Optimal. Leaf size=15 \[ \frac{\sinh ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0137316, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2564, 30} \[ \frac{\sinh ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 30
Rubi steps
\begin{align*} \int \cosh (a+b x) \sinh (a+b x) \, dx &=-\frac{\operatorname{Subst}(\int x \, dx,x,i \sinh (a+b x))}{b}\\ &=\frac{\sinh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [B] time = 0.0096956, size = 37, normalized size = 2.47 \[ \frac{1}{2} \left (\frac{\sinh (2 a) \sinh (2 b x)}{2 b}+\frac{\cosh (2 a) \cosh (2 b x)}{2 b}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0907, size = 18, normalized size = 1.2 \begin{align*} \frac{\cosh \left (b x + a\right )^{2}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99963, size = 58, normalized size = 3.87 \begin{align*} \frac{\cosh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{2}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.22203, size = 19, normalized size = 1.27 \begin{align*} \begin{cases} \frac{\cosh ^{2}{\left (a + b x \right )}}{2 b} & \text{for}\: b \neq 0 \\x \sinh{\left (a \right )} \cosh{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12654, size = 32, normalized size = 2.13 \begin{align*} \frac{e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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