Integrand size = 17, antiderivative size = 8 \[ \int \frac {1+\cosh ^2(x)}{1-\cosh ^2(x)} \, dx=-x+2 \coth (x) \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3250, 3254, 3852, 8} \[ \int \frac {1+\cosh ^2(x)}{1-\cosh ^2(x)} \, dx=2 \coth (x)-x \]
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Rule 8
Rule 3250
Rule 3254
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -x+2 \int \frac {1}{1-\cosh ^2(x)} \, dx \\ & = -x-2 \int \text {csch}^2(x) \, dx \\ & = -x+2 i \text {Subst}(\int 1 \, dx,x,-i \coth (x)) \\ & = -x+2 \coth (x) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.25 \[ \int \frac {1+\cosh ^2(x)}{1-\cosh ^2(x)} \, dx=\coth (x)+\coth (x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\tanh ^2(x)\right ) \]
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Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12
method | result | size |
parallelrisch | \(-x +2 \coth \left (x \right )\) | \(9\) |
risch | \(-x +\frac {4}{{\mathrm e}^{2 x}-1}\) | \(15\) |
default | \(\tanh \left (\frac {x}{2}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )}\) | \(28\) |
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (8) = 16\).
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.12 \[ \int \frac {1+\cosh ^2(x)}{1-\cosh ^2(x)} \, dx=-\frac {{\left (x + 2\right )} \sinh \left (x\right ) - 2 \, \cosh \left (x\right )}{\sinh \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (5) = 10\).
Time = 0.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.50 \[ \int \frac {1+\cosh ^2(x)}{1-\cosh ^2(x)} \, dx=- x + \tanh {\left (\frac {x}{2} \right )} + \frac {1}{\tanh {\left (\frac {x}{2} \right )}} \]
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none
Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {1+\cosh ^2(x)}{1-\cosh ^2(x)} \, dx=-x - \frac {4}{e^{\left (-2 \, x\right )} - 1} \]
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none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {1+\cosh ^2(x)}{1-\cosh ^2(x)} \, dx=-x + \frac {4}{e^{\left (2 \, x\right )} - 1} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {1+\cosh ^2(x)}{1-\cosh ^2(x)} \, dx=\frac {4}{{\mathrm {e}}^{2\,x}-1}-x \]
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