Optimal. Leaf size=35 \[ \frac {\tanh ^5(\pi x)}{5 \pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh (\pi x)}{\pi } \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3767} \[ \frac {\tanh ^5(\pi x)}{5 \pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh (\pi x)}{\pi } \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin {align*} \int \text {sech}^6(\pi x) \, dx &=\frac {i \operatorname {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*}
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Mathematica [A] time = 0.00, size = 35, normalized size = 1.00 \[ \frac {\tanh ^5(\pi x)}{5 \pi }-\frac {2 \tanh ^3(\pi x)}{3 \pi }+\frac {\tanh (\pi x)}{\pi } \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 280, normalized size = 8.00 \[ -\frac {16 \, {\left (11 \, \cosh \left (\pi x\right )^{2} + 18 \, \cosh \left (\pi x\right ) \sinh \left (\pi x\right ) + 11 \, \sinh \left (\pi x\right )^{2} + 5\right )}}{15 \, {\left (5 \, \pi + \pi \cosh \left (\pi x\right )^{8} + 8 \, \pi \cosh \left (\pi x\right ) \sinh \left (\pi x\right )^{7} + \pi \sinh \left (\pi x\right )^{8} + 5 \, \pi \cosh \left (\pi x\right )^{6} + {\left (5 \, \pi + 28 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{6} + 2 \, {\left (28 \, \pi \cosh \left (\pi x\right )^{3} + 15 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )^{5} + 10 \, \pi \cosh \left (\pi x\right )^{4} + 5 \, {\left (2 \, \pi + 14 \, \pi \cosh \left (\pi x\right )^{4} + 15 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{4} + 4 \, {\left (14 \, \pi \cosh \left (\pi x\right )^{5} + 25 \, \pi \cosh \left (\pi x\right )^{3} + 10 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )^{3} + 11 \, \pi \cosh \left (\pi x\right )^{2} + {\left (11 \, \pi + 28 \, \pi \cosh \left (\pi x\right )^{6} + 75 \, \pi \cosh \left (\pi x\right )^{4} + 60 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{2} + 2 \, {\left (4 \, \pi \cosh \left (\pi x\right )^{7} + 15 \, \pi \cosh \left (\pi x\right )^{5} + 20 \, \pi \cosh \left (\pi x\right )^{3} + 9 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 30, normalized size = 0.86 \[ -\frac {16 \, {\left (10 \, e^{\left (4 \, \pi x\right )} + 5 \, e^{\left (2 \, \pi x\right )} + 1\right )}}{15 \, \pi {\left (e^{\left (2 \, \pi x\right )} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 27, normalized size = 0.77 \[ \frac {\left (\frac {8}{15}+\frac {\mathrm {sech}\left (\pi x \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (\pi x \right )^{2}}{15}\right ) \tanh \left (\pi x \right )}{\pi } \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.08, size = 137, normalized size = 3.91 \[ \frac {16 \, e^{\left (-2 \, \pi x\right )}}{3 \, \pi {\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} + \frac {32 \, e^{\left (-4 \, \pi x\right )}}{3 \, \pi {\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} + \frac {16}{15 \, \pi {\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.52, size = 30, normalized size = 0.86 \[ -\frac {16\,\left (5\,{\mathrm {e}}^{2\,\Pi \,x}+10\,{\mathrm {e}}^{4\,\Pi \,x}+1\right )}{15\,\Pi \,{\left ({\mathrm {e}}^{2\,\Pi \,x}+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {sech}^{6}{\left (\pi x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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