Integrand size = 11, antiderivative size = 34 \[ \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx=\frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\coth ^{-1}(\tanh (a+b x))^5}{20 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2199, 2188, 30} \[ \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx=\frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\coth ^{-1}(\tanh (a+b x))^5}{20 b^2} \]
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Rule 30
Rule 2188
Rule 2199
Rubi steps \begin{align*} \text {integral}& = \frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\int \coth ^{-1}(\tanh (a+b x))^4 \, dx}{4 b} \\ & = \frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\text {Subst}\left (\int x^4 \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{4 b^2} \\ & = \frac {x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\coth ^{-1}(\tanh (a+b x))^5}{20 b^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(99\) vs. \(2(34)=68\).
Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.91 \[ \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx=\frac {(a+b x) \left ((4 a-b x) (a+b x)^3-5 (3 a-b x) (a+b x)^2 \coth ^{-1}(\tanh (a+b x))+10 \left (2 a^2+a b x-b^2 x^2\right ) \coth ^{-1}(\tanh (a+b x))^2-10 (a-b x) \coth ^{-1}(\tanh (a+b x))^3\right )}{20 b^2} \]
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Time = 25.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.59
method | result | size |
parallelrisch | \(-\frac {b^{3} x^{5}}{20}+\frac {x^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}{2}+\frac {b^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) x^{4}}{4}-\frac {b \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} x^{3}}{2}\) | \(54\) |
risch | \(\text {Expression too large to display}\) | \(18111\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.56 \[ \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx=\frac {1}{5} \, b^{3} x^{5} + \frac {3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} - \frac {1}{16} i \, \pi ^{3} x^{2} + \frac {1}{2} \, a^{3} x^{2} - \frac {1}{8} \, \pi ^{2} {\left (2 \, b x^{3} + 3 \, a x^{2}\right )} + \frac {1}{8} i \, \pi {\left (3 \, b^{2} x^{4} + 8 \, a b x^{3} + 6 \, a^{2} x^{2}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx=\begin {cases} \frac {x \operatorname {acoth}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{4 b} - \frac {\operatorname {acoth}^{5}{\left (\tanh {\left (a + b x \right )} \right )}}{20 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {acoth}^{3}{\left (\tanh {\left (a \right )} \right )}}{2} & \text {otherwise} \end {cases} \]
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none
Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.59 \[ \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx=-\frac {1}{2} \, b x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{20} \, {\left (b^{2} x^{5} - 5 \, b x^{4} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )\right )} b \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.26 \[ \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx=\frac {1}{5} \, b^{3} x^{5} - \frac {3}{8} \, {\left (-i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{4} - \frac {1}{4} \, {\left (\pi ^{2} b - 4 i \, \pi a b - 4 \, a^{2} b\right )} x^{3} - \frac {1}{16} \, {\left (i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}\right )} x^{2} \]
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Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx=-\frac {b^3\,x^5}{20}+\frac {b^2\,x^4\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{4}-\frac {b\,x^3\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{2}+\frac {x^2\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{2} \]
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