Optimal. Leaf size=297 \[ \frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{6 c^6}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )+\frac{b x^3 (3 d-e)}{54 c^3}+\frac{b x (3 d-e)}{18 c^5}-\frac{b (3 d-e) \tanh ^{-1}(c x)}{18 c^6}-\frac{47 b e x^3}{540 c^3}+\frac{b e x^5 \log \left (1-c^2 x^2\right )}{30 c}+\frac{b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac{b e x \log \left (1-c^2 x^2\right )}{6 c^5}-\frac{137 b e x}{180 c^5}+\frac{137 b e \tanh ^{-1}(c x)}{180 c^6}+\frac{b x^5 (3 d-e)}{90 c}-\frac{b e x^5}{75 c} \]
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Rubi [A] time = 0.385512, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {2454, 2395, 43, 6084, 321, 207, 302, 2528, 2448, 206, 2455} \[ \frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{6 c^6}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )+\frac{b x^3 (3 d-e)}{54 c^3}+\frac{b x (3 d-e)}{18 c^5}-\frac{b (3 d-e) \tanh ^{-1}(c x)}{18 c^6}-\frac{47 b e x^3}{540 c^3}+\frac{b e x^5 \log \left (1-c^2 x^2\right )}{30 c}+\frac{b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac{b e x \log \left (1-c^2 x^2\right )}{6 c^5}-\frac{137 b e x}{180 c^5}+\frac{137 b e \tanh ^{-1}(c x)}{180 c^6}+\frac{b x^5 (3 d-e)}{90 c}-\frac{b e x^5}{75 c} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 43
Rule 6084
Rule 321
Rule 207
Rule 302
Rule 2528
Rule 2448
Rule 206
Rule 2455
Rubi steps
\begin{align*} \int x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx &=-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac{e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-(b c) \int \left (\frac{e x^2}{6 c^4 \left (-1+c^2 x^2\right )}+\frac{e x^4}{12 c^2 \left (-1+c^2 x^2\right )}-\frac{d \left (1-\frac{e}{3 d}\right ) x^6}{6 \left (-1+c^2 x^2\right )}-\frac{e \left (1+c^2 x^2+c^4 x^4\right ) \log \left (1-c^2 x^2\right )}{6 c^6}\right ) \, dx\\ &=-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac{e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{1}{18} (b c (3 d-e)) \int \frac{x^6}{-1+c^2 x^2} \, dx+\frac{(b e) \int \left (1+c^2 x^2+c^4 x^4\right ) \log \left (1-c^2 x^2\right ) \, dx}{6 c^5}-\frac{(b e) \int \frac{x^2}{-1+c^2 x^2} \, dx}{6 c^3}-\frac{(b e) \int \frac{x^4}{-1+c^2 x^2} \, dx}{12 c}\\ &=-\frac{b e x}{6 c^5}-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac{e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{1}{18} (b c (3 d-e)) \int \left (\frac{1}{c^6}+\frac{x^2}{c^4}+\frac{x^4}{c^2}+\frac{1}{c^6 \left (-1+c^2 x^2\right )}\right ) \, dx-\frac{(b e) \int \frac{1}{-1+c^2 x^2} \, dx}{6 c^5}+\frac{(b e) \int \left (\log \left (1-c^2 x^2\right )+c^2 x^2 \log \left (1-c^2 x^2\right )+c^4 x^4 \log \left (1-c^2 x^2\right )\right ) \, dx}{6 c^5}-\frac{(b e) \int \left (\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (-1+c^2 x^2\right )}\right ) \, dx}{12 c}\\ &=\frac{b (3 d-e) x}{18 c^5}-\frac{b e x}{4 c^5}+\frac{b (3 d-e) x^3}{54 c^3}-\frac{b e x^3}{36 c^3}+\frac{b (3 d-e) x^5}{90 c}-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )+\frac{b e \tanh ^{-1}(c x)}{6 c^6}-\frac{e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{(b (3 d-e)) \int \frac{1}{-1+c^2 x^2} \, dx}{18 c^5}-\frac{(b e) \int \frac{1}{-1+c^2 x^2} \, dx}{12 c^5}+\frac{(b e) \int \log \left (1-c^2 x^2\right ) \, dx}{6 c^5}+\frac{(b e) \int x^2 \log \left (1-c^2 x^2\right ) \, dx}{6 c^3}+\frac{(b e) \int x^4 \log \left (1-c^2 x^2\right ) \, dx}{6 c}\\ &=\frac{b (3 d-e) x}{18 c^5}-\frac{b e x}{4 c^5}+\frac{b (3 d-e) x^3}{54 c^3}-\frac{b e x^3}{36 c^3}+\frac{b (3 d-e) x^5}{90 c}-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (3 d-e) \tanh ^{-1}(c x)}{18 c^6}+\frac{b e \tanh ^{-1}(c x)}{4 c^6}+\frac{b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac{b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac{b e x^5 \log \left (1-c^2 x^2\right )}{30 c}-\frac{e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{(b e) \int \frac{x^2}{1-c^2 x^2} \, dx}{3 c^3}+\frac{(b e) \int \frac{x^4}{1-c^2 x^2} \, dx}{9 c}+\frac{1}{15} (b c e) \int \frac{x^6}{1-c^2 x^2} \, dx\\ &=\frac{b (3 d-e) x}{18 c^5}-\frac{7 b e x}{12 c^5}+\frac{b (3 d-e) x^3}{54 c^3}-\frac{b e x^3}{36 c^3}+\frac{b (3 d-e) x^5}{90 c}-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (3 d-e) \tanh ^{-1}(c x)}{18 c^6}+\frac{b e \tanh ^{-1}(c x)}{4 c^6}+\frac{b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac{b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac{b e x^5 \log \left (1-c^2 x^2\right )}{30 c}-\frac{e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{(b e) \int \frac{1}{1-c^2 x^2} \, dx}{3 c^5}+\frac{(b e) \int \left (-\frac{1}{c^4}-\frac{x^2}{c^2}+\frac{1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx}{9 c}+\frac{1}{15} (b c e) \int \left (-\frac{1}{c^6}-\frac{x^2}{c^4}-\frac{x^4}{c^2}+\frac{1}{c^6 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac{b (3 d-e) x}{18 c^5}-\frac{137 b e x}{180 c^5}+\frac{b (3 d-e) x^3}{54 c^3}-\frac{47 b e x^3}{540 c^3}+\frac{b (3 d-e) x^5}{90 c}-\frac{b e x^5}{75 c}-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (3 d-e) \tanh ^{-1}(c x)}{18 c^6}+\frac{7 b e \tanh ^{-1}(c x)}{12 c^6}+\frac{b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac{b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac{b e x^5 \log \left (1-c^2 x^2\right )}{30 c}-\frac{e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{(b e) \int \frac{1}{1-c^2 x^2} \, dx}{15 c^5}+\frac{(b e) \int \frac{1}{1-c^2 x^2} \, dx}{9 c^5}\\ &=\frac{b (3 d-e) x}{18 c^5}-\frac{137 b e x}{180 c^5}+\frac{b (3 d-e) x^3}{54 c^3}-\frac{47 b e x^3}{540 c^3}+\frac{b (3 d-e) x^5}{90 c}-\frac{b e x^5}{75 c}-\frac{e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac{e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac{1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (3 d-e) \tanh ^{-1}(c x)}{18 c^6}+\frac{137 b e \tanh ^{-1}(c x)}{180 c^6}+\frac{b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac{b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac{b e x^5 \log \left (1-c^2 x^2\right )}{30 c}-\frac{e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac{1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )\\ \end{align*}
Mathematica [A] time = 0.171192, size = 236, normalized size = 0.79 \[ \frac{20 e \log \left (1-c^2 x^2\right ) \left (15 a c^6 x^6+b c x \left (3 c^4 x^4+5 c^2 x^2+15\right )+15 b \left (c^6 x^6-1\right ) \coth ^{-1}(c x)\right )+15 \log (1-c x) (-20 a e+10 b d-49 b e)-15 \log (c x+1) (20 a e+10 b d-49 b e)+100 a c^6 x^6 (3 d-e)-150 a c^4 e x^4-300 a c^2 e x^2+4 b c^5 x^5 (15 d-11 e)+10 b c^3 x^3 (10 d-19 e)-50 b c^2 x^2 \coth ^{-1}(c x) \left (e \left (2 c^4 x^4+3 c^2 x^2+6\right )-6 c^4 d x^4\right )+30 b c x (10 d-49 e)}{1800 c^6} \]
Antiderivative was successfully verified.
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Maple [C] time = 10.293, size = 4034, normalized size = 13.6 \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 = 1.11322, size = 447, normalized size = 1.51 \begin{align*} \frac{1}{6} \, a d x^{6} + \frac{1}{36} \,{\left (6 \, x^{6} \log \left (-c^{2} x^{2} + 1\right ) - c^{2}{\left (\frac{2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac{6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b e \operatorname{arcoth}\left (c x\right ) + \frac{1}{180} \,{\left (30 \, x^{6} \operatorname{arcoth}\left (c x\right ) + c{\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 d + \frac{1}{36} \,{\left (6 \, x^{6} \log \left (-c^{2} x^{2} + 1\right ) - c^{2}{\left (\frac{2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac{6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} a e - \frac{{\left (4 \,{\left (-15 i \, \pi c^{5} + 11 \, c^{5}\right )} x^{5} + 10 \,{\left (-10 i \, \pi c^{3} + 19 \, c^{3}\right )} x^{3} + 30 \,{\left (-10 i \, \pi c + 49 \, c\right )} x -{\left (-150 i \, \pi + 60 \, c^{5} x^{5} + 100 \, c^{3} x^{3} + 300 \, c x + 735\right )} \log \left (c x + 1\right ) -{\left (150 i \, \pi + 60 \, c^{5} x^{5} + 100 \, c^{3} x^{3} + 300 \, c x - 735\right )} \log \left (c x - 1\right )\right )} b e}{1800 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69584, size = 579, normalized size = 1.95 \begin{align*} -\frac{150 \, a c^{4} e x^{4} - 100 \,{\left (3 \, a c^{6} d - a c^{6} e\right )} x^{6} + 300 \, a c^{2} e x^{2} - 4 \,{\left (15 \, b c^{5} d - 11 \, b c^{5} e\right )} x^{5} - 10 \,{\left (10 \, b c^{3} d - 19 \, b c^{3} e\right )} x^{3} - 30 \,{\left (10 \, b c d - 49 \, b c e\right )} x - 20 \,{\left (15 \, a c^{6} e x^{6} + 3 \, b c^{5} e x^{5} + 5 \, b c^{3} e x^{3} + 15 \, b c e x - 15 \, a e\right )} \log \left (-c^{2} x^{2} + 1\right ) + 5 \,{\left (15 \, b c^{4} e x^{4} - 10 \,{\left (3 \, b c^{6} d - b c^{6} e\right )} x^{6} + 30 \, b c^{2} e x^{2} + 30 \, b d - 147 \, b e - 30 \,{\left (b c^{6} e x^{6} - b e\right )} \log \left (-c^{2} x^{2} + 1\right )\right )} \log \left (\frac{c x + 1}{c x - 1}\right )}{1800 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 113.193, size = 362, normalized size = 1.22 \begin{align*} \begin{cases} \frac{a d x^{6}}{6} + \frac{a e x^{6} \log{\left (- c^{2} x^{2} + 1 \right )}}{6} - \frac{a e x^{6}}{18} - \frac{a e x^{4}}{12 c^{2}} - \frac{a e x^{2}}{6 c^{4}} - \frac{a e \log{\left (- c^{2} x^{2} + 1 \right )}}{6 c^{6}} + \frac{b d x^{6} \operatorname{acoth}{\left (c x \right )}}{6} + \frac{b e x^{6} \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{acoth}{\left (c x \right )}}{6} - \frac{b e x^{6} \operatorname{acoth}{\left (c x \right )}}{18} + \frac{b d x^{5}}{30 c} + \frac{b e x^{5} \log{\left (- c^{2} x^{2} + 1 \right )}}{30 c} - \frac{11 b e x^{5}}{450 c} - \frac{b e x^{4} \operatorname{acoth}{\left (c x \right )}}{12 c^{2}} + \frac{b d x^{3}}{18 c^{3}} + \frac{b e x^{3} \log{\left (- c^{2} x^{2} + 1 \right )}}{18 c^{3}} - \frac{19 b e x^{3}}{180 c^{3}} - \frac{b e x^{2} \operatorname{acoth}{\left (c x \right )}}{6 c^{4}} + \frac{b d x}{6 c^{5}} + \frac{b e x \log{\left (- c^{2} x^{2} + 1 \right )}}{6 c^{5}} - \frac{49 b e x}{60 c^{5}} - \frac{b d \operatorname{acoth}{\left (c x \right )}}{6 c^{6}} - \frac{b e \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{acoth}{\left (c x \right )}}{6 c^{6}} + \frac{49 b e \operatorname{acoth}{\left (c x \right )}}{60 c^{6}} & \text{for}\: c \neq 0 \\\frac{d x^{6} \left (a + \frac{i \pi b}{2}\right )}{6} & \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^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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