Integrand size = 11, antiderivative size = 3 \[ \int \frac {1}{\cosh ^2(x)+\sinh ^2(x)} \, dx=\arctan (\tanh (x)) \]
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Time = 0.01 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {209} \[ \int \frac {1}{\cosh ^2(x)+\sinh ^2(x)} \, dx=\arctan (\tanh (x)) \]
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Rule 209
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tanh (x)\right ) \\ & = \arctan (\tanh (x)) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(9\) vs. \(2(3)=6\).
Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 3.00 \[ \int \frac {1}{\cosh ^2(x)+\sinh ^2(x)} \, dx=\frac {1}{2} \arctan (\sinh (2 x)) \]
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Result contains complex when optimal does not.
Time = 1.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 8.00
method | result | size |
risch | \(\frac {i \ln \left ({\mathrm e}^{2 x}+i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{2 x}-i\right )}{2}\) | \(24\) |
default | \(-\frac {\sqrt {2}\, \left (2+\sqrt {2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}}-\frac {\left (-2+\sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2 \sqrt {2}-2}\right )}{2 \sqrt {2}-2}\) | \(72\) |
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (3) = 6\).
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 6.33 \[ \int \frac {1}{\cosh ^2(x)+\sinh ^2(x)} \, dx=-\arctan \left (-\frac {\cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (3) = 6\).
Time = 3.77 (sec) , antiderivative size = 172, normalized size of antiderivative = 57.33 \[ \int \frac {1}{\cosh ^2(x)+\sinh ^2(x)} \, dx=\frac {47321 \sqrt {3 - 2 \sqrt {2}} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )}}{13860 \sqrt {2} + 19601} + \frac {33461 \sqrt {2} \sqrt {3 - 2 \sqrt {2}} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )}}{13860 \sqrt {2} + 19601} - \frac {5741 \sqrt {2} \sqrt {2 \sqrt {2} + 3} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )}}{13860 \sqrt {2} + 19601} - \frac {8119 \sqrt {2 \sqrt {2} + 3} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )}}{13860 \sqrt {2} + 19601} \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (3) = 6\).
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 11.67 \[ \int \frac {1}{\cosh ^2(x)+\sinh ^2(x)} \, dx=\arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-x\right )}\right )}\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\cosh ^2(x)+\sinh ^2(x)} \, dx=\arctan \left (e^{\left (2 \, x\right )}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\cosh ^2(x)+\sinh ^2(x)} \, dx=\mathrm {atan}\left ({\mathrm {e}}^{2\,x}\right ) \]
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