Optimal. Leaf size=26 \[ \frac{\tanh (a+b x)}{b}-\frac{\tanh ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.011633, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3767} \[ \frac{\tanh (a+b x)}{b}-\frac{\tanh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \text{sech}^4(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (a+b x)\right )}{b}\\ &=\frac{\tanh (a+b x)}{b}-\frac{\tanh ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0064148, size = 26, normalized size = 1. \[ \frac{\tanh (a+b x)}{b}-\frac{\tanh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 23, normalized size = 0.9 \begin{align*}{\frac{\tanh \left ( bx+a \right ) }{b} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01786, size = 122, normalized size = 4.69 \begin{align*} \frac{4 \, e^{\left (-2 \, b x - 2 \, a\right )}}{b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} + \frac{4}{3 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96322, size = 448, normalized size = 17.23 \begin{align*} -\frac{8 \,{\left (2 \, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{5} + 5 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + b \sinh \left (b x + a\right )^{5} + 3 \, b \cosh \left (b x + a\right )^{3} +{\left (10 \, b \cosh \left (b x + a\right )^{2} + 3 \, b\right )} \sinh \left (b x + a\right )^{3} +{\left (10 \, b \cosh \left (b x + a\right )^{3} + 9 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 4 \, b \cosh \left (b x + a\right ) +{\left (5 \, b \cosh \left (b x + a\right )^{4} + 9 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\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 (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15398, size = 42, normalized size = 1.62 \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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