Integrand size = 10, antiderivative size = 61 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=a \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {2}{3} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+\frac {1}{5} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x) \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852} \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=a \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)}+\frac {1}{5} a \sinh ^2(x) \tanh ^3(x) \sqrt {a \text {sech}^4(x)}-\frac {2}{3} a \sinh ^2(x) \tanh (x) \sqrt {a \text {sech}^4(x)} \]
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Rule 3852
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \left (a \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \text {sech}^6(x) \, dx \\ & = \left (i a \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (x)\right ) \\ & = a \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {2}{3} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+\frac {1}{5} a \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=\frac {1}{15} \cosh (x) (8+6 \cosh (2 x)+\cosh (4 x)) \left (a \text {sech}^4(x)\right )^{3/2} \sinh (x) \]
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Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {16 a \,{\mathrm e}^{-2 x} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (10 \,{\mathrm e}^{4 x}+5 \,{\mathrm e}^{2 x}+1\right )}{15 \left (1+{\mathrm e}^{2 x}\right )^{3}}\) | \(46\) |
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Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (51) = 102\).
Time = 0.26 (sec) , antiderivative size = 516, normalized size of antiderivative = 8.46 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=-\frac {16 \, {\left (10 \, a \cosh \left (x\right )^{4} + 10 \, {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{4} + 40 \, {\left (a \cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 5 \, a \cosh \left (x\right )^{2} + 5 \, {\left (12 \, a \cosh \left (x\right )^{2} + {\left (12 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (12 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} + {\left (10 \, a \cosh \left (x\right )^{4} + 5 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (10 \, a \cosh \left (x\right )^{4} + 5 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 10 \, {\left (4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right ) + {\left (4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (4 \, x\right )} + 2 \, {\left (4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + a\right )} \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}} e^{\left (2 \, x\right )}}{15 \, {\left (10 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right )^{9} + e^{\left (2 \, x\right )} \sinh \left (x\right )^{10} + 5 \, {\left (9 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{8} + 40 \, {\left (3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{7} + 10 \, {\left (21 \, \cosh \left (x\right )^{4} + 14 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{6} + 4 \, {\left (63 \, \cosh \left (x\right )^{5} + 70 \, \cosh \left (x\right )^{3} + 15 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{5} + 10 \, {\left (21 \, \cosh \left (x\right )^{6} + 35 \, \cosh \left (x\right )^{4} + 15 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{4} + 40 \, {\left (3 \, \cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{3} + 5 \, {\left (9 \, \cosh \left (x\right )^{8} + 28 \, \cosh \left (x\right )^{6} + 30 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{2} + 10 \, {\left (\cosh \left (x\right )^{9} + 4 \, \cosh \left (x\right )^{7} + 6 \, \cosh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{10} + 5 \, \cosh \left (x\right )^{8} + 10 \, \cosh \left (x\right )^{6} + 10 \, \cosh \left (x\right )^{4} + 5 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )}\right )}} \]
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\[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=\int \left (a \operatorname {sech}^{4}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (51) = 102\).
Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.97 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=\frac {16 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} + \frac {32 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} + \frac {16 \, a^{\frac {3}{2}}}{15 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.44 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=-\frac {16 \, a^{\frac {3}{2}} {\left (10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} + 1\right )}}{15 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \]
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Time = 1.96 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \left (a \text {sech}^4(x)\right )^{3/2} \, dx=-\frac {4\,a\,{\mathrm {e}}^{-2\,x}\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+1\right )}{15\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \]
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