Integrand size = 12, antiderivative size = 57 \[ \int \frac {1}{\sqrt {b \coth (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}+\frac {\text {arctanh}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d} \]
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Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3557, 335, 218, 212, 209} \[ \int \frac {1}{\sqrt {b \coth (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}+\frac {\text {arctanh}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 3557
Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-b^2+x^2\right )} \, dx,x,b \coth (c+d x)\right )}{d} \\ & = -\frac {(2 b) \text {Subst}\left (\int \frac {1}{-b^2+x^4} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}+\frac {\text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d} \\ & = \frac {\arctan \left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}+\frac {\text {arctanh}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {b \coth (c+d x)}} \, dx=\frac {\left (\arctan \left (\sqrt {\coth (c+d x)}\right )+\text {arctanh}\left (\sqrt {\coth (c+d x)}\right )\right ) \sqrt {\coth (c+d x)}}{d \sqrt {b \coth (c+d x)}} \]
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Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{d \sqrt {b}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{d \sqrt {b}}\) | \(46\) |
default | \(\frac {\arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{d \sqrt {b}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{d \sqrt {b}}\) | \(46\) |
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (45) = 90\).
Time = 0.29 (sec) , antiderivative size = 598, normalized size of antiderivative = 10.49 \[ \int \frac {1}{\sqrt {b \coth (c+d x)}} \, dx=\left [-\frac {2 \, \sqrt {-b} \arctan \left (\frac {{\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}}}{b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + b}\right ) + \sqrt {-b} \log \left (-\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, b \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \sqrt {-b} \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}} - 2 \, b}{\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4}}\right )}{4 \, b d}, \frac {2 \, \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}}}{b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + b}\right ) + \sqrt {b} \log \left (2 \, b \cosh \left (d x + c\right )^{4} + 8 \, b \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 12 \, b \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 8 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, b \sinh \left (d x + c\right )^{4} + 2 \, {\left (\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4} + {\left (6 \, \cosh \left (d x + c\right )^{2} - 1\right )} \sinh \left (d x + c\right )^{2} - \cosh \left (d x + c\right )^{2} + 2 \, {\left (2 \, \cosh \left (d x + c\right )^{3} - \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b} \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}} - b\right )}{4 \, b d}\right ] \]
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\[ \int \frac {1}{\sqrt {b \coth (c+d x)}} \, dx=\int \frac {1}{\sqrt {b \coth {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {b \coth (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \coth \left (d x + c\right )}} \,d x } \]
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Exception generated. \[ \int \frac {1}{\sqrt {b \coth (c+d x)}} \, dx=\text {Exception raised: TypeError} \]
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Time = 1.97 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {b \coth (c+d x)}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{\sqrt {b}}\right )+\mathrm {atanh}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{\sqrt {b}\,d} \]
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