Integrand size = 25, antiderivative size = 140 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {b (d-e) x}{2 c}-\frac {b e x}{c}+\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}+\frac {b e \text {arctanh}(c x)}{c^2}+\frac {b e x \log \left (1-c^2 x^2\right )}{2 c}-\frac {e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{2 c^2} \]
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Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2504, 2436, 2332, 6231, 327, 213, 2498, 212} \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {e \left (1-c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{2 c^2}+\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}+\frac {b e \text {arctanh}(c x)}{c^2}+\frac {b e x \log \left (1-c^2 x^2\right )}{2 c}+\frac {b x (d-e)}{2 c}-\frac {b e x}{c} \]
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Rule 212
Rule 213
Rule 327
Rule 2332
Rule 2436
Rule 2498
Rule 2504
Rule 6231
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{2 c^2}-(b c) \int \left (-\frac {(d-e) x^2}{2 \left (-1+c^2 x^2\right )}-\frac {e \log \left (1-c^2 x^2\right )}{2 c^2}\right ) \, dx \\ & = \frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{2 c^2}+\frac {1}{2} (b c (d-e)) \int \frac {x^2}{-1+c^2 x^2} \, dx+\frac {(b e) \int \log \left (1-c^2 x^2\right ) \, dx}{2 c} \\ & = \frac {b (d-e) x}{2 c}+\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac {b e x \log \left (1-c^2 x^2\right )}{2 c}-\frac {e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{2 c^2}+\frac {(b (d-e)) \int \frac {1}{-1+c^2 x^2} \, dx}{2 c}+(b c e) \int \frac {x^2}{1-c^2 x^2} \, dx \\ & = \frac {b (d-e) x}{2 c}-\frac {b e x}{c}+\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}+\frac {b e x \log \left (1-c^2 x^2\right )}{2 c}-\frac {e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{2 c^2}+\frac {(b e) \int \frac {1}{1-c^2 x^2} \, dx}{c} \\ & = \frac {b (d-e) x}{2 c}-\frac {b e x}{c}+\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}+\frac {b e \text {arctanh}(c x)}{c^2}+\frac {b e x \log \left (1-c^2 x^2\right )}{2 c}-\frac {e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{2 c^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.92 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {2 b c (d-3 e) x+2 a c^2 (d-e) x^2+2 b c^2 (d-e) x^2 \coth ^{-1}(c x)+(b (d-3 e)-2 a e) \log (1-c x)-(b (d-3 e)+2 a e) \log (1+c x)+2 e \left (c x (b+a c x)+b \left (-1+c^2 x^2\right ) \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^2} \]
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Time = 1.63 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.24
method | result | size |
parallelrisch | \(\frac {\ln \left (-c^{2} x^{2}+1\right ) \operatorname {arccoth}\left (c x \right ) b \,c^{2} e \,x^{2}+\operatorname {arccoth}\left (c x \right ) b \,c^{2} d \,x^{2}-\operatorname {arccoth}\left (c x \right ) b \,c^{2} e \,x^{2}+\ln \left (-c^{2} x^{2}+1\right ) a \,c^{2} e \,x^{2}+a \,c^{2} d \,x^{2}-a \,c^{2} e \,x^{2}+\ln \left (-c^{2} x^{2}+1\right ) b c e x +b c d x -3 b x e c -\operatorname {arccoth}\left (c x \right ) \ln \left (-c^{2} x^{2}+1\right ) b e -\operatorname {arccoth}\left (c x \right ) b d +3 \,\operatorname {arccoth}\left (c x \right ) b e -\ln \left (-c^{2} x^{2}+1\right ) a e}{2 c^{2}}\) | \(174\) |
default | \(\text {Expression too large to display}\) | \(2474\) |
parts | \(\text {Expression too large to display}\) | \(2474\) |
risch | \(\text {Expression too large to display}\) | \(6696\) |
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Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.99 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {2 \, {\left (a c^{2} d - a c^{2} e\right )} x^{2} + 2 \, {\left (b c d - 3 \, b c e\right )} x + 2 \, {\left (a c^{2} e x^{2} + b c e x - a e\right )} \log \left (-c^{2} x^{2} + 1\right ) + {\left ({\left (b c^{2} d - b c^{2} e\right )} x^{2} - b d + 3 \, b e + {\left (b c^{2} e x^{2} - b e\right )} \log \left (-c^{2} x^{2} + 1\right )\right )} \log \left (\frac {c x + 1}{c x - 1}\right )}{4 \, c^{2}} \]
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Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.49 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\begin {cases} \frac {a d x^{2}}{2} + \frac {a e x^{2} \log {\left (- c^{2} x^{2} + 1 \right )}}{2} - \frac {a e x^{2}}{2} - \frac {a e \log {\left (- c^{2} x^{2} + 1 \right )}}{2 c^{2}} + \frac {b d x^{2} \operatorname {acoth}{\left (c x \right )}}{2} + \frac {b e x^{2} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{2} - \frac {b e x^{2} \operatorname {acoth}{\left (c x \right )}}{2} + \frac {b d x}{2 c} + \frac {b e x \log {\left (- c^{2} x^{2} + 1 \right )}}{2 c} - \frac {3 b e x}{2 c} - \frac {b d \operatorname {acoth}{\left (c x \right )}}{2 c^{2}} - \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{2 c^{2}} + \frac {3 b e \operatorname {acoth}{\left (c x \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\\frac {d x^{2} \left (a + \frac {i \pi b}{2}\right )}{2} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.22 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcoth}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b d - \frac {{\left (c^{2} x^{2} - {\left (c^{2} x^{2} - 1\right )} \log \left (-c^{2} x^{2} + 1\right ) - 1\right )} b e \operatorname {arcoth}\left (c x\right )}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - {\left (c^{2} x^{2} - 1\right )} \log \left (-c^{2} x^{2} + 1\right ) - 1\right )} a e}{2 \, c^{2}} - \frac {{\left (3 \, c x - {\left (c x + 1\right )} \log \left (c x + 1\right ) - {\left (c x - 1\right )} \log \left (-c x + 1\right )\right )} b e}{2 \, c^{2}} \]
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Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.72 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {1}{4} \, b e x^{2} \log \left (-c x + 1\right )^{2} - \frac {1}{4} \, {\left (-i \, \pi b d + i \, \pi b e - 2 \, a d + 2 \, a e\right )} x^{2} + \frac {1}{4} \, {\left (b e x^{2} - \frac {b e}{c^{2}}\right )} \log \left (c x + 1\right )^{2} - \frac {1}{4} \, {\left ({\left (-i \, \pi b e - b d - 2 \, a e + b e\right )} x^{2} - \frac {2 \, b e x}{c}\right )} \log \left (c x + 1\right ) - \frac {b e \log \left (c x - 1\right )^{2}}{4 \, c^{2}} - \frac {1}{4} \, {\left ({\left (-i \, \pi b e + b d - 2 \, a e - b e\right )} x^{2} - \frac {2 \, b e x}{c} - \frac {2 \, b e \log \left (c x - 1\right )}{c^{2}}\right )} \log \left (-c x + 1\right ) + \frac {{\left (b d - 3 \, b e\right )} x}{2 \, c} + \frac {{\left (-i \, \pi b e - b d - 2 \, a e + 3 \, b e\right )} \log \left (c x + 1\right )}{4 \, c^{2}} + \frac {{\left (-i \, \pi b e + b d - 2 \, a e - 3 \, b e\right )} \log \left (c x - 1\right )}{4 \, c^{2}} \]
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Time = 5.60 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.35 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\ln \left (1-\frac {1}{c\,x}\right )\,\left (\frac {\frac {b\,d\,x^3}{2}-\frac {b\,c^2\,d\,x^5}{2}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}-\frac {\frac {b\,e\,x^3}{2}-\frac {b\,c^2\,e\,x^5}{2}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e\,x^3}{2}-\frac {b\,c^2\,e\,x^5}{2}\right )}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}-\frac {b\,e\,\ln \left (1-c^2\,x^2\right )\,\left (x-c^2\,x^3\right )}{4\,c^2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}\right )+\ln \left (1-c^2\,x^2\right )\,\left (\frac {a\,e\,x^2}{2}+\frac {b\,e\,x}{2\,c}\right )-\ln \left (\frac {1}{c\,x}+1\right )\,\left (\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e}{4\,c^2}-\frac {b\,e\,x^2}{4}\right )-\frac {b\,d\,x^2}{4}+\frac {b\,e\,x^2}{4}\right )+\frac {a\,x^2\,\left (d-e\right )}{2}-\frac {\ln \left (c\,x+1\right )\,\left (2\,a\,e+b\,d-3\,b\,e\right )}{4\,c^2}-\frac {\ln \left (c\,x-1\right )\,\left (2\,a\,e-b\,d+3\,b\,e\right )}{4\,c^2}+\frac {b\,x\,\left (d-3\,e\right )}{2\,c} \]
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