Integrand size = 6, antiderivative size = 35 \[ \int \text {sech}^6(\pi x) \, dx=\frac {\tanh (\pi x)}{\pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh ^5(\pi x)}{5 \pi } \]
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Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3852} \[ \int \text {sech}^6(\pi x) \, dx=\frac {\tanh ^5(\pi x)}{5 \pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh (\pi x)}{\pi } \]
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Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {i \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (\pi x)\right )}{\pi } \\ & = \frac {\tanh (\pi x)}{\pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh ^5(\pi x)}{5 \pi } \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \text {sech}^6(\pi x) \, dx=\frac {\tanh (\pi x)}{\pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh ^5(\pi x)}{5 \pi } \]
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Time = 0.77 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (\pi x \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (\pi x \right )^{2}}{15}\right ) \tanh \left (\pi x \right )}{\pi }\) | \(27\) |
default | \(\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (\pi x \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (\pi x \right )^{2}}{15}\right ) \tanh \left (\pi x \right )}{\pi }\) | \(27\) |
risch | \(-\frac {16 \left (10 \,{\mathrm e}^{4 \pi x}+5 \,{\mathrm e}^{2 \pi x}+1\right )}{15 \pi \left ({\mathrm e}^{2 \pi x}+1\right )^{5}}\) | \(31\) |
parallelrisch | \(\frac {2 \tanh \left (\frac {\pi x}{2}\right )+2 \tanh \left (\frac {\pi x}{2}\right )^{9}+\frac {116 \tanh \left (\frac {\pi x}{2}\right )^{5}}{15}+\frac {8 \tanh \left (\frac {\pi x}{2}\right )^{3}}{3}+\frac {8 \tanh \left (\frac {\pi x}{2}\right )^{7}}{3}}{\pi \left (1+\tanh \left (\frac {\pi x}{2}\right )^{2}\right )^{5}}\) | \(61\) |
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Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 280, normalized size of antiderivative = 8.00 \[ \int \text {sech}^6(\pi x) \, dx=-\frac {16 \, {\left (11 \, \cosh \left (\pi x\right )^{2} + 18 \, \cosh \left (\pi x\right ) \sinh \left (\pi x\right ) + 11 \, \sinh \left (\pi x\right )^{2} + 5\right )}}{15 \, {\left (5 \, \pi + \pi \cosh \left (\pi x\right )^{8} + 8 \, \pi \cosh \left (\pi x\right ) \sinh \left (\pi x\right )^{7} + \pi \sinh \left (\pi x\right )^{8} + 5 \, \pi \cosh \left (\pi x\right )^{6} + {\left (5 \, \pi + 28 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{6} + 2 \, {\left (28 \, \pi \cosh \left (\pi x\right )^{3} + 15 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )^{5} + 10 \, \pi \cosh \left (\pi x\right )^{4} + 5 \, {\left (2 \, \pi + 14 \, \pi \cosh \left (\pi x\right )^{4} + 15 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{4} + 4 \, {\left (14 \, \pi \cosh \left (\pi x\right )^{5} + 25 \, \pi \cosh \left (\pi x\right )^{3} + 10 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )^{3} + 11 \, \pi \cosh \left (\pi x\right )^{2} + {\left (11 \, \pi + 28 \, \pi \cosh \left (\pi x\right )^{6} + 75 \, \pi \cosh \left (\pi x\right )^{4} + 60 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{2} + 2 \, {\left (4 \, \pi \cosh \left (\pi x\right )^{7} + 15 \, \pi \cosh \left (\pi x\right )^{5} + 20 \, \pi \cosh \left (\pi x\right )^{3} + 9 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )\right )}} \]
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\[ \int \text {sech}^6(\pi x) \, dx=\int \operatorname {sech}^{6}{\left (\pi x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (31) = 62\).
Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 3.91 \[ \int \text {sech}^6(\pi x) \, dx=\frac {16 \, e^{\left (-2 \, \pi x\right )}}{3 \, \pi {\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} + \frac {32 \, e^{\left (-4 \, \pi x\right )}}{3 \, \pi {\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} + \frac {16}{15 \, \pi {\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \text {sech}^6(\pi x) \, dx=-\frac {16 \, {\left (10 \, e^{\left (4 \, \pi x\right )} + 5 \, e^{\left (2 \, \pi x\right )} + 1\right )}}{15 \, \pi {\left (e^{\left (2 \, \pi x\right )} + 1\right )}^{5}} \]
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Time = 2.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \text {sech}^6(\pi x) \, dx=-\frac {16\,\left (5\,{\mathrm {e}}^{2\,\Pi \,x}+10\,{\mathrm {e}}^{4\,\Pi \,x}+1\right )}{15\,\Pi \,{\left ({\mathrm {e}}^{2\,\Pi \,x}+1\right )}^5} \]
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