Optimal. Leaf size=83 \[ -\frac{e^{-3 a-3 b x}}{48 b}-\frac{e^{-a-b x}}{4 b}+\frac{3 e^{a+b x}}{8 b}+\frac{e^{3 a+3 b x}}{12 b}+\frac{e^{5 a+5 b x}}{80 b} \]
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Rubi [A] time = 0.0386232, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2282, 12, 270} \[ -\frac{e^{-3 a-3 b x}}{48 b}-\frac{e^{-a-b x}}{4 b}+\frac{3 e^{a+b x}}{8 b}+\frac{e^{3 a+3 b x}}{12 b}+\frac{e^{5 a+5 b x}}{80 b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 270
Rubi steps
\begin{align*} \int e^{a+b x} \cosh ^4(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^4}{16 x^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^4}{x^4} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (6+\frac{1}{x^4}+\frac{4}{x^2}+4 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}}{4 b}+\frac{3 e^{a+b x}}{8 b}+\frac{e^{3 a+3 b x}}{12 b}+\frac{e^{5 a+5 b x}}{80 b}\\ \end{align*}
Mathematica [A] time = 0.0416132, size = 62, normalized size = 0.75 \[ \frac{e^{-3 (a+b x)} \left (-60 e^{2 (a+b x)}+90 e^{4 (a+b x)}+20 e^{6 (a+b x)}+3 e^{8 (a+b x)}-5\right )}{240 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 77, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{5}}+{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{5}}+{\frac{\cosh \left ( bx+a \right ) }{5}}+ \left ({\frac{8}{15}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{5}}+{\frac{4\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{15}} \right ) \sinh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06531, size = 92, normalized size = 1.11 \begin{align*} \frac{e^{\left (5 \, b x + 5 \, a\right )}}{80 \, b} + \frac{e^{\left (3 \, b x + 3 \, a\right )}}{12 \, b} + \frac{3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac{e^{\left (-b x - a\right )}}{4 \, b} - \frac{e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79349, size = 325, normalized size = 3.92 \begin{align*} -\frac{\cosh \left (b x + a\right )^{4} - 16 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} + 10\right )} \sinh \left (b x + a\right )^{2} + 20 \, \cosh \left (b x + a\right )^{2} - 16 \,{\left (\cosh \left (b x + a\right )^{3} + 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 45}{120 \,{\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 = 82.5895, size = 139, normalized size = 1.67 \begin{align*} \begin{cases} \frac{8 e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )}}{15 b} - \frac{8 e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{15 b} - \frac{4 e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{5 b} + \frac{4 e^{a} e^{b x} \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{5 b} + \frac{e^{a} e^{b x} \cosh ^{4}{\left (a + b x \right )}}{5 b} & \text{for}\: b \neq 0 \\x e^{a} \cosh ^{4}{\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 = 1.32643, size = 81, normalized size = 0.98 \begin{align*} -\frac{5 \,{\left (12 \, 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 )} - 20 \, e^{\left (3 \, b x + 3 \, a\right )} - 90 \, e^{\left (b x + a\right )}}{240 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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