Integrand size = 10, antiderivative size = 15 \[ \int \sqrt {a \text {sech}^4(x)} \, dx=\cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x) \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 3852, 8} \[ \int \sqrt {a \text {sech}^4(x)} \, dx=\sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)} \]
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Rule 8
Rule 3852
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \text {sech}^2(x) \, dx \\ & = \left (i \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \text {Subst}(\int 1 \, dx,x,-i \tanh (x)) \\ & = \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \sqrt {a \text {sech}^4(x)} \, dx=\cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(13)=26\).
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.93
method | result | size |
risch | \(-2 \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (13) = 26\).
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 5.40 \[ \int \sqrt {a \text {sech}^4(x)} \, dx=-\frac {2 \, \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )}}{2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right ) + e^{\left (2 \, x\right )} \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )}} \]
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\[ \int \sqrt {a \text {sech}^4(x)} \, dx=\int \sqrt {a \operatorname {sech}^{4}{\left (x \right )}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \sqrt {a \text {sech}^4(x)} \, dx=\frac {2 \, \sqrt {a}}{e^{\left (-2 \, x\right )} + 1} \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \sqrt {a \text {sech}^4(x)} \, dx=-\frac {2 \, \sqrt {a}}{e^{\left (2 \, x\right )} + 1} \]
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Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.73 \[ \int \sqrt {a \text {sech}^4(x)} \, dx=-\frac {\sqrt {a}\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (2\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+2\,{\mathrm {e}}^{6\,x}+\frac {{\mathrm {e}}^{8\,x}}{2}+\frac {1}{2}\right )}{\left ({\mathrm {e}}^{2\,x}+1\right )\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]
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