Optimal. Leaf size=20 \[ a x+\frac{b \sinh ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0187967, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2564, 30} \[ a x+\frac{b \sinh ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 30
Rubi steps
\begin{align*} \int (a+b \cosh (c+d x) \sinh (c+d x)) \, dx &=a x+b \int \cosh (c+d x) \sinh (c+d x) \, dx\\ &=a x-\frac{b \operatorname{Subst}(\int x \, dx,x,i \sinh (c+d x))}{d}\\ &=a x+\frac{b \sinh ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0074473, size = 38, normalized size = 1.9 \[ a x+\frac{b \sinh (2 c) \sinh (2 d x)}{4 d}+\frac{b \cosh (2 c) \cosh (2 d x)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 19, normalized size = 1. \begin{align*} ax+{\frac{b \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23071, size = 24, normalized size = 1.2 \begin{align*} a x + \frac{b \cosh \left (d x + c\right )^{2}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48052, size = 77, normalized size = 3.85 \begin{align*} \frac{4 \, a d x + b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.288004, size = 24, normalized size = 1.2 \begin{align*} a x + b \left (\begin{cases} \frac{\sinh ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \sinh{\left (c \right )} \cosh{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13355, size = 39, normalized size = 1.95 \begin{align*} a x + \frac{b{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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