Optimal. Leaf size=204 \[ \frac{a^2 b \cosh ^3(c+d x)}{d}-\frac{3 a^2 b \cosh (c+d x)}{d}+a^3 x+\frac{a b^2 \sinh ^5(c+d x) \cosh (c+d x)}{2 d}-\frac{5 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{8 d}+\frac{15 a b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{15}{16} a b^2 x+\frac{b^3 \cosh ^9(c+d x)}{9 d}-\frac{4 b^3 \cosh ^7(c+d x)}{7 d}+\frac{6 b^3 \cosh ^5(c+d x)}{5 d}-\frac{4 b^3 \cosh ^3(c+d x)}{3 d}+\frac{b^3 \cosh (c+d x)}{d} \]
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Rubi [A] time = 0.127754, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3213, 2633, 2635, 8} \[ \frac{a^2 b \cosh ^3(c+d x)}{d}-\frac{3 a^2 b \cosh (c+d x)}{d}+a^3 x+\frac{a b^2 \sinh ^5(c+d x) \cosh (c+d x)}{2 d}-\frac{5 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{8 d}+\frac{15 a b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{15}{16} a b^2 x+\frac{b^3 \cosh ^9(c+d x)}{9 d}-\frac{4 b^3 \cosh ^7(c+d x)}{7 d}+\frac{6 b^3 \cosh ^5(c+d x)}{5 d}-\frac{4 b^3 \cosh ^3(c+d x)}{3 d}+\frac{b^3 \cosh (c+d x)}{d} \]
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 )^3 \, dx &=\int \left (a^3+3 a^2 b \sinh ^3(c+d x)+3 a b^2 \sinh ^6(c+d x)+b^3 \sinh ^9(c+d x)\right ) \, dx\\ &=a^3 x+\left (3 a^2 b\right ) \int \sinh ^3(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^6(c+d x) \, dx+b^3 \int \sinh ^9(c+d x) \, dx\\ &=a^3 x+\frac{a b^2 \cosh (c+d x) \sinh ^5(c+d x)}{2 d}-\frac{1}{2} \left (5 a b^2\right ) \int \sinh ^4(c+d x) \, dx-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac{b^3 \operatorname{Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=a^3 x-\frac{3 a^2 b \cosh (c+d x)}{d}+\frac{b^3 \cosh (c+d x)}{d}+\frac{a^2 b \cosh ^3(c+d x)}{d}-\frac{4 b^3 \cosh ^3(c+d x)}{3 d}+\frac{6 b^3 \cosh ^5(c+d x)}{5 d}-\frac{4 b^3 \cosh ^7(c+d x)}{7 d}+\frac{b^3 \cosh ^9(c+d x)}{9 d}-\frac{5 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{8 d}+\frac{a b^2 \cosh (c+d x) \sinh ^5(c+d x)}{2 d}+\frac{1}{8} \left (15 a b^2\right ) \int \sinh ^2(c+d x) \, dx\\ &=a^3 x-\frac{3 a^2 b \cosh (c+d x)}{d}+\frac{b^3 \cosh (c+d x)}{d}+\frac{a^2 b \cosh ^3(c+d x)}{d}-\frac{4 b^3 \cosh ^3(c+d x)}{3 d}+\frac{6 b^3 \cosh ^5(c+d x)}{5 d}-\frac{4 b^3 \cosh ^7(c+d x)}{7 d}+\frac{b^3 \cosh ^9(c+d x)}{9 d}+\frac{15 a b^2 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac{5 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{8 d}+\frac{a b^2 \cosh (c+d x) \sinh ^5(c+d x)}{2 d}-\frac{1}{16} \left (15 a b^2\right ) \int 1 \, dx\\ &=a^3 x-\frac{15}{16} a b^2 x-\frac{3 a^2 b \cosh (c+d x)}{d}+\frac{b^3 \cosh (c+d x)}{d}+\frac{a^2 b \cosh ^3(c+d x)}{d}-\frac{4 b^3 \cosh ^3(c+d x)}{3 d}+\frac{6 b^3 \cosh ^5(c+d x)}{5 d}-\frac{4 b^3 \cosh ^7(c+d x)}{7 d}+\frac{b^3 \cosh ^9(c+d x)}{9 d}+\frac{15 a b^2 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac{5 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{8 d}+\frac{a b^2 \cosh (c+d x) \sinh ^5(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.251226, size = 159, normalized size = 0.78 \[ \frac{5670 b \left (7 b^2-32 a^2\right ) \cosh (c+d x)+1260 \left (16 a^2 b-7 b^3\right ) \cosh (3 (c+d x))+80640 a^3 c+80640 a^3 d x+56700 a b^2 \sinh (2 (c+d x))-11340 a b^2 \sinh (4 (c+d x))+1260 a b^2 \sinh (6 (c+d x))-75600 a b^2 c-75600 a b^2 d x+2268 b^3 \cosh (5 (c+d x))-405 b^3 \cosh (7 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 141, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ({\frac{128}{315}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{9}}-{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{63}}+{\frac{16\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{105}}-{\frac{64\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{315}} \right ) \cosh \left ( dx+c \right ) +3\,a{b}^{2} \left ( \left ( 1/6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}-{\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 ) +3\,{a}^{2}b \left ( -2/3+1/3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) \cosh \left ( dx+c \right ) +{a}^{3} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17724, size = 378, normalized size = 1.85 \begin{align*} a^{3} x - \frac{1}{161280} \, b^{3}{\left (\frac{{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac{39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} - \frac{1}{128} \, a 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}{8} \, a^{2} 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 [B] time = 1.81282, size = 1017, normalized size = 4.99 \begin{align*} \frac{35 \, b^{3} \cosh \left (d x + c\right )^{9} + 315 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} - 405 \, b^{3} \cosh \left (d x + c\right )^{7} + 7560 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2268 \, b^{3} \cosh \left (d x + c\right )^{5} + 105 \,{\left (28 \, b^{3} \cosh \left (d x + c\right )^{3} - 27 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 315 \,{\left (14 \, b^{3} \cosh \left (d x + c\right )^{5} - 45 \, b^{3} \cosh \left (d x + c\right )^{3} + 36 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 1260 \,{\left (16 \, a^{2} b - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 5040 \,{\left (5 \, a b^{2} \cosh \left (d x + c\right )^{3} - 9 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 5040 \,{\left (16 \, a^{3} - 15 \, a b^{2}\right )} d x + 315 \,{\left (4 \, b^{3} \cosh \left (d x + c\right )^{7} - 27 \, b^{3} \cosh \left (d x + c\right )^{5} + 72 \, b^{3} \cosh \left (d x + c\right )^{3} + 12 \,{\left (16 \, a^{2} b - 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 5670 \,{\left (32 \, a^{2} b - 7 \, b^{3}\right )} \cosh \left (d x + c\right ) + 7560 \,{\left (a b^{2} \cosh \left (d x + c\right )^{5} - 6 \, a b^{2} \cosh \left (d x + c\right )^{3} + 15 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{80640 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 33.4784, size = 340, normalized size = 1.67 \begin{align*} \begin{cases} a^{3} x + \frac{3 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 a^{2} b \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{15 a b^{2} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac{45 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac{45 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac{15 a b^{2} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac{33 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{16 d} - \frac{5 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{2 d} + \frac{15 a b^{2} \sinh{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} + \frac{b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{8 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{16 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{64 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac{128 b^{3} \cosh ^{9}{\left (c + d x \right )}}{315 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16286, size = 405, normalized size = 1.99 \begin{align*} \frac{35 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 405 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 1260 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 2268 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 11340 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 20160 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 8820 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 56700 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 181440 \, a^{2} b e^{\left (d x + c\right )} + 39690 \, b^{3} e^{\left (d x + c\right )} + 10080 \,{\left (16 \, a^{3} - 15 \, a b^{2}\right )}{\left (d x + c\right )} -{\left (56700 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 11340 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 2268 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1260 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 405 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, b^{3} + 5670 \,{\left (32 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )} - 1260 \,{\left (16 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-9 \, d x - 9 \, c\right )}}{161280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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