Integrand size = 14, antiderivative size = 104 \[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\frac {\arctan \left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}+\frac {\text {arctanh}\left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2 \sqrt {b \coth ^3(c+d x)} \tanh (c+d x)}{d} \]
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Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3739, 3554, 3557, 335, 218, 212, 209} \[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\frac {\arctan \left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}+\frac {\text {arctanh}\left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2 \tanh (c+d x) \sqrt {b \coth ^3(c+d x)}}{d} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 3554
Rule 3557
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b \coth ^3(c+d x)} \int \coth ^{\frac {3}{2}}(c+d x) \, dx}{\coth ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {2 \sqrt {b \coth ^3(c+d x)} \tanh (c+d x)}{d}+\frac {\sqrt {b \coth ^3(c+d x)} \int \frac {1}{\sqrt {\coth (c+d x)}} \, dx}{\coth ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {2 \sqrt {b \coth ^3(c+d x)} \tanh (c+d x)}{d}-\frac {\sqrt {b \coth ^3(c+d x)} \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\coth (c+d x)\right )}{d \coth ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {2 \sqrt {b \coth ^3(c+d x)} \tanh (c+d x)}{d}-\frac {\left (2 \sqrt {b \coth ^3(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\coth (c+d x)}\right )}{d \coth ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {2 \sqrt {b \coth ^3(c+d x)} \tanh (c+d x)}{d}+\frac {\sqrt {b \coth ^3(c+d x)} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\coth (c+d x)}\right )}{d \coth ^{\frac {3}{2}}(c+d x)}+\frac {\sqrt {b \coth ^3(c+d x)} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\coth (c+d x)}\right )}{d \coth ^{\frac {3}{2}}(c+d x)} \\ & = \frac {\arctan \left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}+\frac {\text {arctanh}\left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2 \sqrt {b \coth ^3(c+d x)} \tanh (c+d x)}{d} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.61 \[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\frac {\left (\arctan \left (\sqrt {\coth (c+d x)}\right )+\text {arctanh}\left (\sqrt {\coth (c+d x)}\right )-2 \sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \coth \left (d x +c \right )^{3}}\, \left (2 \sqrt {b \coth \left (d x +c \right )}-\sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )-\sqrt {b}\, \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )\right )}{d \coth \left (d x +c \right ) \sqrt {b \coth \left (d x +c \right )}}\) | \(89\) |
default | \(-\frac {\sqrt {b \coth \left (d x +c \right )^{3}}\, \left (2 \sqrt {b \coth \left (d x +c \right )}-\sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )-\sqrt {b}\, \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )\right )}{d \coth \left (d x +c \right ) \sqrt {b \coth \left (d x +c \right )}}\) | \(89\) |
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (90) = 180\).
Time = 0.29 (sec) , antiderivative size = 633, normalized size of antiderivative = 6.09 \[ \int \sqrt {b \coth ^3(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 ) + 8 \, \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}}}{4 \, 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 ) - 8 \, \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}}}{4 \, d}\right ] \]
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\[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\int \sqrt {b \coth ^{3}{\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\int { \sqrt {b \coth \left (d x + c\right )^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (90) = 180\).
Time = 0.39 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.59 \[ \int \sqrt {b \coth ^3(c+d x)} \, dx=-\frac {2 \, \sqrt {b} \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}}{\sqrt {b}}\right ) \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) + \sqrt {b} \log \left ({\left | -\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} + \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} \right |}\right ) \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) - \frac {8 \, b \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}{\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} - \sqrt {b}}}{2 \, d} \]
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Timed out. \[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\int \sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^3} \,d x \]
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