Optimal. Leaf size=40 \[ -\frac{e^{2 a+2 b x}}{4 b}+\frac{e^{4 a+4 b x}}{16 b}+\frac{x}{4} \]
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Rubi [A] time = 0.035378, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2282, 12, 266, 43} \[ -\frac{e^{2 a+2 b x}}{4 b}+\frac{e^{4 a+4 b x}}{16 b}+\frac{x}{4} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \sinh ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{4 x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1-x)^2}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2+\frac{1}{x}+x\right ) \, dx,x,e^{2 a+2 b x}\right )}{8 b}\\ &=-\frac{e^{2 a+2 b x}}{4 b}+\frac{e^{4 a+4 b x}}{16 b}+\frac{x}{4}\\ \end{align*}
Mathematica [A] time = 0.0191898, size = 32, normalized size = 0.8 \[ \frac{-4 e^{2 (a+b x)}+e^{4 (a+b x)}+4 b x}{16 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 61, normalized size = 1.5 \begin{align*}{\frac{x}{4}}-{\frac{\sinh \left ( 2\,bx+2\,a \right ) }{4\,b}}+{\frac{\sinh \left ( 4\,bx+4\,a \right ) }{16\,b}}-{\frac{\cosh \left ( 2\,bx+2\,a \right ) }{4\,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 = 1.25528, size = 50, normalized size = 1.25 \begin{align*} \frac{1}{4} \, x - \frac{{\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{16 \, b} + \frac{a}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.80609, size = 254, normalized size = 6.35 \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} - 4}{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 = 7.28974, size = 139, normalized size = 3.48 \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 ^{2}{\left (a + b x \right )}}{2 b} - \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 ^{2}{\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 = 1.14819, size = 41, normalized size = 1.02 \begin{align*} \frac{4 \, b x + e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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