Integrand size = 14, antiderivative size = 31 \[ \int \frac {1}{\sqrt {-\tanh ^2(c+d x)}} \, dx=\frac {\log (\sinh (c+d x)) \tanh (c+d x)}{d \sqrt {-\tanh ^2(c+d x)}} \]
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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} \[ \int \frac {1}{\sqrt {-\tanh ^2(c+d x)}} \, dx=\frac {\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt {-\tanh ^2(c+d x)}} \]
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Rule 3556
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\tanh (c+d x) \int \coth (c+d x) \, dx}{\sqrt {-\tanh ^2(c+d x)}} \\ & = \frac {\log (\sinh (c+d x)) \tanh (c+d x)}{d \sqrt {-\tanh ^2(c+d x)}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {-\tanh ^2(c+d x)}} \, dx=\frac {(\log (\cosh (c+d x))+\log (\tanh (c+d x))) \tanh (c+d x)}{d \sqrt {-\tanh ^2(c+d x)}} \]
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Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68
method | result | size |
derivativedivides | \(-\frac {\tanh \left (d x +c \right ) \left (\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (\tanh \left (d x +c \right )+1\right )-2 \ln \left (\tanh \left (d x +c \right )\right )\right )}{2 d \sqrt {-\tanh \left (d x +c \right )^{2}}}\) | \(52\) |
default | \(-\frac {\tanh \left (d x +c \right ) \left (\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (\tanh \left (d x +c \right )+1\right )-2 \ln \left (\tanh \left (d x +c \right )\right )\right )}{2 d \sqrt {-\tanh \left (d x +c \right )^{2}}}\) | \(52\) |
risch | \(\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right ) x}{\sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}+1\right )}-\frac {2 \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (d x +c \right )}{\sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}+1\right ) d}+\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right ) \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{\sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}+1\right ) d}\) | \(192\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {-\tanh ^2(c+d x)}} \, dx=\frac {i \, d x - i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}{d} \]
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\[ \int \frac {1}{\sqrt {-\tanh ^2(c+d x)}} \, dx=\int \frac {1}{\sqrt {- \tanh ^{2}{\left (c + d x \right )}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\sqrt {-\tanh ^2(c+d x)}} \, dx=\frac {i \, {\left (d x + c\right )}}{d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {1}{\sqrt {-\tanh ^2(c+d x)}} \, dx=-\frac {\frac {i \, d x + i \, c}{\mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )} - \frac {i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}{\mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )}}{d} \]
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Time = 1.84 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt {-\tanh ^2(c+d x)}} \, dx=\frac {\mathrm {atan}\left (\frac {\mathrm {tanh}\left (c+d\,x\right )}{\sqrt {-{\mathrm {tanh}\left (c+d\,x\right )}^2}}\right )}{d} \]
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