Optimal. Leaf size=69 \[ \frac{e^{-3 a-3 b x}}{48 b}-\frac{e^{-a-b x}}{8 b}-\frac{e^{3 a+3 b x}}{24 b}+\frac{e^{5 a+5 b x}}{80 b} \]
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Rubi [A] time = 0.0510697, antiderivative size = 69, 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, 448} \[ \frac{e^{-3 a-3 b x}}{48 b}-\frac{e^{-a-b x}}{8 b}-\frac{e^{3 a+3 b x}}{24 b}+\frac{e^{5 a+5 b x}}{80 b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 448
Rubi steps
\begin{align*} \int e^{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^4} \, 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^4} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{x^4}+\frac{2}{x^2}-2 x^2+x^4\right ) \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac{e^{-3 a-3 b x}}{48 b}-\frac{e^{-a-b x}}{8 b}-\frac{e^{3 a+3 b x}}{24 b}+\frac{e^{5 a+5 b x}}{80 b}\\ \end{align*}
Mathematica [A] time = 0.0618103, size = 51, normalized size = 0.74 \[ \frac{e^{-3 (a+b x)} \left (-30 e^{2 (a+b x)}-10 e^{6 (a+b x)}+3 e^{8 (a+b x)}+5\right )}{240 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 90, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{5}}-{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{5}}+{\frac{\sinh \left ( bx+a \right ) }{5}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{5}}-{\frac{2\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{15}}-{\frac{2\,\cosh \left ( bx+a \right ) }{15}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994325, size = 76, normalized size = 1.1 \begin{align*} -\frac{{\left (6 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, b} + \frac{3 \, e^{\left (5 \, b x + 5 \, a\right )} - 10 \, e^{\left (3 \, b x + 3 \, a\right )}}{240 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56103, size = 302, normalized size = 4.38 \begin{align*} \frac{\cosh \left (b x + a\right )^{4} - \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} +{\left (6 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right )^{2} - 5 \, \cosh \left (b x + a\right )^{2} -{\left (\cosh \left (b x + a\right )^{3} - 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{30 \,{\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 92.6981, size = 139, normalized size = 2.01 \begin{align*} \begin{cases} \frac{e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )}}{5 b} - \frac{e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{5 b} + \frac{e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{5 b} + \frac{2 e^{a} e^{b x} \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{15 b} - \frac{2 e^{a} e^{b x} \cosh ^{4}{\left (a + b x \right )}}{15 b} & \text{for}\: b \neq 0 \\x e^{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 = 1.14918, size = 70, normalized size = 1.01 \begin{align*} -\frac{5 \,{\left (6 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, e^{\left (5 \, b x + 5 \, a\right )} + 10 \, e^{\left (3 \, b x + 3 \, a\right )}}{240 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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