Optimal. Leaf size=23 \[ \frac{e^{4 a+4 b x}}{16 b}-\frac{x}{4} \]
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Rubi [A] time = 0.0229656, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2282, 12, 14} \[ \frac{e^{4 a+4 b x}}{16 b}-\frac{x}{4} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 14
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \cosh (a+b x) \sinh (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+x^4}{4 x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{-1+x^4}{x} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{x}+x^3\right ) \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac{e^{4 a+4 b x}}{16 b}-\frac{x}{4}\\ \end{align*}
Mathematica [A] time = 0.0151823, size = 25, normalized size = 1.09 \[ \frac{1}{4} \left (\frac{e^{4 a+4 b x}}{4 b}-x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 33, normalized size = 1.4 \begin{align*} -{\frac{x}{4}}+{\frac{\sinh \left ( 4\,bx+4\,a \right ) }{16\,b}}+{\frac{\cosh \left ( 4\,bx+4\,a \right ) }{16\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00212, size = 32, normalized size = 1.39 \begin{align*} -\frac{1}{4} \, x - \frac{a}{4 \, b} + \frac{e^{\left (4 \, b x + 4 \, a\right )}}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74641, size = 250, normalized size = 10.87 \begin{align*} -\frac{{\left (4 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} - 2 \,{\left (4 \, b x + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) +{\left (4 \, b x - 1\right )} \sinh \left (b x + a\right )^{2}}{16 \,{\left (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}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.81299, size = 117, normalized size = 5.09 \begin{align*} \begin{cases} - \frac{x e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )}}{4} + \frac{x e^{2 a} e^{2 b x} \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2} - \frac{x e^{2 a} e^{2 b x} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac{e^{2 a} e^{2 b x} \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{4 b} & \text{for}\: b \neq 0 \\x e^{2 a} \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 = 1.1319, size = 24, normalized size = 1.04 \begin{align*} -\frac{1}{4} \, x + \frac{e^{\left (4 \, b x + 4 \, a\right )}}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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