Optimal. Leaf size=53 \[ \frac{(a-2 b) \cosh ^3(c+d x)}{3 d}-\frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0557225, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3013, 373} \[ \frac{(a-2 b) \cosh ^3(c+d x)}{3 d}-\frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3013
Rule 373
Rubi steps
\begin{align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a-b+b x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a \left (1-\frac{b}{a}\right )-(a-2 b) x^2-b x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{(a-b) \cosh (c+d x)}{d}+\frac{(a-2 b) \cosh ^3(c+d x)}{3 d}+\frac{b \cosh ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.0255729, size = 77, normalized size = 1.45 \[ -\frac{3 a \cosh (c+d x)}{4 d}+\frac{a \cosh (3 (c+d x))}{12 d}+\frac{5 b \cosh (c+d x)}{8 d}-\frac{5 b \cosh (3 (c+d x))}{48 d}+\frac{b \cosh (5 (c+d x))}{80 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 56, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{8}{15}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( dx+c \right ) +a \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04109, size = 190, normalized size = 3.58 \begin{align*} \frac{1}{480} \, b{\left (\frac{3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac{25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{150 \, e^{\left (d x + c\right )}}{d} + \frac{150 \, e^{\left (-d x - c\right )}}{d} - \frac{25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac{1}{24} \, a{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82341, size = 273, normalized size = 5.15 \begin{align*} \frac{3 \, b \cosh \left (d x + c\right )^{5} + 15 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \,{\left (4 \, a - 5 \, b\right )} \cosh \left (d x + c\right )^{3} + 15 \,{\left (2 \, b \cosh \left (d x + c\right )^{3} +{\left (4 \, a - 5 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 30 \,{\left (6 \, a - 5 \, b\right )} \cosh \left (d x + c\right )}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.47998, size = 105, normalized size = 1.98 \begin{align*} \begin{cases} \frac{a \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 a \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{b \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{8 b \cosh ^{5}{\left (c + d x \right )}}{15 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \sinh ^{3}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25488, size = 166, normalized size = 3.13 \begin{align*} \frac{3 \, b e^{\left (5 \, d x + 5 \, c\right )} + 20 \, a e^{\left (3 \, d x + 3 \, c\right )} - 25 \, b e^{\left (3 \, d x + 3 \, c\right )} - 180 \, a e^{\left (d x + c\right )} + 150 \, b e^{\left (d x + c\right )} -{\left (180 \, a e^{\left (4 \, d x + 4 \, c\right )} - 150 \, b e^{\left (4 \, d x + 4 \, c\right )} - 20 \, a e^{\left (2 \, d x + 2 \, c\right )} + 25 \, b e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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