Optimal. Leaf size=10 \[ \frac{\tanh (a+b x)}{b} \]
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Rubi [A] time = 0.0100053, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3767, 8} \[ \frac{\tanh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \text{sech}^2(a+b x) \, dx &=\frac{i \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (a+b x))}{b}\\ &=\frac{\tanh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0028304, size = 10, normalized size = 1. \[ \frac{\tanh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 11, normalized size = 1.1 \begin{align*}{\frac{\tanh \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991004, size = 24, normalized size = 2.4 \begin{align*} \frac{2}{b{\left (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 = 2.07124, size = 111, normalized size = 11.1 \begin{align*} -\frac{2}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b} \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}^{2}{\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 = 1.13855, size = 24, normalized size = 2.4 \begin{align*} -\frac{2}{b{\left (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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