Optimal. Leaf size=85 \[ \frac{b (2 a-3 b) \cosh ^5(c+d x)}{5 d}+\frac{(a-3 b) (a-b) \cosh ^3(c+d x)}{3 d}-\frac{(a-b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.0974927, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3186, 373} \[ \frac{b (2 a-3 b) \cosh ^5(c+d x)}{5 d}+\frac{(a-3 b) (a-b) \cosh ^3(c+d x)}{3 d}-\frac{(a-b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 373
Rubi steps
\begin{align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a-b+b x^2\right )^2 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left ((a-b)^2+(a-3 b) (-a+b) x^2-(2 a-3 b) b x^4-b^2 x^6\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{(a-b)^2 \cosh (c+d x)}{d}+\frac{(a-3 b) (a-b) \cosh ^3(c+d x)}{3 d}+\frac{(2 a-3 b) b \cosh ^5(c+d x)}{5 d}+\frac{b^2 \cosh ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.0384168, size = 154, normalized size = 1.81 \[ -\frac{3 a^2 \cosh (c+d x)}{4 d}+\frac{a^2 \cosh (3 (c+d x))}{12 d}+\frac{5 a b \cosh (c+d x)}{4 d}-\frac{5 a b \cosh (3 (c+d x))}{24 d}+\frac{a b \cosh (5 (c+d x))}{40 d}-\frac{35 b^2 \cosh (c+d x)}{64 d}+\frac{7 b^2 \cosh (3 (c+d x))}{64 d}-\frac{7 b^2 \cosh (5 (c+d x))}{320 d}+\frac{b^2 \cosh (7 (c+d x))}{448 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 102, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( -{\frac{16}{35}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \cosh \left ( dx+c \right ) +2\,ab \left ({\frac{8}{15}}+1/5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( dx+c \right ) +{a}^{2} \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 = 1.05969, size = 333, normalized size = 3.92 \begin{align*} -\frac{1}{4480} \, b^{2}{\left (\frac{{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac{1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac{1}{240} \, a 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^{2}{\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 = 2.24913, size = 554, normalized size = 6.52 \begin{align*} \frac{15 \, b^{2} \cosh \left (d x + c\right )^{7} + 105 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 21 \,{\left (8 \, a b - 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 105 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} +{\left (8 \, a b - 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 35 \,{\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 105 \,{\left (3 \, b^{2} \cosh \left (d x + c\right )^{5} + 2 \,{\left (8 \, a b - 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} +{\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 105 \,{\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} \cosh \left (d x + c\right )}{6720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.16706, size = 204, normalized size = 2.4 \begin{align*} \begin{cases} \frac{a^{2} \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 a^{2} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{2 a b \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{8 a b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{16 a b \cosh ^{5}{\left (c + d x \right )}}{15 d} + \frac{b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{8 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{16 b^{2} \cosh ^{7}{\left (c + d x \right )}}{35 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} \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 = 1.32485, size = 332, normalized size = 3.91 \begin{align*} \frac{15 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 168 \, a b e^{\left (5 \, d x + 5 \, c\right )} - 147 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 560 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 1400 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 735 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 5040 \, a^{2} e^{\left (d x + c\right )} + 8400 \, a b e^{\left (d x + c\right )} - 3675 \, b^{2} e^{\left (d x + c\right )} -{\left (5040 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 8400 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 3675 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 560 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 1400 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 735 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 168 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 147 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{2}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{13440 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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