Optimal. Leaf size=297 \[ -\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}+\frac {1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {e \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{6 c^6}-\frac {1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (3 d-e) \tanh ^{-1}(c x)}{18 c^6}+\frac {137 b e \tanh ^{-1}(c x)}{180 c^6}+\frac {b x (3 d-e)}{18 c^5}-\frac {137 b e x}{180 c^5}+\frac {b x^3 (3 d-e)}{54 c^3}-\frac {47 b e x^3}{540 c^3}+\frac {b e x^5 \log \left (1-c^2 x^2\right )}{30 c}+\frac {b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac {b x^5 (3 d-e)}{90 c}-\frac {b e x^5}{75 c} \]
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Rubi [A] time = 0.39, antiderivative size = 297, normalized size of antiderivative = 1.00, 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 43
Rule 206
Rule 207
Rule 302
Rule 321
Rule 2395
Rule 2448
Rule 2454
Rule 2455
Rule 2528
Rule 6084
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*}
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Mathematica [A] time = 0.17, size = 236, normalized size = 0.79 \[ \frac {20 e \log \left (1-c^2 x^2\right ) \left (15 a c^6 x^6+15 b \left (c^6 x^6-1\right ) \coth ^{-1}(c x)+b c x \left (3 c^4 x^4+5 c^2 x^2+15\right )\right )+15 \log (1-c x) (-20 a e+10 b d-49 b e)-15 \log (c x+1) (20 a e+10 b d-49 b e)+100 a c^6 x^6 (3 d-e)-150 a c^4 e x^4-300 a c^2 e x^2+4 b c^5 x^5 (15 d-11 e)+10 b c^3 x^3 (10 d-19 e)-50 b c^2 x^2 \coth ^{-1}(c x) \left (e \left (2 c^4 x^4+3 c^2 x^2+6\right )-6 c^4 d x^4\right )+30 b c x (10 d-49 e)}{1800 c^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 246, normalized size = 0.83 \[ -\frac {150 \, a c^{4} e x^{4} - 100 \, {\left (3 \, a c^{6} d - a c^{6} e\right )} x^{6} + 300 \, a c^{2} e x^{2} - 4 \, {\left (15 \, b c^{5} d - 11 \, b c^{5} e\right )} x^{5} - 10 \, {\left (10 \, b c^{3} d - 19 \, b c^{3} e\right )} x^{3} - 30 \, {\left (10 \, b c d - 49 \, b c e\right )} x - 20 \, {\left (15 \, a c^{6} e x^{6} + 3 \, b c^{5} e x^{5} + 5 \, b c^{3} e x^{3} + 15 \, b c e x - 15 \, a e\right )} \log \left (-c^{2} x^{2} + 1\right ) + 5 \, {\left (15 \, b c^{4} e x^{4} - 10 \, {\left (3 \, b c^{6} d - b c^{6} e\right )} x^{6} + 30 \, b c^{2} e x^{2} + 30 \, b d - 147 \, b e - 30 \, {\left (b c^{6} e x^{6} - b e\right )} \log \left (-c^{2} x^{2} + 1\right )\right )} \log \left (\frac {c x + 1}{c x - 1}\right )}{1800 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 16.95, size = 4034, normalized size = 13.58 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.35, size = 328, normalized size = 1.10 \[ \frac {1}{6} \, a d x^{6} + \frac {1}{36} \, {\left (6 \, x^{6} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b e \operatorname {arcoth}\left (c x\right ) + \frac {1}{180} \, {\left (30 \, x^{6} \operatorname {arcoth}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b d + \frac {1}{36} \, {\left (6 \, x^{6} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} a e + \frac {{\left ({\left (60 i \, \pi c^{5} - 44 \, c^{5}\right )} x^{5} + {\left (100 i \, \pi c^{3} - 190 \, c^{3}\right )} x^{3} + {\left (300 i \, \pi c - 1470 \, c\right )} x - 5 \, {\left (30 i \, \pi - 12 \, c^{5} x^{5} - 20 \, c^{3} x^{3} - 60 \, c x - 147\right )} \log \left (c x + 1\right ) - 5 \, {\left (-30 i \, \pi - 12 \, c^{5} x^{5} - 20 \, c^{3} x^{3} - 60 \, c x + 147\right )} \log \left (c x - 1\right )\right )} b e}{1800 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.29, size = 510, normalized size = 1.72 \[ \ln \left (1-c^2\,x^2\right )\,\left (\frac {a\,e\,x^6}{6}+\frac {b\,e\,x}{6\,c^5}+\frac {b\,e\,x^5}{30\,c}+\frac {b\,e\,x^3}{18\,c^3}\right )-\ln \left (\frac {1}{c\,x}+1\right )\,\left (\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e}{12\,c^6}-\frac {b\,e\,x^6}{12}\right )-\frac {b\,d\,x^6}{12}+\frac {b\,e\,x^6}{36}+\frac {b\,e\,x^4}{24\,c^2}+\frac {b\,e\,x^2}{12\,c^4}\right )+\ln \left (1-\frac {1}{c\,x}\right )\,\left (\frac {\frac {b\,d\,x^7}{6}-\frac {b\,c^2\,d\,x^9}{6}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\frac {b\,e\,x^7}{36}+\frac {b\,e\,x^5}{12\,c^2}-\frac {b\,e\,x^3}{6\,c^4}+\frac {b\,c^2\,e\,x^9}{18}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e\,x^7}{6}-\frac {b\,c^2\,e\,x^9}{6}\right )}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}-\frac {b\,e\,\ln \left (1-c^2\,x^2\right )\,\left (x-c^2\,x^3\right )}{12\,c^6\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}\right )+x^4\,\left (\frac {a\,\left (3\,d-e\right )}{12\,c^2}-\frac {a\,d}{4\,c^2}\right )+x^3\,\left (\frac {b\,\left (15\,d-11\,e\right )}{270\,c^3}-\frac {7\,b\,e}{108\,c^3}\right )+x\,\left (\frac {\frac {b\,\left (15\,d-11\,e\right )}{90\,c^3}-\frac {7\,b\,e}{36\,c^3}}{c^2}-\frac {b\,e}{2\,c^5}\right )+\frac {a\,x^6\,\left (3\,d-e\right )}{18}+\frac {x^2\,\left (\frac {a\,\left (3\,d-e\right )}{3\,c^2}-\frac {a\,d}{c^2}\right )}{2\,c^2}-\frac {\ln \left (c\,x-1\right )\,\left (20\,a\,e-10\,b\,d+49\,b\,e\right )}{120\,c^6}-\frac {\ln \left (c\,x+1\right )\,\left (20\,a\,e+10\,b\,d-49\,b\,e\right )}{120\,c^6}+\frac {b\,x^5\,\left (15\,d-11\,e\right )}{450\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.40, size = 362, normalized size = 1.22 \[ \begin {cases} \frac {a d x^{6}}{6} + \frac {a e x^{6} \log {\left (- c^{2} x^{2} + 1 \right )}}{6} - \frac {a e x^{6}}{18} - \frac {a e x^{4}}{12 c^{2}} - \frac {a e x^{2}}{6 c^{4}} - \frac {a e \log {\left (- c^{2} x^{2} + 1 \right )}}{6 c^{6}} + \frac {b d x^{6} \operatorname {acoth}{\left (c x \right )}}{6} + \frac {b e x^{6} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{6} - \frac {b e x^{6} \operatorname {acoth}{\left (c x \right )}}{18} + \frac {b d x^{5}}{30 c} + \frac {b e x^{5} \log {\left (- c^{2} x^{2} + 1 \right )}}{30 c} - \frac {11 b e x^{5}}{450 c} - \frac {b e x^{4} \operatorname {acoth}{\left (c x \right )}}{12 c^{2}} + \frac {b d x^{3}}{18 c^{3}} + \frac {b e x^{3} \log {\left (- c^{2} x^{2} + 1 \right )}}{18 c^{3}} - \frac {19 b e x^{3}}{180 c^{3}} - \frac {b e x^{2} \operatorname {acoth}{\left (c x \right )}}{6 c^{4}} + \frac {b d x}{6 c^{5}} + \frac {b e x \log {\left (- c^{2} x^{2} + 1 \right )}}{6 c^{5}} - \frac {49 b e x}{60 c^{5}} - \frac {b d \operatorname {acoth}{\left (c x \right )}}{6 c^{6}} - \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{6 c^{6}} + \frac {49 b e \operatorname {acoth}{\left (c x \right )}}{60 c^{6}} & \text {for}\: c \neq 0 \\\frac {d x^{6} \left (a + \frac {i \pi b}{2}\right )}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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