Integrand size = 14, antiderivative size = 88 \[ \int \frac {1}{\left (-\tanh ^2(c+d x)\right )^{5/2}} \, dx=-\frac {\coth (c+d x)}{2 d \sqrt {-\tanh ^2(c+d x)}}-\frac {\coth ^3(c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}}+\frac {\log (\sinh (c+d x)) \tanh (c+d x)}{d \sqrt {-\tanh ^2(c+d x)}} \]
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Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3739, 3554, 3556} \[ \int \frac {1}{\left (-\tanh ^2(c+d x)\right )^{5/2}} \, dx=-\frac {\coth ^3(c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}}-\frac {\coth (c+d x)}{2 d \sqrt {-\tanh ^2(c+d x)}}+\frac {\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt {-\tanh ^2(c+d x)}} \]
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Rule 3554
Rule 3556
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\tanh (c+d x) \int \coth ^5(c+d x) \, dx}{\sqrt {-\tanh ^2(c+d x)}} \\ & = -\frac {\coth ^3(c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}}+\frac {\tanh (c+d x) \int \coth ^3(c+d x) \, dx}{\sqrt {-\tanh ^2(c+d x)}} \\ & = -\frac {\coth (c+d x)}{2 d \sqrt {-\tanh ^2(c+d x)}}-\frac {\coth ^3(c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}}+\frac {\tanh (c+d x) \int \coth (c+d x) \, dx}{\sqrt {-\tanh ^2(c+d x)}} \\ & = -\frac {\coth (c+d x)}{2 d \sqrt {-\tanh ^2(c+d x)}}-\frac {\coth ^3(c+d x)}{4 d \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 = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (-\tanh ^2(c+d x)\right )^{5/2}} \, dx=\frac {-2 \coth (c+d x)-\coth ^3(c+d x)+4 (\log (\cosh (c+d x))+\log (\tanh (c+d x))) \tanh (c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}} \]
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Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(-\frac {\tanh \left (d x +c \right ) \left (2 \ln \left (\tanh \left (d x +c \right )-1\right ) \tanh \left (d x +c \right )^{4}+2 \ln \left (\tanh \left (d x +c \right )+1\right ) \tanh \left (d x +c \right )^{4}-4 \ln \left (\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right )^{4}+2 \tanh \left (d x +c \right )^{2}+1\right )}{4 d \left (-\tanh \left (d x +c \right )^{2}\right )^{\frac {5}{2}}}\) | \(91\) |
default | \(-\frac {\tanh \left (d x +c \right ) \left (2 \ln \left (\tanh \left (d x +c \right )-1\right ) \tanh \left (d x +c \right )^{4}+2 \ln \left (\tanh \left (d x +c \right )+1\right ) \tanh \left (d x +c \right )^{4}-4 \ln \left (\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right )^{4}+2 \tanh \left (d x +c \right )^{2}+1\right )}{4 d \left (-\tanh \left (d x +c \right )^{2}\right )^{\frac {5}{2}}}\) | \(91\) |
risch | \(\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right ) x}{\sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}+1\right )}-\frac {2 \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (d x +c \right )}{\sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}+1\right ) d}-\frac {4 \,{\mathrm e}^{2 d x +2 c} \left ({\mathrm e}^{4 d x +4 c}-{\mathrm e}^{2 d x +2 c}+1\right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3} \left ({\mathrm e}^{2 d x +2 c}+1\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, d}+\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right ) \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{\sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}+1\right ) d}\) | \(284\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\left (-\tanh ^2(c+d x)\right )^{5/2}} \, dx=\frac {i \, d x e^{\left (8 \, d x + 8 \, c\right )} + i \, d x - 4 \, {\left (i \, d x - i\right )} e^{\left (6 \, d x + 6 \, c\right )} - 2 \, {\left (-3 i \, d x + 2 i\right )} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, {\left (i \, d x - i\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-i \, e^{\left (8 \, d x + 8 \, c\right )} + 4 i \, e^{\left (6 \, d x + 6 \, c\right )} - 6 i \, e^{\left (4 \, d x + 4 \, c\right )} + 4 i \, e^{\left (2 \, d x + 2 \, c\right )} - i\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}{d e^{\left (8 \, d x + 8 \, c\right )} - 4 \, d e^{\left (6 \, d x + 6 \, c\right )} + 6 \, d e^{\left (4 \, d x + 4 \, c\right )} - 4 \, d e^{\left (2 \, d x + 2 \, c\right )} + d} \]
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\[ \int \frac {1}{\left (-\tanh ^2(c+d x)\right )^{5/2}} \, dx=\int \frac {1}{\left (- \tanh ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\left (-\tanh ^2(c+d x)\right )^{5/2}} \, dx=\frac {i \, {\left (d x + c\right )}}{d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {4 \, {\left (-i \, e^{\left (-2 \, d x - 2 \, c\right )} + i \, e^{\left (-4 \, d x - 4 \, c\right )} - i \, e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\left (-\tanh ^2(c+d x)\right )^{5/2}} \, dx=-\frac {\frac {i \, d x + i \, c}{\mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )} - \frac {i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}{\mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )} + \frac {4 \, {\left (i \, e^{\left (6 \, d x + 6 \, c\right )} - i \, e^{\left (4 \, d x + 4 \, c\right )} + i \, e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4} \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )}}{d} \]
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Timed out. \[ \int \frac {1}{\left (-\tanh ^2(c+d x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (-{\mathrm {tanh}\left (c+d\,x\right )}^2\right )}^{5/2}} \,d x \]
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