Optimal. Leaf size=31 \[ \frac{\cosh ^5(a+b x)}{5 b}-\frac{\cosh ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0348252, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2565, 14} \[ \frac{\cosh ^5(a+b x)}{5 b}-\frac{\cosh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2565
Rule 14
Rubi steps
\begin{align*} \int \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac{\cosh ^3(a+b x)}{3 b}+\frac{\cosh ^5(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.069528, size = 27, normalized size = 0.87 \[ \frac{\cosh ^3(a+b x) (3 \cosh (2 (a+b x))-7)}{30 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 48, normalized size = 1.6 \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}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02092, size = 105, normalized size = 3.39 \begin{align*} -\frac{{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 30 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3\right )} e^{\left (5 \, b x + 5 \, a\right )}}{480 \, b} - \frac{30 \, e^{\left (-b x - a\right )} + 5 \, e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, e^{\left (-5 \, b x - 5 \, a\right )}}{480 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.884, size = 216, normalized size = 6.97 \begin{align*} \frac{3 \, \cosh \left (b x + a\right )^{5} + 15 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 5 \, \cosh \left (b x + a\right )^{3} + 15 \,{\left (2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 30 \, \cosh \left (b x + a\right )}{240 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.12358, size = 44, normalized size = 1.42 \begin{align*} \begin{cases} \frac{\sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac{2 \cosh ^{5}{\left (a + b x \right )}}{15 b} & \text{for}\: b \neq 0 \\x \sinh ^{3}{\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 [B] time = 1.15514, size = 95, normalized size = 3.06 \begin{align*} -\frac{{\left (30 \, e^{\left (4 \, b x + 4 \, a\right )} + 5 \, e^{\left (2 \, b x + 2 \, a\right )} - 3\right )} e^{\left (-5 \, b x - 5 \, a\right )} - 3 \, e^{\left (5 \, b x + 5 \, a\right )} + 5 \, e^{\left (3 \, b x + 3 \, a\right )} + 30 \, e^{\left (b x + a\right )}}{480 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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