Integrand size = 8, antiderivative size = 27 \[ \int \coth ^3(a+b x) \, dx=-\frac {\coth ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556} \[ \int \coth ^3(a+b x) \, dx=\frac {\log (\sinh (a+b x))}{b}-\frac {\coth ^2(a+b x)}{2 b} \]
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Rule 3554
Rule 3556
Rubi steps \begin{align*} \text {integral}& = -\frac {\coth ^2(a+b x)}{2 b}+\int \coth (a+b x) \, dx \\ & = -\frac {\coth ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \coth ^3(a+b x) \, dx=-\frac {\coth ^2(a+b x)-2 \log (\cosh (a+b x))-2 \log (\tanh (a+b x))}{2 b} \]
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Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41
method | result | size |
derivativedivides | \(\frac {-\frac {\coth \left (b x +a \right )^{2}}{2}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}-\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(38\) |
default | \(\frac {-\frac {\coth \left (b x +a \right )^{2}}{2}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}-\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(38\) |
parallelrisch | \(\frac {-2 b x +2 \ln \left (\tanh \left (b x +a \right )\right )-2 \ln \left (1-\tanh \left (b x +a \right )\right )-\coth \left (b x +a \right )^{2}}{2 b}\) | \(43\) |
risch | \(-x -\frac {2 a}{b}-\frac {2 \,{\mathrm e}^{2 b x +2 a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(54\) |
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Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 346, normalized size of antiderivative = 12.81 \[ \int \coth ^3(a+b x) \, dx=-\frac {b x \cosh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} - 2 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x + 1\right )} \sinh \left (b x + a\right )^{2} + b x - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \, {\left (b x \cosh \left (b x + a\right )^{3} - {\left (b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (20) = 40\).
Time = 0.69 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.15 \[ \int \coth ^3(a+b x) \, dx=\begin {cases} x \coth ^{3}{\left (a \right )} & \text {for}\: b = 0 \\- \frac {\log {\left (- e^{- b x} \right )} \coth ^{3}{\left (b x + \log {\left (- e^{- b x} \right )} \right )}}{b} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \\- \frac {\log {\left (e^{- b x} \right )} \coth ^{3}{\left (b x + \log {\left (e^{- b x} \right )} \right )}}{b} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} + \frac {\log {\left (\tanh {\left (a + b x \right )} \right )}}{b} - \frac {1}{2 b \tanh ^{2}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (25) = 50\).
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.93 \[ \int \coth ^3(a+b x) \, dx=x + \frac {a}{b} + \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, e^{\left (-2 \, b x - 2 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \coth ^3(a+b x) \, dx=-\frac {b x + a + \frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} - \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \]
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Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \coth ^3(a+b x) \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-x-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {2}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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