Optimal. Leaf size=204 \[ a^3 x+\frac {a^2 b \cosh ^3(c+d x)}{d}-\frac {3 a^2 b \cosh (c+d x)}{d}+\frac {a b^2 \sinh ^5(c+d x) \cosh (c+d x)}{2 d}-\frac {5 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{8 d}+\frac {15 a b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {15}{16} a b^2 x+\frac {b^3 \cosh ^9(c+d x)}{9 d}-\frac {4 b^3 \cosh ^7(c+d x)}{7 d}+\frac {6 b^3 \cosh ^5(c+d x)}{5 d}-\frac {4 b^3 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh (c+d x)}{d} \]
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Rubi [A] time = 0.13, antiderivative size = 204, normalized size of antiderivative = 1.00, 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 8
Rule 2633
Rule 2635
Rule 3213
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*}
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Mathematica [A] time = 0.26, size = 159, normalized size = 0.78 \[ \frac {80640 a^3 c+80640 a^3 d x+1260 \left (16 a^2 b-7 b^3\right ) \cosh (3 (c+d x))+5670 b \left (7 b^2-32 a^2\right ) \cosh (c+d x)+56700 a b^2 \sinh (2 (c+d x))-11340 a b^2 \sinh (4 (c+d x))+1260 a b^2 \sinh (6 (c+d x))-75600 a b^2 c-75600 a b^2 d x+2268 b^3 \cosh (5 (c+d x))-405 b^3 \cosh (7 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 380, normalized size = 1.86 \[ \frac {35 \, b^{3} \cosh \left (d x + c\right )^{9} + 315 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} - 405 \, b^{3} \cosh \left (d x + c\right )^{7} + 7560 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2268 \, b^{3} \cosh \left (d x + c\right )^{5} + 105 \, {\left (28 \, b^{3} \cosh \left (d x + c\right )^{3} - 27 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 315 \, {\left (14 \, b^{3} \cosh \left (d x + c\right )^{5} - 45 \, b^{3} \cosh \left (d x + c\right )^{3} + 36 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 1260 \, {\left (16 \, a^{2} b - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 5040 \, {\left (5 \, a b^{2} \cosh \left (d x + c\right )^{3} - 9 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 5040 \, {\left (16 \, a^{3} - 15 \, a b^{2}\right )} d x + 315 \, {\left (4 \, b^{3} \cosh \left (d x + c\right )^{7} - 27 \, b^{3} \cosh \left (d x + c\right )^{5} + 72 \, b^{3} \cosh \left (d x + c\right )^{3} + 12 \, {\left (16 \, a^{2} b - 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 5670 \, {\left (32 \, a^{2} b - 7 \, b^{3}\right )} \cosh \left (d x + c\right ) + 7560 \, {\left (a b^{2} \cosh \left (d x + c\right )^{5} - 6 \, a b^{2} \cosh \left (d x + c\right )^{3} + 15 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 327, normalized size = 1.60 \[ \frac {b^{3} e^{\left (9 \, d x + 9 \, c\right )}}{4608 \, d} - \frac {9 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )}}{3584 \, d} + \frac {a b^{2} e^{\left (6 \, d x + 6 \, c\right )}}{128 \, d} + \frac {9 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} - \frac {9 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {45 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {45 \, a b^{2} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} + \frac {9 \, a b^{2} e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} + \frac {9 \, b^{3} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} - \frac {a b^{2} e^{\left (-6 \, d x - 6 \, c\right )}}{128 \, d} - \frac {9 \, b^{3} e^{\left (-7 \, d x - 7 \, c\right )}}{3584 \, d} + \frac {b^{3} e^{\left (-9 \, d x - 9 \, c\right )}}{4608 \, d} + \frac {1}{16} \, {\left (16 \, a^{3} - 15 \, a b^{2}\right )} x + \frac {{\left (16 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{128 \, d} - \frac {9 \, {\left (32 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (d x + c\right )}}{256 \, d} - \frac {9 \, {\left (32 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (-d x - c\right )}}{256 \, d} + \frac {{\left (16 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 141, normalized size = 0.69 \[ \frac {b^{3} \left (\frac {128}{315}+\frac {\left (\sinh ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sinh ^{6}\left (d x +c \right )\right )}{63}+\frac {16 \left (\sinh ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sinh ^{2}\left (d x +c \right )\right )}{315}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+3 a^{2} b \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )+a^{3} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 280, normalized size = 1.37 \[ a^{3} x - \frac {1}{161280} \, b^{3} {\left (\frac {{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} - \frac {1}{128} \, a b^{2} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} + \frac {1}{8} \, a^{2} b {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 164, normalized size = 0.80 \[ \frac {d\,x\,a^3+a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3-3\,a^2\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {\mathrm {sinh}\left (c+d\,x\right )\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{2}-\frac {13\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{8}+\frac {33\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b^2\,\mathrm {cosh}\left (c+d\,x\right )}{16}-\frac {15\,d\,x\,a\,b^2}{16}+\frac {b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-\frac {4\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {6\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {4\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^3\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.92, size = 340, normalized size = 1.67 \[ \begin {cases} a^{3} x + \frac {3 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a^{2} b \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {15 a b^{2} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {45 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {45 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {15 a b^{2} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {33 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{2 d} + \frac {15 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} + \frac {b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {64 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 b^{3} \cosh ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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