Integrand size = 13, antiderivative size = 24 \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2800, 65, 213} \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rule 65
Rule 213
Rule 2800
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \cosh (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \cosh (x)}\right ) \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(18)=36\).
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.33
method | result | size |
default | \(-\frac {\ln \left (\frac {4 \cosh \left (\frac {x}{2}\right ) b \sqrt {2}+4 \sqrt {a}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}+4 a -4 b}{2 \cosh \left (\frac {x}{2}\right )-\sqrt {2}}\right )+\ln \left (-\frac {4 \left (\cosh \left (\frac {x}{2}\right ) b \sqrt {2}-\sqrt {a}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}-a +b \right )}{2 \cosh \left (\frac {x}{2}\right )+\sqrt {2}}\right )}{\sqrt {a}}\) | \(104\) |
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (18) = 36\).
Time = 0.29 (sec) , antiderivative size = 356, normalized size of antiderivative = 14.83 \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\left [\frac {\log \left (\frac {b^{2} \cosh \left (x\right )^{4} + b^{2} \sinh \left (x\right )^{4} + 16 \, a b \cosh \left (x\right )^{3} + 4 \, {\left (b^{2} \cosh \left (x\right ) + 4 \, a b\right )} \sinh \left (x\right )^{3} + 16 \, a b \cosh \left (x\right ) + 2 \, {\left (16 \, a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} + 24 \, a b \cosh \left (x\right ) + 16 \, a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} - 8 \, {\left (b \cosh \left (x\right )^{3} + b \sinh \left (x\right )^{3} + 4 \, a \cosh \left (x\right )^{2} + {\left (3 \, b \cosh \left (x\right ) + 4 \, a\right )} \sinh \left (x\right )^{2} + b \cosh \left (x\right ) + {\left (3 \, b \cosh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) + b\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a} \sqrt {a} + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} + 12 \, a b \cosh \left (x\right )^{2} + 4 \, a b + {\left (16 \, a^{2} + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right ) + b\right )} \sqrt {b \cosh \left (x\right ) + a} \sqrt {-a}}{2 \, {\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) + a b + 2 \, {\left (a b \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )}}\right )}{a}\right ] \]
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\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int \frac {\tanh {\left (x \right )}}{\sqrt {a + b \cosh {\left (x \right )}}}\, dx \]
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\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {b \cosh \left (x\right ) + a}} \,d x } \]
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\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {b \cosh \left (x\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int \frac {\mathrm {tanh}\left (x\right )}{\sqrt {a+b\,\mathrm {cosh}\left (x\right )}} \,d x \]
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