Optimal. Leaf size=19 \[ \frac {1}{7} \tanh (7 x)-\frac {1}{21} \tanh ^3(7 x) \]
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Rubi [A] time = 0.01, antiderivative size = 19, 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} \[ \frac {1}{7} \tanh (7 x)-\frac {1}{21} \tanh ^3(7 x) \]
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*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \[ \frac {1}{7} \tanh (7 x)-\frac {1}{21} \tanh ^3(7 x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 116, normalized size = 6.11 \[ -\frac {8 \, {\left (2 \, \cosh \left (7 \, x\right ) + \sinh \left (7 \, x\right )\right )}}{21 \, {\left (\cosh \left (7 \, x\right )^{5} + 5 \, \cosh \left (7 \, x\right ) \sinh \left (7 \, x\right )^{4} + \sinh \left (7 \, x\right )^{5} + {\left (10 \, \cosh \left (7 \, x\right )^{2} + 3\right )} \sinh \left (7 \, x\right )^{3} + 3 \, \cosh \left (7 \, x\right )^{3} + {\left (10 \, \cosh \left (7 \, x\right )^{3} + 9 \, \cosh \left (7 \, x\right )\right )} \sinh \left (7 \, x\right )^{2} + {\left (5 \, \cosh \left (7 \, x\right )^{4} + 9 \, \cosh \left (7 \, x\right )^{2} + 2\right )} \sinh \left (7 \, x\right ) + 4 \, \cosh \left (7 \, x\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 18, normalized size = 0.95 \[ -\frac {4 \, {\left (3 \, e^{\left (14 \, x\right )} + 1\right )}}{21 \, {\left (e^{\left (14 \, x\right )} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 17, normalized size = 0.89 \[ \frac {\left (\frac {2}{3}+\frac {\mathrm {sech}\left (7 x \right )^{2}}{3}\right ) \tanh \left (7 x \right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 49, normalized size = 2.58 \[ \frac {4 \, e^{\left (-14 \, x\right )}}{7 \, {\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} + \frac {4}{21 \, {\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 30, normalized size = 1.58 \[ -\frac {2\,\left (3\,{\mathrm {e}}^{14\,x}-3\,{\mathrm {e}}^{28\,x}-{\mathrm {e}}^{42\,x}+1\right )}{21\,{\left ({\mathrm {e}}^{14\,x}+1\right )}^3} \]
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 \[ \int \operatorname {sech}^{4}{\left (7 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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