Optimal. Leaf size=57 \[ \frac{2 b (a-b) \cosh ^3(c+d x)}{3 d}+\frac{(a-b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0600295, antiderivative size = 57, 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 = {3186, 194} \[ \frac{2 b (a-b) \cosh ^3(c+d x)}{3 d}+\frac{(a-b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 194
Rubi steps
\begin{align*} \int \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a-b+b x^2\right )^2 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 \left (1+\frac{b (-2 a+b)}{a^2}\right )+2 a b \left (1-\frac{b}{a}\right ) x^2+b^2 x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{(a-b)^2 \cosh (c+d x)}{d}+\frac{2 (a-b) b \cosh ^3(c+d x)}{3 d}+\frac{b^2 \cosh ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.0362276, size = 111, normalized size = 1.95 \[ \frac{a^2 \sinh (c) \sinh (d x)}{d}+\frac{a^2 \cosh (c) \cosh (d x)}{d}-\frac{3 a b \cosh (c+d x)}{2 d}+\frac{a b \cosh (3 (c+d x))}{6 d}+\frac{5 b^2 \cosh (c+d x)}{8 d}-\frac{5 b^2 \cosh (3 (c+d x))}{48 d}+\frac{b^2 \cosh (5 (c+d x))}{80 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 70, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({b}^{2} \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 ) +2\,ab \left ( -2/3+1/3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) \cosh \left ( dx+c \right ) +{a}^{2}\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 = 1.04599, size = 212, normalized size = 3.72 \begin{align*} \frac{1}{480} \, b^{2}{\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}{12} \, a b{\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 )} + \frac{a^{2} \cosh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01757, size = 309, normalized size = 5.42 \begin{align*} \frac{3 \, b^{2} \cosh \left (d x + c\right )^{5} + 15 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 15 \,{\left (2 \, b^{2} \cosh \left (d x + c\right )^{3} +{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 30 \,{\left (8 \, a^{2} - 12 \, a b + 5 \, b^{2}\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 = 2.35011, size = 128, normalized size = 2.25 \begin{align*} \begin{cases} \frac{a^{2} \cosh{\left (c + d x \right )}}{d} + \frac{2 a b \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 a b \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{8 b^{2} \cosh ^{5}{\left (c + d x \right )}}{15 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} \sinh{\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 = 1.25033, size = 220, normalized size = 3.86 \begin{align*} \frac{3 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 40 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 25 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 240 \, a^{2} e^{\left (d x + c\right )} - 360 \, a b e^{\left (d x + c\right )} + 150 \, b^{2} e^{\left (d x + c\right )} +{\left (240 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 360 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 150 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 40 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 25 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2}\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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