Integrand size = 14, antiderivative size = 65 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3739, 3554, 3556} \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \]
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Rule 3554
Rule 3556
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\coth (c+d x) \int \tanh ^3(c+d x) \, dx}{b \sqrt {b \coth ^2(c+d x)}} \\ & = -\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}}+\frac {\coth (c+d x) \int \tanh (c+d x) \, dx}{b \sqrt {b \coth ^2(c+d x)}} \\ & = \frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\frac {2 \coth (c+d x) \log (\cosh (c+d x))-\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \]
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Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(-\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )+1\right ) \coth \left (d x +c \right )^{2}+\ln \left (\coth \left (d x +c \right )-1\right ) \coth \left (d x +c \right )^{2}-2 \ln \left (\coth \left (d x +c \right )\right ) \coth \left (d x +c \right )^{2}+1\right )}{2 d \left (\coth \left (d x +c \right )^{2} b \right )^{\frac {3}{2}}}\) | \(79\) |
default | \(-\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )+1\right ) \coth \left (d x +c \right )^{2}+\ln \left (\coth \left (d x +c \right )-1\right ) \coth \left (d x +c \right )^{2}-2 \ln \left (\coth \left (d x +c \right )\right ) \coth \left (d x +c \right )^{2}+1\right )}{2 d \left (\coth \left (d x +c \right )^{2} b \right )^{\frac {3}{2}}}\) | \(79\) |
risch | \(\frac {-{\mathrm e}^{4 d x +4 c} d x +{\mathrm e}^{4 d x +4 c} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )-2 \,{\mathrm e}^{4 d x +4 c} c -2 \,{\mathrm e}^{2 d x +2 c} d x +2 \,{\mathrm e}^{2 d x +2 c} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )-4 \,{\mathrm e}^{2 d x +2 c} c -d x +2 \,{\mathrm e}^{2 d x +2 c}+\ln \left ({\mathrm e}^{2 d x +2 c}+1\right )-2 c}{b \left ({\mathrm e}^{2 d x +2 c}+1\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2} b}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, d}\) | \(190\) |
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Leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (59) = 118\).
Time = 0.27 (sec) , antiderivative size = 817, normalized size of antiderivative = 12.57 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{\left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {b} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-4 \, d x - 4 \, c\right )} + b^{2}\right )} d} - \frac {d x + c}{b^{\frac {3}{2}} d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{\frac {3}{2}} d} \]
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Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=-\frac {\frac {d x + c}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {2 \, e^{\left (2 \, d x + 2 \, c\right )}}{\sqrt {b} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}}{b d} \]
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Timed out. \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{3/2}} \,d x \]
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