Integrand size = 13, antiderivative size = 1 \[ \int \frac {1}{\left (\cosh ^2(x)-\sinh ^2(x)\right )^2} \, dx=x \]
[Out]
Time = 0.01 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4465, 8} \[ \int \frac {1}{\left (\cosh ^2(x)-\sinh ^2(x)\right )^2} \, dx=x \]
[In]
[Out]
Rule 8
Rule 4465
Rubi steps \begin{align*} \text {integral}& = \int 1 \, dx \\ & = x \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\cosh ^2(x)-\sinh ^2(x)\right )^2} \, dx=x \]
[In]
[Out]
Time = 11.54 (sec) , antiderivative size = 2, normalized size of antiderivative = 2.00
method | result | size |
risch | \(x\) | \(2\) |
default | \(2 \,\operatorname {arctanh}\left (\tanh \left (\frac {x}{2}\right )\right )\) | \(8\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\cosh ^2(x)-\sinh ^2(x)\right )^2} \, dx=x \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (0) = 0\).
Time = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 22.00 \[ \int \frac {1}{\left (\cosh ^2(x)-\sinh ^2(x)\right )^2} \, dx=\frac {x}{\sinh ^{4}{\left (x \right )} - 2 \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )} + \cosh ^{4}{\left (x \right )}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\cosh ^2(x)-\sinh ^2(x)\right )^2} \, dx=x \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\cosh ^2(x)-\sinh ^2(x)\right )^2} \, dx=x \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\cosh ^2(x)-\sinh ^2(x)\right )^2} \, dx=x \]
[In]
[Out]