Optimal. Leaf size=49 \[ -\frac{e^{-3 a-3 b x}}{48 b}-\frac{e^{a+b x}}{8 b}+\frac{e^{5 a+5 b x}}{80 b} \]
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Rubi [A] time = 0.0530814, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2282, 12, 270} \[ -\frac{e^{-3 a-3 b x}}{48 b}-\frac{e^{a+b x}}{8 b}+\frac{e^{5 a+5 b x}}{80 b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 270
Rubi steps
\begin{align*} \int e^{a+b x} \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^4\right )^2}{16 x^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^4\right )^2}{x^4} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2+\frac{1}{x^4}+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^{5 a+5 b x}}{80 b}\\ \end{align*}
Mathematica [A] time = 0.025766, size = 40, normalized size = 0.82 \[ \frac{e^{-3 (a+b x)} \left (-30 e^{4 (a+b x)}+3 e^{8 (a+b x)}-5\right )}{240 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 84, normalized size = 1.7 \begin{align*}{\frac{1}{b} \left ({\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}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4}\sinh \left ( bx+a \right ) }{5}}-{\frac{\sinh \left ( bx+a \right ) }{5} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992282, size = 51, normalized size = 1.04 \begin{align*} \frac{e^{\left (5 \, b x + 5 \, a\right )} - 10 \, e^{\left (b x + a\right )}}{80 \, 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 [B] time = 1.55048, size = 258, normalized size = 5.27 \begin{align*} -\frac{\cosh \left (b x + a\right )^{4} - 16 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - 16 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 15}{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 = 137.372, size = 144, normalized size = 2.94 \begin{align*} \begin{cases} - \frac{2 e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )}}{15 b} + \frac{2 e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{15 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 ^{2}{\left (a \right )} \cosh ^{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 = 1.14842, size = 49, normalized size = 1. \begin{align*} \frac{3 \, e^{\left (5 \, b x + 5 \, a\right )} - 30 \, e^{\left (b x + a\right )} - 5 \, 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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