Optimal. Leaf size=115 \[ \frac{b^2 (3 a-4 b) \cosh ^7(c+d x)}{7 d}+\frac{3 b (a-2 b) (a-b) \cosh ^5(c+d x)}{5 d}+\frac{(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}-\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^9(c+d x)}{9 d} \]
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Rubi [A] time = 0.129173, antiderivative size = 115, 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} \[ \frac{b^2 (3 a-4 b) \cosh ^7(c+d x)}{7 d}+\frac{3 b (a-2 b) (a-b) \cosh ^5(c+d x)}{5 d}+\frac{(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}-\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^9(c+d x)}{9 d} \]
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 )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a-b+b x^2\right )^3 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left ((a-b)^3-(a-4 b) (a-b)^2 x^2+3 (a-2 b) b (-a+b) x^4-(3 a-4 b) b^2 x^6-b^3 x^8\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}+\frac{3 (a-2 b) (a-b) b \cosh ^5(c+d x)}{5 d}+\frac{(3 a-4 b) b^2 \cosh ^7(c+d x)}{7 d}+\frac{b^3 \cosh ^9(c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 0.796978, size = 127, normalized size = 1.1 \[ \frac{-1890 (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right ) \cosh (c+d x)+420 \left (-60 a^2 b+16 a^3+63 a b^2-21 b^3\right ) \cosh (3 (c+d x))+135 b^2 (4 a-3 b) \cosh (7 (c+d x))+756 b (4 a-3 b) (a-b) \cosh (5 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 158, normalized size = 1.4 \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 ( -{\frac{16}{35}}+1/7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}-{\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 ) +3\,{a}^{2}b \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}^{3} \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 = 1.08331, size = 508, normalized size = 4.42 \begin{align*} -\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{3}{4480} \, a 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}{160} \, a^{2} 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^{3}{\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 = 1.82041, size = 940, normalized size = 8.17 \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} + 135 \,{\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 105 \,{\left (28 \, b^{3} \cosh \left (d x + c\right )^{3} + 9 \,{\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 756 \,{\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 315 \,{\left (14 \, b^{3} \cosh \left (d x + c\right )^{5} + 15 \,{\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 12 \,{\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 420 \,{\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 315 \,{\left (4 \, b^{3} \cosh \left (d x + c\right )^{7} + 9 \,{\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 24 \,{\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \,{\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 1890 \,{\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} \cosh \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 = 21.724, size = 330, normalized size = 2.87 \begin{align*} \begin{cases} \frac{a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 a^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{3 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{8 a^{2} b \cosh ^{5}{\left (c + d x \right )}}{5 d} + \frac{3 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{6 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{24 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{48 a b^{2} \cosh ^{7}{\left (c + d x \right )}}{35 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 ^{2}{\left (c \right )}\right )^{3} \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 = 1.41424, size = 544, normalized size = 4.73 \begin{align*} \frac{35 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 540 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 405 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 3024 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} - 5292 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 2268 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 6720 \, a^{3} e^{\left (3 \, d x + 3 \, c\right )} - 25200 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 26460 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 8820 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 60480 \, a^{3} e^{\left (d x + c\right )} + 151200 \, a^{2} b e^{\left (d x + c\right )} - 132300 \, a b^{2} e^{\left (d x + c\right )} + 39690 \, b^{3} e^{\left (d x + c\right )} -{\left (60480 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 151200 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 132300 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 39690 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 6720 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 25200 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 26460 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8820 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 3024 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 5292 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2268 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 540 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 405 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, b^{3}\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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