Integrand size = 14, antiderivative size = 31 \[ \int \sqrt {b \coth ^2(c+d x)} \, dx=\frac {\sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x)) \tanh (c+d x)}{d} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3739, 3556} \[ \int \sqrt {b \coth ^2(c+d x)} \, dx=\frac {\tanh (c+d x) \sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x))}{d} \]
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Rule 3556
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {b \coth ^2(c+d x)} \tanh (c+d x)\right ) \int \coth (c+d x) \, dx \\ & = \frac {\sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x)) \tanh (c+d x)}{d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \sqrt {b \coth ^2(c+d x)} \, dx=\frac {\sqrt {b \coth ^2(c+d x)} (\log (\cosh (c+d x))+\log (\tanh (c+d x))) \tanh (c+d x)}{d} \]
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Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45
method | result | size |
derivativedivides | \(-\frac {\sqrt {\coth \left (d x +c \right )^{2} b}\, \left (\ln \left (\coth \left (d x +c \right )-1\right )+\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )}\) | \(45\) |
default | \(-\frac {\sqrt {\coth \left (d x +c \right )^{2} b}\, \left (\ln \left (\coth \left (d x +c \right )-1\right )+\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )}\) | \(45\) |
risch | \(\frac {\sqrt {\frac {\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2} b}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right ) x}{{\mathrm e}^{2 d x +2 c}+1}-\frac {2 \sqrt {\frac {\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2} b}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (d x +c \right )}{\left ({\mathrm e}^{2 d x +2 c}+1\right ) d}+\frac {\sqrt {\frac {\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2} b}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right ) \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{\left ({\mathrm e}^{2 d x +2 c}+1\right ) d}\) | \(192\) |
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (29) = 58\).
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.03 \[ \int \sqrt {b \coth ^2(c+d x)} \, dx=-\frac {{\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x - {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )\right )} \sqrt {\frac {b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}}}{d e^{\left (2 \, d x + 2 \, c\right )} + d} \]
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\[ \int \sqrt {b \coth ^2(c+d x)} \, dx=\int \sqrt {b \coth ^{2}{\left (c + d x \right )}}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \sqrt {b \coth ^2(c+d x)} \, dx=-\frac {{\left (d x + c\right )} \sqrt {b}}{d} - \frac {\sqrt {b} \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\sqrt {b} \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \sqrt {b \coth ^2(c+d x)} \, dx=-\frac {{\left ({\left (d x + c\right )} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) - \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )\right )} \sqrt {b}}{d} \]
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Timed out. \[ \int \sqrt {b \coth ^2(c+d x)} \, dx=\int \sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^2} \,d x \]
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