Integrand size = 11, antiderivative size = 1 \[ \int \frac {1}{\text {sech}^2(x)+\tanh ^2(x)} \, dx=x \]
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Time = 0.01 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4466, 8} \[ \int \frac {1}{\text {sech}^2(x)+\tanh ^2(x)} \, dx=x \]
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Rule 8
Rule 4466
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 \[ \int \frac {1}{\text {sech}^2(x)+\tanh ^2(x)} \, dx=x \]
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Time = 1.07 (sec) , antiderivative size = 2, normalized size of antiderivative = 2.00
method | result | size |
risch | \(x\) | \(2\) |
default | \(2 \,\operatorname {arctanh}\left (\tanh \left (\frac {x}{2}\right )\right )\) | \(8\) |
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none
Time = 0.25 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\text {sech}^2(x)+\tanh ^2(x)} \, dx=x \]
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\[ \int \frac {1}{\text {sech}^2(x)+\tanh ^2(x)} \, dx=\int \frac {1}{\tanh ^{2}{\left (x \right )} + \operatorname {sech}^{2}{\left (x \right )}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\text {sech}^2(x)+\tanh ^2(x)} \, dx=x \]
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none
Time = 0.26 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\text {sech}^2(x)+\tanh ^2(x)} \, dx=x \]
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Time = 2.26 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\text {sech}^2(x)+\tanh ^2(x)} \, dx=x \]
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