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 = 0.013558, antiderivative size = 35, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0040945, size = 35, normalized size = 1. \[ \frac{\tanh ^5(\pi x)}{5 \pi }-\frac{2 \tanh ^3(\pi x)}{3 \pi }+\frac{\tanh (\pi x)}{\pi } \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 27, normalized size = 0.8 \begin{align*}{\frac{\tanh \left ( \pi \,x \right ) }{\pi } \left ({\frac{8}{15}}+{\frac{ \left ({\rm sech} \left (\pi \,x\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm sech} \left (\pi \,x\right ) \right ) ^{2}}{15}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02779, size = 185, normalized size = 5.29 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12901, size = 859, normalized size = 24.54 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}^{6}{\left (\pi x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14291, size = 41, normalized size = 1.17 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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