Optimal. Leaf size=70 \[ \frac{a \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{a x}{2}+\frac{b \cosh ^5(c+d x)}{5 d}-\frac{2 b \cosh ^3(c+d x)}{3 d}+\frac{b \cosh (c+d x)}{d} \]
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Rubi [A] time = 0.0708939, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3220, 2635, 8, 2633} \[ \frac{a \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{a x}{2}+\frac{b \cosh ^5(c+d x)}{5 d}-\frac{2 b \cosh ^3(c+d x)}{3 d}+\frac{b \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=-\int \left (-a \sinh ^2(c+d x)-b \sinh ^5(c+d x)\right ) \, dx\\ &=a \int \sinh ^2(c+d x) \, dx+b \int \sinh ^5(c+d x) \, dx\\ &=\frac{a \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{1}{2} a \int 1 \, dx+\frac{b \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a x}{2}+\frac{b \cosh (c+d x)}{d}-\frac{2 b \cosh ^3(c+d x)}{3 d}+\frac{b \cosh ^5(c+d x)}{5 d}+\frac{a \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0954682, size = 79, normalized size = 1.13 \[ \frac{a (-c-d x)}{2 d}+\frac{a \sinh (2 (c+d x))}{4 d}+\frac{5 b \cosh (c+d x)}{8 d}-\frac{5 b \cosh (3 (c+d x))}{48 d}+\frac{b \cosh (5 (c+d x))}{80 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 60, normalized size = 0.9 \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{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11831, size = 162, normalized size = 2.31 \begin{align*} -\frac{1}{8} \, a{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0538, size = 302, normalized size = 4.31 \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} - 25 \, b \cosh \left (d x + c\right )^{3} - 120 \, a d x + 120 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + 15 \,{\left (2 \, b \cosh \left (d x + c\right )^{3} - 5 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 150 \, b \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 = 2.30679, size = 117, normalized size = 1.67 \begin{align*} \begin{cases} \frac{a x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac{a x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{a \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 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 ^{3}{\left (c \right )}\right ) \sinh ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16479, size = 149, normalized size = 2.13 \begin{align*} -\frac{240 \,{\left (d x + c\right )} a - 3 \, b e^{\left (5 \, d x + 5 \, c\right )} + 25 \, b e^{\left (3 \, d x + 3 \, c\right )} - 60 \, a e^{\left (2 \, d x + 2 \, c\right )} - 150 \, b e^{\left (d x + c\right )} -{\left (150 \, b e^{\left (4 \, d x + 4 \, c\right )} - 60 \, a e^{\left (3 \, d x + 3 \, 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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