Optimal. Leaf size=29 \[ \frac{2 e^{4 a+4 b x}}{b \left (e^{2 a+2 b x}+1\right )^2} \]
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Rubi [A] time = 0.0265666, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2282, 12, 264} \[ \frac{2 e^{4 a+4 b x}}{b \left (e^{2 a+2 b x}+1\right )^2} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 264
Rubi steps
\begin{align*} \int e^{a+b x} \text{sech}^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{8 x^3}{\left (1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{8 \operatorname{Subst}\left (\int \frac{x^3}{\left (1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{2 e^{4 a+4 b x}}{b \left (1+e^{2 a+2 b x}\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0177166, size = 29, normalized size = 1. \[ \frac{2 e^{4 a+4 b x}}{b \left (e^{2 a+2 b x}+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 30, normalized size = 1. \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}+\tanh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06647, size = 92, normalized size = 3.17 \begin{align*} -\frac{4 \, e^{\left (2 \, b x + 2 \, a\right )}}{b{\left (e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} - \frac{2}{b{\left (e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.65953, size = 238, normalized size = 8.21 \begin{align*} -\frac{2 \,{\left (3 \, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) +{\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\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*} e^{a} \int e^{b x} \operatorname{sech}^{3}{\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 = 1.28117, size = 42, normalized size = 1.45 \begin{align*} -\frac{2 \,{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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