Optimal. Leaf size=19 \[ \frac{1}{7} \tanh (7 x)-\frac{1}{21} \tanh ^3(7 x) \]
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Rubi [A] time = 0.0096991, antiderivative size = 19, 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} \[ \frac{1}{7} \tanh (7 x)-\frac{1}{21} \tanh ^3(7 x) \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \text{sech}^4(7 x) \, dx &=\frac{1}{7} i \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (7 x)\right )\\ &=\frac{1}{7} \tanh (7 x)-\frac{1}{21} \tanh ^3(7 x)\\ \end{align*}
Mathematica [A] time = 0.0036122, size = 19, normalized size = 1. \[ \frac{1}{7} \tanh (7 x)-\frac{1}{21} \tanh ^3(7 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 17, normalized size = 0.9 \begin{align*}{\frac{\tanh \left ( 7\,x \right ) }{7} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (7\,x\right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04548, size = 66, normalized size = 3.47 \begin{align*} \frac{4 \, e^{\left (-14 \, x\right )}}{7 \,{\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} + \frac{4}{21 \,{\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10968, size = 331, normalized size = 17.42 \begin{align*} -\frac{8 \,{\left (2 \, \cosh \left (7 \, x\right ) + \sinh \left (7 \, x\right )\right )}}{21 \,{\left (\cosh \left (7 \, x\right )^{5} + 5 \, \cosh \left (7 \, x\right ) \sinh \left (7 \, x\right )^{4} + \sinh \left (7 \, x\right )^{5} +{\left (10 \, \cosh \left (7 \, x\right )^{2} + 3\right )} \sinh \left (7 \, x\right )^{3} + 3 \, \cosh \left (7 \, x\right )^{3} +{\left (10 \, \cosh \left (7 \, x\right )^{3} + 9 \, \cosh \left (7 \, x\right )\right )} \sinh \left (7 \, x\right )^{2} +{\left (5 \, \cosh \left (7 \, x\right )^{4} + 9 \, \cosh \left (7 \, x\right )^{2} + 2\right )} \sinh \left (7 \, x\right ) + 4 \, \cosh \left (7 \, 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}^{4}{\left (7 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13372, size = 24, normalized size = 1.26 \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (14 \, x\right )} + 1\right )}}{21 \,{\left (e^{\left (14 \, x\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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