Integrand size = 12, antiderivative size = 75 \[ \int (b \tanh (c+d x))^{3/2} \, dx=\frac {b^{3/2} \arctan \left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {2 b \sqrt {b \tanh (c+d x)}}{d} \]
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Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3554, 3557, 335, 218, 212, 209} \[ \int (b \tanh (c+d x))^{3/2} \, dx=\frac {b^{3/2} \arctan \left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {2 b \sqrt {b \tanh (c+d x)}}{d} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 3554
Rule 3557
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b \sqrt {b \tanh (c+d x)}}{d}+b^2 \int \frac {1}{\sqrt {b \tanh (c+d x)}} \, dx \\ & = -\frac {2 b \sqrt {b \tanh (c+d x)}}{d}-\frac {b^3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-b^2+x^2\right )} \, dx,x,b \tanh (c+d x)\right )}{d} \\ & = -\frac {2 b \sqrt {b \tanh (c+d x)}}{d}-\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{-b^2+x^4} \, dx,x,\sqrt {b \tanh (c+d x)}\right )}{d} \\ & = -\frac {2 b \sqrt {b \tanh (c+d x)}}{d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \tanh (c+d x)}\right )}{d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \tanh (c+d x)}\right )}{d} \\ & = \frac {b^{3/2} \arctan \left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {2 b \sqrt {b \tanh (c+d x)}}{d} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88 \[ \int (b \tanh (c+d x))^{3/2} \, dx=-\frac {\left (-\arctan \left (\sqrt {\tanh (c+d x)}\right )-\text {arctanh}\left (\sqrt {\tanh (c+d x)}\right )+2 \sqrt {\tanh (c+d x)}\right ) (b \tanh (c+d x))^{3/2}}{d \tanh ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b \tanh \left (d x +c \right )}}{\sqrt {b}}\right )}{d}+\frac {b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \tanh \left (d x +c \right )}}{\sqrt {b}}\right )}{d}-\frac {2 b \sqrt {b \tanh \left (d x +c \right )}}{d}\) | \(62\) |
default | \(\frac {b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b \tanh \left (d x +c \right )}}{\sqrt {b}}\right )}{d}+\frac {b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \tanh \left (d x +c \right )}}{\sqrt {b}}\right )}{d}-\frac {2 b \sqrt {b \tanh \left (d x +c \right )}}{d}\) | \(62\) |
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (61) = 122\).
Time = 0.27 (sec) , antiderivative size = 638, normalized size of antiderivative = 8.51 \[ \int (b \tanh (c+d x))^{3/2} \, dx=\left [-\frac {2 \, \sqrt {-b} 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 \sinh \left (d x + c\right )}{\cosh \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} 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 \sinh \left (d x + c\right )}{\cosh \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 \, b \sqrt {\frac {b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )}}}{4 \, d}, -\frac {2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b} \sqrt {\frac {b \sinh \left (d x + c\right )}{\cosh \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 ) - b^{\frac {3}{2}} \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 \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )}} - b\right ) + 8 \, b \sqrt {\frac {b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )}}}{4 \, d}\right ] \]
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\[ \int (b \tanh (c+d x))^{3/2} \, dx=\int \left (b \tanh {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (b \tanh (c+d x))^{3/2} \, dx=\int { \left (b \tanh \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (61) = 122\).
Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.75 \[ \int (b \tanh (c+d x))^{3/2} \, dx=\frac {{\left (2 \, \sqrt {b} \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}}{\sqrt {b}}\right ) - \sqrt {b} \log \left ({\left | -\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} + \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} \right |}\right ) - \frac {8 \, b}{\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} + \sqrt {b}}\right )} b}{2 \, d} \]
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Time = 1.80 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.81 \[ \int (b \tanh (c+d x))^{3/2} \, dx=\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,\mathrm {tanh}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{d}-\frac {2\,b\,\sqrt {b\,\mathrm {tanh}\left (c+d\,x\right )}}{d}+\frac {b^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,\mathrm {tanh}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{d} \]
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