Integrand size = 27, antiderivative size = 225 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b e x^3}{18 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {2 b e \text {arctanh}(c x)}{3 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \]
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Time = 0.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2504, 2442, 45, 6231, 470, 327, 212, 2521, 2498, 2505, 308} \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {e \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{4 c^4}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {2 b e \text {arctanh}(c x)}{3 c^4}+\frac {b x (2 d-3 e)}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b x^3 (2 d-e)}{24 c}-\frac {b e x^3}{18 c} \]
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Rule 45
Rule 212
Rule 308
Rule 327
Rule 470
Rule 2442
Rule 2498
Rule 2504
Rule 2505
Rule 2521
Rule 6231
Rubi steps \begin{align*} \text {integral}& = -\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \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 )}{4 c^4}+\frac {1}{4} x^4 \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 {x^2 \left (-2 e+c^2 (2 d-e) x^2\right )}{8 c^2 \left (1-c^2 x^2\right )}-\frac {e \left (1+c^2 x^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}\right ) \, dx \\ & = -\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \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 )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {b \int \frac {x^2 \left (-2 e+c^2 (2 d-e) x^2\right )}{1-c^2 x^2} \, dx}{8 c}+\frac {(b e) \int \left (1+c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \, dx}{4 c^3} \\ & = \frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \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 )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {(b (2 d-3 e)) \int \frac {x^2}{1-c^2 x^2} \, dx}{8 c}+\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 )\right ) \, dx}{4 c^3} \\ & = \frac {b (2 d-3 e) x}{8 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \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 )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {(b (2 d-3 e)) \int \frac {1}{1-c^2 x^2} \, dx}{8 c^3}+\frac {(b e) \int \log \left (1-c^2 x^2\right ) \, dx}{4 c^3}+\frac {(b e) \int x^2 \log \left (1-c^2 x^2\right ) \, dx}{4 c} \\ & = \frac {b (2 d-3 e) x}{8 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \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}{2 c}+\frac {1}{6} (b c e) \int \frac {x^4}{1-c^2 x^2} \, dx \\ & = \frac {b (2 d-3 e) x}{8 c^3}-\frac {b e x}{2 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \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}{2 c^3}+\frac {1}{6} (b c 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 \\ & = \frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b e x^3}{18 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {b e \text {arctanh}(c x)}{2 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \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}{6 c^3} \\ & = \frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b e x^3}{18 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {2 b e \text {arctanh}(c x)}{3 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.85 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {6 b c (6 d-25 e) x-36 a c^2 e x^2+2 b c^3 (6 d-7 e) x^3+18 a c^4 (2 d-e) x^4-18 b c^2 x^2 \left (-2 c^2 d x^2+e \left (2+c^2 x^2\right )\right ) \coth ^{-1}(c x)+3 (6 b d-12 a e-25 b e) \log (1-c x)-3 (6 b d+12 a e-25 b e) \log (1+c x)+12 e \left (3 a c^4 x^4+b c x \left (3+c^2 x^2\right )+3 b \left (-1+c^4 x^4\right ) \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{144 c^4} \]
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Time = 2.51 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.11
method | result | size |
parallelrisch | \(\frac {18 b e \ln \left (-c^{2} x^{2}+1\right ) \operatorname {arccoth}\left (c x \right ) x^{4} c^{4}-18 \,\operatorname {arccoth}\left (c x \right ) b d -18 \ln \left (-c^{2} x^{2}+1\right ) a e -18 \,\operatorname {arccoth}\left (c x \right ) \ln \left (-c^{2} x^{2}+1\right ) b e -18 a \,c^{2} e \,x^{2}+18 b c d x -18 \,\operatorname {arccoth}\left (c x \right ) b \,c^{2} e \,x^{2}+18 \ln \left (-c^{2} x^{2}+1\right ) b c e x -75 b x e c +6 b \,c^{3} d \,x^{3}-7 b e \,x^{3} c^{3}-18 a e -9 e b \,\operatorname {arccoth}\left (c x \right ) x^{4} c^{4}+6 e b \,x^{3} \ln \left (-c^{2} x^{2}+1\right ) c^{3}+18 a e \ln \left (-c^{2} x^{2}+1\right ) x^{4} c^{4}+18 b \,\operatorname {arccoth}\left (c x \right ) x^{4} c^{4} d +75 \,\operatorname {arccoth}\left (c x \right ) b e +18 a \,x^{4} d \,c^{4}-9 a e \,x^{4} c^{4}}{72 c^{4}}\) | \(249\) |
default | \(\text {Expression too large to display}\) | \(3170\) |
parts | \(\text {Expression too large to display}\) | \(3170\) |
risch | \(\text {Expression too large to display}\) | \(7050\) |
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Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.87 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {36 \, a c^{2} e x^{2} - 18 \, {\left (2 \, a c^{4} d - a c^{4} e\right )} x^{4} - 2 \, {\left (6 \, b c^{3} d - 7 \, b c^{3} e\right )} x^{3} - 6 \, {\left (6 \, b c d - 25 \, b c e\right )} x - 12 \, {\left (3 \, a c^{4} e x^{4} + b c^{3} e x^{3} + 3 \, b c e x - 3 \, a e\right )} \log \left (-c^{2} x^{2} + 1\right ) + 3 \, {\left (6 \, b c^{2} e x^{2} - 3 \, {\left (2 \, b c^{4} d - b c^{4} e\right )} x^{4} + 6 \, b d - 25 \, b e - 6 \, {\left (b c^{4} e x^{4} - b e\right )} \log \left (-c^{2} x^{2} + 1\right )\right )} \log \left (\frac {c x + 1}{c x - 1}\right )}{144 \, c^{4}} \]
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Result contains complex when optimal does not.
Time = 1.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.27 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{4} \log {\left (- c^{2} x^{2} + 1 \right )}}{4} - \frac {a e x^{4}}{8} - \frac {a e x^{2}}{4 c^{2}} - \frac {a e \log {\left (- c^{2} x^{2} + 1 \right )}}{4 c^{4}} + \frac {b d x^{4} \operatorname {acoth}{\left (c x \right )}}{4} + \frac {b e x^{4} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{4} - \frac {b e x^{4} \operatorname {acoth}{\left (c x \right )}}{8} + \frac {b d x^{3}}{12 c} + \frac {b e x^{3} \log {\left (- c^{2} x^{2} + 1 \right )}}{12 c} - \frac {7 b e x^{3}}{72 c} - \frac {b e x^{2} \operatorname {acoth}{\left (c x \right )}}{4 c^{2}} + \frac {b d x}{4 c^{3}} + \frac {b e x \log {\left (- c^{2} x^{2} + 1 \right )}}{4 c^{3}} - \frac {25 b e x}{24 c^{3}} - \frac {b d \operatorname {acoth}{\left (c x \right )}}{4 c^{4}} - \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{4 c^{4}} + \frac {25 b e \operatorname {acoth}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {d x^{4} \left (a + \frac {i \pi b}{2}\right )}{4} & \text {otherwise} \end {cases} \]
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Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.20 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {1}{4} \, a d x^{4} + \frac {1}{8} \, {\left (2 \, x^{4} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b e \operatorname {arcoth}\left (c x\right ) + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {arcoth}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d + \frac {1}{8} \, {\left (2 \, x^{4} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} a e - \frac {{\left (2 \, {\left (-6 i \, \pi c^{3} + 7 \, c^{3}\right )} x^{3} + 6 \, {\left (-6 i \, \pi c + 25 \, c\right )} x + 3 \, {\left (6 i \, \pi - 4 \, c^{3} x^{3} - 12 \, c x - 25\right )} \log \left (c x + 1\right ) + 3 \, {\left (-6 i \, \pi - 4 \, c^{3} x^{3} - 12 \, c x + 25\right )} \log \left (c x - 1\right )\right )} b e}{144 \, c^{4}} \]
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Exception generated. \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\text {Exception raised: TypeError} \]
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Time = 5.42 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.84 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\ln \left (1-\frac {1}{c\,x}\right )\,\left (\frac {\frac {b\,e\,x^5}{8}-\frac {b\,e\,x^3}{4\,c^2}+\frac {b\,c^2\,e\,x^7}{8}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\frac {b\,d\,x^5}{4}-\frac {b\,c^2\,d\,x^7}{4}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e\,x^5}{4}-\frac {b\,c^2\,e\,x^7}{4}\right )}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}-\frac {b\,e\,\ln \left (1-c^2\,x^2\right )\,\left (x-c^2\,x^3\right )}{8\,c^4\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}\right )+x\,\left (\frac {b\,\left (6\,d-7\,e\right )}{24\,c^3}-\frac {3\,b\,e}{4\,c^3}\right )+\ln \left (1-c^2\,x^2\right )\,\left (\frac {a\,e\,x^4}{4}+\frac {b\,e\,x}{4\,c^3}+\frac {b\,e\,x^3}{12\,c}\right )-\ln \left (\frac {1}{c\,x}+1\right )\,\left (\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e}{8\,c^4}-\frac {b\,e\,x^4}{8}\right )-\frac {b\,d\,x^4}{8}+\frac {b\,e\,x^4}{16}+\frac {b\,e\,x^2}{8\,c^2}\right )+x^2\,\left (\frac {a\,\left (2\,d-e\right )}{4\,c^2}-\frac {a\,d}{2\,c^2}\right )+\frac {a\,x^4\,\left (2\,d-e\right )}{8}-\frac {\ln \left (c\,x-1\right )\,\left (12\,a\,e-6\,b\,d+25\,b\,e\right )}{48\,c^4}-\frac {\ln \left (c\,x+1\right )\,\left (12\,a\,e+6\,b\,d-25\,b\,e\right )}{48\,c^4}+\frac {b\,x^3\,\left (6\,d-7\,e\right )}{72\,c} \]
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