Optimal. Leaf size=315 \[ \frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{e (4 a+3 b) \log (1-c x)}{20 c^5}+\frac{e (4 a-3 b) \log (c x+1)}{20 c^5}-\frac{2 a e x^3}{15 c^2}-\frac{2 a e x}{5 c^4}-\frac{2}{25} a e x^5+\frac{b x^4 \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 c}+\frac{b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^3}+\frac{b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^5}-\frac{77 b e x^2}{300 c^3}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}-\frac{23 b e \log \left (1-c^2 x^2\right )}{75 c^5}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{9 b e x^4}{200 c}-\frac{2}{25} b e x^5 \coth ^{-1}(c x) \]
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Rubi [A] time = 0.749021, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5917, 266, 43, 6086, 6725, 1802, 633, 31, 5981, 5911, 260, 5949, 2475, 2390, 2301} \[ \frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{e (4 a+3 b) \log (1-c x)}{20 c^5}+\frac{e (4 a-3 b) \log (c x+1)}{20 c^5}-\frac{2 a e x^3}{15 c^2}-\frac{2 a e x}{5 c^4}-\frac{2}{25} a e x^5+\frac{b x^4 \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 c}+\frac{b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^3}+\frac{b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^5}-\frac{77 b e x^2}{300 c^3}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}-\frac{23 b e \log \left (1-c^2 x^2\right )}{75 c^5}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{9 b e x^4}{200 c}-\frac{2}{25} b e x^5 \coth ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5917
Rule 266
Rule 43
Rule 6086
Rule 6725
Rule 1802
Rule 633
Rule 31
Rule 5981
Rule 5911
Rule 260
Rule 5949
Rule 2475
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\left (2 c^2 e\right ) \int \left (\frac{-2 b x^3-b c^2 x^5-4 a c^3 x^6-4 b c^3 x^6 \coth ^{-1}(c x)}{20 c^3 \left (-1+c^2 x^2\right )}-\frac{b x \log \left (1-c^2 x^2\right )}{10 c^5 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{(b e) \int \frac{x \log \left (1-c^2 x^2\right )}{-1+c^2 x^2} \, dx}{5 c^3}+\frac{e \int \frac{-2 b x^3-b c^2 x^5-4 a c^3 x^6-4 b c^3 x^6 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx}{10 c}\\ &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (1-c^2 x\right )}{-1+c^2 x} \, dx,x,x^2\right )}{10 c^3}+\frac{e \int \left (\frac{x^3 \left (2 b+b c^2 x^2+4 a c^3 x^3\right )}{1-c^2 x^2}-\frac{4 b c^3 x^6 \coth ^{-1}(c x)}{-1+c^2 x^2}\right ) \, dx}{10 c}\\ &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c^2 x^2\right )}{10 c^5}+\frac{e \int \frac{x^3 \left (2 b+b c^2 x^2+4 a c^3 x^3\right )}{1-c^2 x^2} \, dx}{10 c}-\frac{1}{5} \left (2 b c^2 e\right ) \int \frac{x^6 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx\\ &=-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{1}{5} (2 b e) \int x^4 \coth ^{-1}(c x) \, dx-\frac{1}{5} (2 b e) \int \frac{x^4 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac{e \int \left (-\frac{4 a}{c^3}-\frac{3 b x}{c^2}-\frac{4 a x^2}{c}-b x^3-4 a c x^4+\frac{4 a+3 b c x}{c^3 \left (1-c^2 x^2\right )}\right ) \, dx}{10 c}\\ &=-\frac{2 a e x}{5 c^4}-\frac{3 b e x^2}{20 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{b e x^4}{40 c}-\frac{2}{25} a e x^5-\frac{2}{25} b e x^5 \coth ^{-1}(c x)-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\frac{e \int \frac{4 a+3 b c x}{1-c^2 x^2} \, dx}{10 c^4}-\frac{(2 b e) \int x^2 \coth ^{-1}(c x) \, dx}{5 c^2}-\frac{(2 b e) \int \frac{x^2 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx}{5 c^2}+\frac{1}{25} (2 b c e) \int \frac{x^5}{1-c^2 x^2} \, dx\\ &=-\frac{2 a e x}{5 c^4}-\frac{3 b e x^2}{20 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{b e x^4}{40 c}-\frac{2}{25} a e x^5-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \coth ^{-1}(c x)-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{(2 b e) \int \coth ^{-1}(c x) \, dx}{5 c^4}-\frac{(2 b e) \int \frac{\coth ^{-1}(c x)}{-1+c^2 x^2} \, dx}{5 c^4}-\frac{((4 a-3 b) e) \int \frac{1}{-c-c^2 x} \, dx}{20 c^3}+\frac{((4 a+3 b) e) \int \frac{1}{c-c^2 x} \, dx}{20 c^3}+\frac{(2 b e) \int \frac{x^3}{1-c^2 x^2} \, dx}{15 c}+\frac{1}{25} (b c e) \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{2 a e x}{5 c^4}-\frac{3 b e x^2}{20 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{b e x^4}{40 c}-\frac{2}{25} a e x^5-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac{(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\frac{(2 b e) \int \frac{x}{1-c^2 x^2} \, dx}{5 c^3}+\frac{(b e) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )}{15 c}+\frac{1}{25} (b c e) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x}{c^2}-\frac{1}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{2 a e x}{5 c^4}-\frac{19 b e x^2}{100 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{9 b e x^4}{200 c}-\frac{2}{25} a e x^5-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac{(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac{6 b e \log \left (1-c^2 x^2\right )}{25 c^5}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\frac{(b e) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{15 c}\\ &=-\frac{2 a e x}{5 c^4}-\frac{77 b e x^2}{300 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{9 b e x^4}{200 c}-\frac{2}{25} a e x^5-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac{(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac{23 b e \log \left (1-c^2 x^2\right )}{75 c^5}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}\\ \end{align*}
Mathematica [A] time = 0.148721, size = 236, normalized size = 0.75 \[ \frac{30 c^2 e x^2 \log \left (1-c^2 x^2\right ) \left (4 a c^3 x^3+b \left (c^2 x^2+2\right )+4 b c^3 x^3 \coth ^{-1}(c x)\right )+2 \log (1-c x) (-60 a e+30 b d-137 b e)+2 \log (c x+1) (60 a e+30 b d-137 b e)+24 a c^5 x^5 (5 d-2 e)-80 a c^3 e x^3-240 a c e x+3 b c^4 x^4 (10 d-9 e)+2 b c^2 x^2 (30 d-77 e)-8 b c x \coth ^{-1}(c x) \left (2 e \left (3 c^4 x^4+5 c^2 x^2+15\right )-15 c^4 d x^4\right )+30 b e \log ^2\left (1-c^2 x^2\right )+120 b e \coth ^{-1}(c x)^2}{600 c^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.72, size = 4194, normalized size = 13.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.10989, size = 428, normalized size = 1.36 \begin{align*} \frac{1}{5} \, a d x^{5} + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2}{\left (\frac{2 \,{\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac{15 \, \log \left (c x + 1\right )}{c^{7}} + \frac{15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b e \operatorname{arcoth}\left (c x\right ) + \frac{1}{20} \,{\left (4 \, x^{5} \operatorname{arcoth}\left (c x\right ) + c{\left (\frac{c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b d + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2}{\left (\frac{2 \,{\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac{15 \, \log \left (c x + 1\right )}{c^{7}} + \frac{15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} a e - \frac{{\left (3 \,{\left (-10 i \, \pi c^{4} + 9 \, c^{4}\right )} x^{4} + 2 \,{\left (-30 i \, \pi c^{2} + 77 \, c^{2}\right )} x^{2} -{\left (60 i \, \pi + 30 \, c^{4} x^{4} + 60 \, c^{2} x^{2} + 120 \, \log \left (c x - 1\right ) - 274\right )} \log \left (c x + 1\right ) -{\left (60 i \, \pi + 30 \, c^{4} x^{4} + 60 \, c^{2} x^{2} - 274\right )} \log \left (c x - 1\right )\right )} b e}{600 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67589, size = 587, normalized size = 1.86 \begin{align*} -\frac{80 \, a c^{3} e x^{3} - 24 \,{\left (5 \, a c^{5} d - 2 \, a c^{5} e\right )} x^{5} - 3 \,{\left (10 \, b c^{4} d - 9 \, b c^{4} e\right )} x^{4} + 240 \, a c e x - 30 \, b e \log \left (-c^{2} x^{2} + 1\right )^{2} - 30 \, b e \log \left (\frac{c x + 1}{c x - 1}\right )^{2} - 2 \,{\left (30 \, b c^{2} d - 77 \, b c^{2} e\right )} x^{2} - 2 \,{\left (60 \, a c^{5} e x^{5} + 15 \, b c^{4} e x^{4} + 30 \, b c^{2} e x^{2} + 30 \, b d - 137 \, b e\right )} \log \left (-c^{2} x^{2} + 1\right ) - 4 \,{\left (15 \, b c^{5} e x^{5} \log \left (-c^{2} x^{2} + 1\right ) - 10 \, b c^{3} e x^{3} + 3 \,{\left (5 \, b c^{5} d - 2 \, b c^{5} e\right )} x^{5} - 30 \, b c e x + 30 \, a e\right )} \log \left (\frac{c x + 1}{c x - 1}\right )}{600 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 43.4042, size = 345, normalized size = 1.1 \begin{align*} \begin{cases} \frac{a d x^{5}}{5} + \frac{a e x^{5} \log{\left (- c^{2} x^{2} + 1 \right )}}{5} - \frac{2 a e x^{5}}{25} - \frac{2 a e x^{3}}{15 c^{2}} - \frac{2 a e x}{5 c^{4}} + \frac{2 a e \operatorname{acoth}{\left (c x \right )}}{5 c^{5}} + \frac{b d x^{5} \operatorname{acoth}{\left (c x \right )}}{5} + \frac{b e x^{5} \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{acoth}{\left (c x \right )}}{5} - \frac{2 b e x^{5} \operatorname{acoth}{\left (c x \right )}}{25} + \frac{b d x^{4}}{20 c} + \frac{b e x^{4} \log{\left (- c^{2} x^{2} + 1 \right )}}{20 c} - \frac{9 b e x^{4}}{200 c} - \frac{2 b e x^{3} \operatorname{acoth}{\left (c x \right )}}{15 c^{2}} + \frac{b d x^{2}}{10 c^{3}} + \frac{b e x^{2} \log{\left (- c^{2} x^{2} + 1 \right )}}{10 c^{3}} - \frac{77 b e x^{2}}{300 c^{3}} - \frac{2 b e x \operatorname{acoth}{\left (c x \right )}}{5 c^{4}} + \frac{b d \log{\left (- c^{2} x^{2} + 1 \right )}}{10 c^{5}} + \frac{b e \log{\left (- c^{2} x^{2} + 1 \right )}^{2}}{20 c^{5}} - \frac{137 b e \log{\left (- c^{2} x^{2} + 1 \right )}}{300 c^{5}} + \frac{b e \operatorname{acoth}^{2}{\left (c x \right )}}{5 c^{5}} & \text{for}\: c \neq 0 \\\frac{d x^{5} \left (a + \frac{i \pi b}{2}\right )}{5} & \text{otherwise} \end{cases} \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{\left (b \operatorname{arcoth}\left (c x\right ) + a\right )}{\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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