Integrand size = 10, antiderivative size = 67 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{3/2}} \, dx=\frac {\cosh (x) \sinh (x)}{a \sqrt {a \cosh ^4(x)}}-\frac {2 \sinh ^2(x) \tanh (x)}{3 a \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt {a \cosh ^4(x)}} \]
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Time = 0.02 (sec) , antiderivative size = 67, 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 = {3286, 3852} \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{3/2}} \, dx=\frac {\sinh (x) \cosh (x)}{a \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt {a \cosh ^4(x)}}-\frac {2 \sinh ^2(x) \tanh (x)}{3 a \sqrt {a \cosh ^4(x)}} \]
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Rule 3286
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh ^2(x) \int \text {sech}^6(x) \, dx}{a \sqrt {a \cosh ^4(x)}} \\ & = \frac {\left (i \cosh ^2(x)\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (x)\right )}{a \sqrt {a \cosh ^4(x)}} \\ & = \frac {\cosh (x) \sinh (x)}{a \sqrt {a \cosh ^4(x)}}-\frac {2 \sinh ^2(x) \tanh (x)}{3 a \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt {a \cosh ^4(x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{3/2}} \, dx=\frac {\cosh (x) (8+6 \cosh (2 x)+\cosh (4 x)) \sinh (x)}{15 \left (a \cosh ^4(x)\right )^{3/2}} \]
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Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {16 \,{\mathrm e}^{-2 x} \left (10 \,{\mathrm e}^{4 x}+5 \,{\mathrm e}^{2 x}+1\right )}{15 a \left (1+{\mathrm e}^{2 x}\right )^{3} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}\) | \(48\) |
default | \(\frac {\sqrt {8}\, \sqrt {2}\, \left (2 \cosh \left (2 x \right )^{2}+6 \cosh \left (2 x \right )+7\right ) \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}}{15 a^{2} \left (1+\cosh \left (2 x \right )\right )^{2} \sinh \left (2 x \right ) \sqrt {\left (1+\cosh \left (2 x \right )\right )^{2} a}}\) | \(80\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1137 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 1137, normalized size of antiderivative = 16.97 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{3/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{3/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.46 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{3/2}} \, dx=\frac {16 \, e^{\left (-2 \, x\right )}}{3 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} + a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{2}}\right )}} + \frac {32 \, e^{\left (-4 \, x\right )}}{3 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} + a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{2}}\right )}} + \frac {16}{15 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} + a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{2}}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{3/2}} \, dx=-\frac {16 \, {\left (10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} + 1\right )}}{15 \, a^{\frac {3}{2}} {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \]
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Time = 1.71 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{3/2}} \, dx=-\frac {64\,{\mathrm {e}}^{2\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+1\right )}{15\,a^2\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7} \]
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