Optimal. Leaf size=114 \[ a^2 x+\frac{2 a b \cosh ^3(c+d x)}{3 d}-\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac{5 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{5 b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{5 b^2 x}{16} \]
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Rubi [A] time = 0.0832559, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3213, 2633, 2635, 8} \[ a^2 x+\frac{2 a b \cosh ^3(c+d x)}{3 d}-\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac{5 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{5 b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{5 b^2 x}{16} \]
Antiderivative was successfully verified.
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Rule 3213
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=\int \left (a^2+2 a b \sinh ^3(c+d x)+b^2 \sinh ^6(c+d x)\right ) \, dx\\ &=a^2 x+(2 a b) \int \sinh ^3(c+d x) \, dx+b^2 \int \sinh ^6(c+d x) \, dx\\ &=a^2 x+\frac{b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac{1}{6} \left (5 b^2\right ) \int \sinh ^4(c+d x) \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=a^2 x-\frac{2 a b \cosh (c+d x)}{d}+\frac{2 a b \cosh ^3(c+d x)}{3 d}-\frac{5 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac{b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}+\frac{1}{8} \left (5 b^2\right ) \int \sinh ^2(c+d x) \, dx\\ &=a^2 x-\frac{2 a b \cosh (c+d x)}{d}+\frac{2 a b \cosh ^3(c+d x)}{3 d}+\frac{5 b^2 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac{5 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac{b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac{1}{16} \left (5 b^2\right ) \int 1 \, dx\\ &=a^2 x-\frac{5 b^2 x}{16}-\frac{2 a b \cosh (c+d x)}{d}+\frac{2 a b \cosh ^3(c+d x)}{3 d}+\frac{5 b^2 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac{5 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac{b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.146479, size = 94, normalized size = 0.82 \[ \frac{192 a^2 c+192 a^2 d x-288 a b \cosh (c+d x)+32 a b \cosh (3 (c+d x))+45 b^2 \sinh (2 (c+d x))-9 b^2 \sinh (4 (c+d x))+b^2 \sinh (6 (c+d x))-60 b^2 c-60 b^2 d x}{192 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 85, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{6}}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{5\,\sinh \left ( dx+c \right ) }{16}} \right ) \cosh \left ( dx+c \right ) -{\frac{5\,dx}{16}}-{\frac{5\,c}{16}} \right ) +2\,ab \left ( -2/3+1/3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) \cosh \left ( dx+c \right ) +{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14157, size = 204, normalized size = 1.79 \begin{align*} a^{2} x - \frac{1}{384} \, b^{2}{\left (\frac{{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac{120 \,{\left (d x + c\right )}}{d} + \frac{45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98662, size = 421, normalized size = 3.69 \begin{align*} \frac{3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 16 \, a b \cosh \left (d x + c\right )^{3} + 48 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} - 9 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \,{\left (16 \, a^{2} - 5 \, b^{2}\right )} d x - 144 \, a b \cosh \left (d x + c\right ) + 3 \,{\left (b^{2} \cosh \left (d x + c\right )^{5} - 6 \, b^{2} \cosh \left (d x + c\right )^{3} + 15 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.48219, size = 212, normalized size = 1.86 \begin{align*} \begin{cases} a^{2} x + \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{5 b^{2} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac{15 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac{15 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac{5 b^{2} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac{11 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{16 d} - \frac{5 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac{5 b^{2} \sinh{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12821, size = 212, normalized size = 1.86 \begin{align*} \frac{b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 9 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 32 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 45 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 288 \, a b e^{\left (d x + c\right )} + 24 \,{\left (16 \, a^{2} - 5 \, b^{2}\right )}{\left (d x + c\right )} -{\left (288 \, a b e^{\left (5 \, d x + 5 \, c\right )} + 45 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 32 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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