Optimal. Leaf size=130 \[ \frac{a^2 \cosh (c+d x)}{d}+\frac{a b \sinh ^3(c+d x) \cosh (c+d x)}{2 d}-\frac{3 a b \sinh (c+d x) \cosh (c+d x)}{4 d}+\frac{3 a b x}{4}+\frac{b^2 \cosh ^7(c+d x)}{7 d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d}+\frac{b^2 \cosh ^3(c+d x)}{d}-\frac{b^2 \cosh (c+d x)}{d} \]
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Rubi [A] time = 0.118934, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3220, 2638, 2635, 8, 2633} \[ \frac{a^2 \cosh (c+d x)}{d}+\frac{a b \sinh ^3(c+d x) \cosh (c+d x)}{2 d}-\frac{3 a b \sinh (c+d x) \cosh (c+d x)}{4 d}+\frac{3 a b x}{4}+\frac{b^2 \cosh ^7(c+d x)}{7 d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d}+\frac{b^2 \cosh ^3(c+d x)}{d}-\frac{b^2 \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 2638
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\left (i \int \left (i a^2 \sinh (c+d x)+2 i a b \sinh ^4(c+d x)+i b^2 \sinh ^7(c+d x)\right ) \, dx\right )\\ &=a^2 \int \sinh (c+d x) \, dx+(2 a b) \int \sinh ^4(c+d x) \, dx+b^2 \int \sinh ^7(c+d x) \, dx\\ &=\frac{a^2 \cosh (c+d x)}{d}+\frac{a b \cosh (c+d x) \sinh ^3(c+d x)}{2 d}-\frac{1}{2} (3 a b) \int \sinh ^2(c+d x) \, dx-\frac{b^2 \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{a^2 \cosh (c+d x)}{d}-\frac{b^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d}+\frac{b^2 \cosh ^7(c+d x)}{7 d}-\frac{3 a b \cosh (c+d x) \sinh (c+d x)}{4 d}+\frac{a b \cosh (c+d x) \sinh ^3(c+d x)}{2 d}+\frac{1}{4} (3 a b) \int 1 \, dx\\ &=\frac{3 a b x}{4}+\frac{a^2 \cosh (c+d x)}{d}-\frac{b^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d}+\frac{b^2 \cosh ^7(c+d x)}{7 d}-\frac{3 a b \cosh (c+d x) \sinh (c+d x)}{4 d}+\frac{a b \cosh (c+d x) \sinh ^3(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.40963, size = 92, normalized size = 0.71 \[ \frac{35 \left (64 a^2-35 b^2\right ) \cosh (c+d x)+b (140 a (12 (c+d x)-8 \sinh (2 (c+d x))+\sinh (4 (c+d x)))+245 b \cosh (3 (c+d x))-49 b \cosh (5 (c+d x))+5 b \cosh (7 (c+d x)))}{2240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 96, normalized size = 0.7 \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 ( \left ( 1/4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}-3/8\,\sinh \left ( dx+c \right ) \right ) \cosh \left ( dx+c \right ) +3/8\,dx+3/8\,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 [A] time = 1.07251, size = 243, normalized size = 1.87 \begin{align*} \frac{1}{32} \, a b{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \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{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 [A] time = 1.90338, size = 594, normalized size = 4.57 \begin{align*} \frac{5 \, b^{2} \cosh \left (d x + c\right )^{7} + 35 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 49 \, b^{2} \cosh \left (d x + c\right )^{5} + 560 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 245 \, b^{2} \cosh \left (d x + c\right )^{3} + 35 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} - 7 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 1680 \, a b d x + 35 \,{\left (3 \, b^{2} \cosh \left (d x + c\right )^{5} - 14 \, b^{2} \cosh \left (d x + c\right )^{3} + 21 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \,{\left (64 \, a^{2} - 35 \, b^{2}\right )} \cosh \left (d x + c\right ) + 560 \,{\left (a b \cosh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.4291, size = 219, normalized size = 1.68 \begin{align*} \begin{cases} \frac{a^{2} \cosh{\left (c + d x \right )}}{d} + \frac{3 a b x \sinh ^{4}{\left (c + d x \right )}}{4} - \frac{3 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{3 a b x \cosh ^{4}{\left (c + d x \right )}}{4} + \frac{5 a b \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{4 d} - \frac{3 a b \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{4 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 ^{3}{\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 [A] time = 1.20251, size = 266, normalized size = 2.05 \begin{align*} \frac{3360 \,{\left (d x + c\right )} a b + 5 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 49 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 140 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 245 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 1120 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 2240 \, a^{2} e^{\left (d x + c\right )} - 1225 \, b^{2} e^{\left (d x + c\right )} +{\left (1120 \, a b e^{\left (5 \, d x + 5 \, c\right )} + 245 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 140 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 49 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{2} + 35 \,{\left (64 \, a^{2} - 35 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{4480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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