Optimal. Leaf size=57 \[ \frac{e^{-2 a-2 b x}}{32 b}-\frac{e^{4 a+4 b x}}{32 b}+\frac{e^{6 a+6 b x}}{96 b}+\frac{x}{8} \]
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Rubi [A] time = 0.053828, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 12, 446, 75} \[ \frac{e^{-2 a-2 b x}}{32 b}-\frac{e^{4 a+4 b x}}{32 b}+\frac{e^{6 a+6 b x}}{96 b}+\frac{x}{8} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 446
Rule 75
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \cosh (a+b x) \sinh ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1-x^2\right ) \left (1-x^2\right )^3}{16 x^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1-x^2\right ) \left (1-x^2\right )^3}{x^3} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-1-x) (1-x)^3}{x^2} \, dx,x,e^{2 a+2 b x}\right )}{32 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{x^2}+\frac{2}{x}-2 x+x^2\right ) \, dx,x,e^{2 a+2 b x}\right )}{32 b}\\ &=\frac{e^{-2 a-2 b x}}{32 b}-\frac{e^{4 a+4 b x}}{32 b}+\frac{e^{6 a+6 b x}}{96 b}+\frac{x}{8}\\ \end{align*}
Mathematica [A] time = 0.0725796, size = 43, normalized size = 0.75 \[ \frac{3 e^{-2 (a+b x)}-3 e^{4 (a+b x)}+e^{6 (a+b x)}+12 b x}{96 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 89, normalized size = 1.6 \begin{align*}{\frac{x}{8}}-{\frac{\sinh \left ( 2\,bx+2\,a \right ) }{32\,b}}-{\frac{\sinh \left ( 4\,bx+4\,a \right ) }{32\,b}}+{\frac{\sinh \left ( 6\,bx+6\,a \right ) }{96\,b}}+{\frac{\cosh \left ( 2\,bx+2\,a \right ) }{32\,b}}-{\frac{\cosh \left ( 4\,bx+4\,a \right ) }{32\,b}}+{\frac{\cosh \left ( 6\,bx+6\,a \right ) }{96\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03007, size = 70, normalized size = 1.23 \begin{align*} -\frac{{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{96 \, b} + \frac{b x + a}{8 \, b} + \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88637, size = 413, normalized size = 7.25 \begin{align*} \frac{4 \, \cosh \left (b x + a\right )^{4} - 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 4 \, \sinh \left (b x + a\right )^{4} + 3 \,{\left (4 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} + 3 \,{\left (4 \, b x + 8 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \,{\left (4 \, \cosh \left (b x + a\right )^{3} + 3 \,{\left (4 \, b x + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \,{\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 = 90.0155, size = 241, normalized size = 4.23 \begin{align*} \begin{cases} - \frac{x e^{2 a} e^{2 b x} \sinh ^{4}{\left (a + b x \right )}}{8} + \frac{x e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{4} - \frac{x e^{2 a} e^{2 b x} \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{4} + \frac{x e^{2 a} e^{2 b x} \cosh ^{4}{\left (a + b x \right )}}{8} + \frac{e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b} + \frac{e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac{7 e^{2 a} e^{2 b x} \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{24 b} + \frac{e^{2 a} e^{2 b x} \cosh ^{4}{\left (a + b x \right )}}{12 b} & \text{for}\: b \neq 0 \\x e^{2 a} \sinh ^{3}{\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 = 1.16822, size = 81, normalized size = 1.42 \begin{align*} \frac{12 \, b x - 3 \,{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} +{\left (e^{\left (6 \, b x + 12 \, a\right )} - 3 \, e^{\left (4 \, b x + 10 \, a\right )}\right )} e^{\left (-6 \, a\right )}}{96 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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