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{d+e \log (x)}{\frac{1}{c^2}-\frac{x}{c^2}} \, dx,x,1-c^2 x^2\right )-\frac{1}{10} \left (b c^5\right ) \operatorname{Subst}\left (\int \frac{d+e \log (x)}{x} \, dx,x,1-c^2 x^2\right )+\frac{1}{20} (b c e) \operatorname{Subst}\left (\int \left (\frac{c^4}{(-1+x)^2}-\frac{c^4}{-1+x}+\frac{c^4}{x}\right ) \, dx,x,1-c^2 x^2\right )+\frac{1}{10} \left (b c^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x}{c^2}} \, dx,x,1-c^2 x^2\right )-\frac{1}{5} \left (b c^5 e\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{5} \left (b c^7 e\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac{7 b c^3 e}{60 x^2}+\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac{2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac{c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}+\frac{1}{5} b c^5 d \log (x)-\frac{5}{6} b c^5 e \log (x)+\frac{19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac{b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac{b c^5 \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{20 e}-\frac{1}{10} \left (b c^3 e\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c^2}-\frac{x}{c^2}} \, dx,x,1-c^2 x^2\right )\\ &=\frac{7 b c^3 e}{60 x^2}+\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac{2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac{c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}+\frac{1}{5} b c^5 d \log (x)-\frac{5}{6} b c^5 e \log (x)+\frac{19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac{b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac{b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac{b c^5 \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{20 e}-\frac{1}{10} b c^5 e \text{Li}_2\left (c^2 x^2\right )\\ \end{align*}
Mathematica [F] time = 0.279638, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 4.318, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccoth} \left (cx\right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{6}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{20} \,{\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac{2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac{4 \, \operatorname{arcoth}\left (c x\right )}{x^{5}}\right )} b d - \frac{1}{15} \,{\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^{2} + \frac{3 \, \log \left (-c^{2} x^{2} + 1\right )}{x^{5}}\right )} a e - \frac{1}{10} \, b e{\left (\frac{\log \left (c x + 1\right )^{2}}{x^{5}} - 5 \, \int -\frac{5 \,{\left (c x + 1\right )} \log \left (c x - 1\right )^{2} -{\left (5 i \, \pi +{\left (5 i \, \pi c + 2 \, c\right )} x\right )} \log \left (c x + 1\right ) -{\left (-5 i \, \pi - 5 i \, \pi c x\right )} \log \left (c x - 1\right )}{5 \,{\left (c x^{7} + x^{6}\right )}}\,{d x}\right )} - \frac{a d}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b d \operatorname{arcoth}\left (c x\right ) + a d +{\left (b e \operatorname{arcoth}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\right )}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]