Integrand size = 2, antiderivative size = 5 \[ \int \text {csch}(x) \, dx=-\text {arctanh}(\cosh (x)) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3855} \[ \int \text {csch}(x) \, dx=-\text {arctanh}(\cosh (x)) \]
[In]
[Out]
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\text {arctanh}(\cosh (x)) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(17\) vs. \(2(5)=10\).
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 3.40 \[ \int \text {csch}(x) \, dx=-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20
method | result | size |
lookup | \(\ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(6\) |
default | \(\ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(6\) |
parallelrisch | \(\ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(6\) |
risch | \(\ln \left (-1+{\mathrm e}^{x}\right )-\ln \left (1+{\mathrm e}^{x}\right )\) | \(14\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (5) = 10\).
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 3.40 \[ \int \text {csch}(x) \, dx=-\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \text {csch}(x) \, dx=\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \text {csch}(x) \, dx=\log \left (\tanh \left (\frac {1}{2} \, x\right )\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (5) = 10\).
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 2.80 \[ \int \text {csch}(x) \, dx=-\log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]
[In]
[Out]
Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \text {csch}(x) \, dx=\ln \left (\mathrm {tanh}\left (\frac {x}{2}\right )\right ) \]
[In]
[Out]