Optimal. Leaf size=15 \[ a x+\frac{b \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.0132529, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3767, 8} \[ a x+\frac{b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \left (a+b \text{sech}^2(c+d x)\right ) \, dx &=a x+b \int \text{sech}^2(c+d x) \, dx\\ &=a x+\frac{(i b) \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{d}\\ &=a x+\frac{b \tanh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0027872, size = 15, normalized size = 1. \[ a x+\frac{b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 16, normalized size = 1.1 \begin{align*} ax+{\frac{b\tanh \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15253, size = 31, normalized size = 2.07 \begin{align*} a x + \frac{2 \, b}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99825, size = 88, normalized size = 5.87 \begin{align*} \frac{{\left (a d x - b\right )} \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{d \cosh \left (d x + c\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 \left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14616, size = 31, normalized size = 2.07 \begin{align*} a x - \frac{2 \, b}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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