Integrand size = 17, antiderivative size = 8 \[ \int \frac {1-\sinh ^2(x)}{1+\sinh ^2(x)} \, dx=-x+2 \tanh (x) \]
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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} \[ \int \frac {1-\sinh ^2(x)}{1+\sinh ^2(x)} \, dx=2 \tanh (x)-x \]
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Rule 8
Rule 3250
Rule 3254
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -x+2 \int \frac {1}{1+\sinh ^2(x)} \, dx \\ & = -x+2 \int \text {sech}^2(x) \, dx \\ & = -x+2 i \text {Subst}(\int 1 \, dx,x,-i \tanh (x)) \\ & = -x+2 \tanh (x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(17\) vs. \(2(8)=16\).
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.12 \[ \int \frac {1-\sinh ^2(x)}{1+\sinh ^2(x)} \, dx=-\frac {x}{2}-\frac {1}{2} \text {arctanh}(\tanh (x))+2 \tanh (x) \]
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Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12
method | result | size |
parallelrisch | \(-x +2 \tanh \left (x \right )\) | \(9\) |
risch | \(-x -\frac {4}{1+{\mathrm e}^{2 x}}\) | \(15\) |
default | \(-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {4 \tanh \left (\frac {x}{2}\right )}{1+\tanh \left (\frac {x}{2}\right )^{2}}+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\) | \(34\) |
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (8) = 16\).
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.12 \[ \int \frac {1-\sinh ^2(x)}{1+\sinh ^2(x)} \, dx=-\frac {{\left (x + 2\right )} \cosh \left (x\right ) - 2 \, \sinh \left (x\right )}{\cosh \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (5) = 10\).
Time = 0.45 (sec) , antiderivative size = 41, normalized size of antiderivative = 5.12 \[ \int \frac {1-\sinh ^2(x)}{1+\sinh ^2(x)} \, dx=- \frac {x \tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} + 1} - \frac {x}{\tanh ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {4 \tanh {\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} + 1} \]
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none
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {1-\sinh ^2(x)}{1+\sinh ^2(x)} \, dx=-x + \frac {4}{e^{\left (-2 \, x\right )} + 1} \]
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {1-\sinh ^2(x)}{1+\sinh ^2(x)} \, dx=-x - \frac {4}{e^{\left (2 \, x\right )} + 1} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {1-\sinh ^2(x)}{1+\sinh ^2(x)} \, dx=-x-\frac {4}{{\mathrm {e}}^{2\,x}+1} \]
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