Integrand size = 14, antiderivative size = 31 \[ \int \sqrt {-\tanh ^2(c+d x)} \, dx=\frac {\coth (c+d x) \log (\cosh (c+d x)) \sqrt {-\tanh ^2(c+d x)}}{d} \]
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Time = 0.01 (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} \[ \int \sqrt {-\tanh ^2(c+d x)} \, dx=\frac {\sqrt {-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
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Rule 3556
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \left (\coth (c+d x) \sqrt {-\tanh ^2(c+d x)}\right ) \int \tanh (c+d x) \, dx \\ & = \frac {\coth (c+d x) \log (\cosh (c+d x)) \sqrt {-\tanh ^2(c+d x)}}{d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \sqrt {-\tanh ^2(c+d x)} \, dx=\frac {\coth (c+d x) \log (\cosh (c+d x)) \sqrt {-\tanh ^2(c+d x)}}{d} \]
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Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45
method | result | size |
derivativedivides | \(-\frac {\sqrt {-\tanh \left (d x +c \right )^{2}}\, \left (\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (\tanh \left (d x +c \right )+1\right )\right )}{2 d \tanh \left (d x +c \right )}\) | \(45\) |
default | \(-\frac {\sqrt {-\tanh \left (d x +c \right )^{2}}\, \left (\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (\tanh \left (d x +c \right )+1\right )\right )}{2 d \tanh \left (d x +c \right )}\) | \(45\) |
risch | \(\frac {\left ({\mathrm e}^{2 d x +2 c}+1\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, x}{{\mathrm e}^{2 d x +2 c}-1}-\frac {2 \left ({\mathrm e}^{2 d x +2 c}+1\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, \left (d x +c \right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) d}+\frac {\left ({\mathrm e}^{2 d x +2 c}+1\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) d}\) | \(192\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \sqrt {-\tanh ^2(c+d x)} \, dx=\frac {-i \, d x + i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d} \]
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\[ \int \sqrt {-\tanh ^2(c+d x)} \, dx=\int \sqrt {- \tanh ^{2}{\left (c + d x \right )}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \sqrt {-\tanh ^2(c+d x)} \, dx=-\frac {i \, {\left (d x + c\right )}}{d} - \frac {i \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \sqrt {-\tanh ^2(c+d x)} \, dx=\frac {i \, {\left (d x + c\right )} \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right ) - i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )}{d} \]
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Timed out. \[ \int \sqrt {-\tanh ^2(c+d x)} \, dx=\int \sqrt {-{\mathrm {tanh}\left (c+d\,x\right )}^2} \,d x \]
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