Integrand size = 10, antiderivative size = 16 \[ \int \sqrt {a \tanh ^2(x)} \, dx=\coth (x) \log (\cosh (x)) \sqrt {a \tanh ^2(x)} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3739, 3556} \[ \int \sqrt {a \tanh ^2(x)} \, dx=\coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x)) \]
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Rule 3556
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \left (\coth (x) \sqrt {a \tanh ^2(x)}\right ) \int \tanh (x) \, dx \\ & = \coth (x) \log (\cosh (x)) \sqrt {a \tanh ^2(x)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {a \tanh ^2(x)} \, dx=\coth (x) \log (\cosh (x)) \sqrt {a \tanh ^2(x)} \]
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Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62
method | result | size |
derivativedivides | \(-\frac {\sqrt {a \tanh \left (x \right )^{2}}\, \left (\ln \left (\tanh \left (x \right )-1\right )+\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )}\) | \(26\) |
default | \(-\frac {\sqrt {a \tanh \left (x \right )^{2}}\, \left (\ln \left (\tanh \left (x \right )-1\right )+\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )}\) | \(26\) |
risch | \(-\frac {\sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) x}{{\mathrm e}^{2 x}-1}+\frac {\sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1}\) | \(81\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (14) = 28\).
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 4.50 \[ \int \sqrt {a \tanh ^2(x)} \, dx=-\frac {{\left (x e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + x\right )} \sqrt {\frac {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{e^{\left (2 \, x\right )} - 1} \]
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\[ \int \sqrt {a \tanh ^2(x)} \, dx=\int \sqrt {a \tanh ^{2}{\left (x \right )}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \sqrt {a \tanh ^2(x)} \, dx=-\sqrt {a} x - \sqrt {a} \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.94 \[ \int \sqrt {a \tanh ^2(x)} \, dx=-{\left (x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )\right )} \sqrt {a} \]
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Timed out. \[ \int \sqrt {a \tanh ^2(x)} \, dx=\int \sqrt {a\,{\mathrm {tanh}\left (x\right )}^2} \,d x \]
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