Integrand size = 4, antiderivative size = 4 \[ \int \text {csch}^2(x) \, dx=-\coth (x) \]
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Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3852, 8} \[ \int \text {csch}^2(x) \, dx=-\coth (x) \]
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Rule 8
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -(i \text {Subst}(\int 1 \, dx,x,-i \coth (x))) \\ & = -\coth (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \text {csch}^2(x) \, dx=-\coth (x) \]
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Time = 0.23 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25
method | result | size |
default | \(-\coth \left (x \right )\) | \(5\) |
parallelrisch | \(-\coth \left (x \right )\) | \(5\) |
risch | \(-\frac {2}{{\mathrm e}^{2 x}-1}\) | \(11\) |
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Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (4) = 8\).
Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 5.00 \[ \int \text {csch}^2(x) \, dx=-\frac {2}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (3) = 6\).
Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 3.75 \[ \int \text {csch}^2(x) \, dx=- \frac {\tanh {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tanh {\left (\frac {x}{2} \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 10 vs. \(2 (4) = 8\).
Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 2.50 \[ \int \text {csch}^2(x) \, dx=\frac {2}{e^{\left (-2 \, x\right )} - 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 10 vs. \(2 (4) = 8\).
Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 2.50 \[ \int \text {csch}^2(x) \, dx=-\frac {2}{e^{\left (2 \, x\right )} - 1} \]
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Time = 0.02 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \text {csch}^2(x) \, dx=-\mathrm {coth}\left (x\right ) \]
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