Integrand size = 9, antiderivative size = 16 \[ \int \coth ^{-1}(\tanh (a+b x))^2 \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^3}{3 b} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2188, 30} \[ \int \coth ^{-1}(\tanh (a+b x))^2 \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^3}{3 b} \]
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Rule 30
Rule 2188
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^2 \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b} \\ & = \frac {\coth ^{-1}(\tanh (a+b x))^3}{3 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \coth ^{-1}(\tanh (a+b x))^2 \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^3}{3 b} \]
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Time = 0.42 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}{3 b}\) | \(15\) |
default | \(\frac {\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}{3 b}\) | \(15\) |
parallelrisch | \(\frac {b^{2} x^{3}}{3}-b \,x^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )+x \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}\) | \(34\) |
risch | \(\text {Expression too large to display}\) | \(14844\) |
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \[ \int \coth ^{-1}(\tanh (a+b x))^2 \, dx=\frac {1}{3} \, b^{2} x^{3} + a b x^{2} - \frac {1}{4} \, \pi ^{2} x + a^{2} x + \frac {1}{2} i \, \pi {\left (b x^{2} + 2 \, a x\right )} \]
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Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \coth ^{-1}(\tanh (a+b x))^2 \, dx=\begin {cases} \frac {\operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{3 b} & \text {for}\: b \neq 0 \\x \operatorname {acoth}^{2}{\left (\tanh {\left (a \right )} \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06 \[ \int \coth ^{-1}(\tanh (a+b x))^2 \, dx=\frac {1}{3} \, b^{2} x^{3} - b x^{2} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right ) + x \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \[ \int \coth ^{-1}(\tanh (a+b x))^2 \, dx=\frac {1}{3} \, b^{2} x^{3} - \frac {1}{2} \, {\left (-i \, \pi b - 2 \, a b\right )} x^{2} - \frac {1}{4} \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} x \]
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Time = 4.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06 \[ \int \coth ^{-1}(\tanh (a+b x))^2 \, dx=\frac {b^2\,x^3}{3}-b\,x^2\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+x\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2 \]
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