Optimal. Leaf size=115 \[ -\frac {(a-b)^3 \cosh (c+d x)}{d}+\frac {(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}+\frac {3 (a-2 b) (a-b) b \cosh ^5(c+d x)}{5 d}+\frac {(3 a-4 b) b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.00, 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 = {3265, 380}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 380
Rule 3265
Rubi steps
\begin {align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=-\frac {\text {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 {\text {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*}
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Mathematica [A]
time = 0.53, size = 127, normalized size = 1.10 \begin {gather*} \frac {-1890 (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right ) \cosh (c+d x)+420 \left (16 a^3-60 a^2 b+63 a b^2-21 b^3\right ) \cosh (3 (c+d x))+756 (4 a-3 b) (a-b) b \cosh (5 (c+d x))+135 (4 a-3 b) b^2 \cosh (7 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.80, size = 147, normalized size = 1.28
method | result | size |
default | \(\frac {\left (-\frac {9}{256} b^{3}+\frac {3}{64} a \,b^{2}\right ) \cosh \left (7 d x +7 c \right )}{7 d}+\frac {\left (\frac {9}{64} b^{3}-\frac {21}{64} a \,b^{2}+\frac {3}{16} a^{2} b \right ) \cosh \left (5 d x +5 c \right )}{5 d}+\frac {\left (-\frac {21}{64} b^{3}+\frac {63}{64} a \,b^{2}-\frac {15}{16} a^{2} b +\frac {1}{4} a^{3}\right ) \cosh \left (3 d x +3 c \right )}{3 d}+\frac {\left (\frac {63}{128} b^{3}-\frac {105}{64} a \,b^{2}+\frac {15}{8} a^{2} b -\frac {3}{4} a^{3}\right ) \cosh \left (d x +c \right )}{d}+\frac {b^{3} \cosh \left (9 d x +9 c \right )}{2304 d}\) | \(147\) |
risch | \(\frac {b^{3} {\mathrm e}^{9 d x +9 c}}{4608 d}+\frac {b^{3} {\mathrm e}^{-9 d x -9 c}}{4608 d}-\frac {9 b^{3} {\mathrm e}^{7 d x +7 c}}{3584 d}+\frac {9 b^{3} {\mathrm e}^{5 d x +5 c}}{640 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{3}}{24 d}-\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{3}}{128 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{3}}{8 d}+\frac {63 \,{\mathrm e}^{d x +c} b^{3}}{256 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{3}}{8 d}+\frac {63 \,{\mathrm e}^{-d x -c} b^{3}}{256 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{3}}{24 d}-\frac {7 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{128 d}+\frac {9 b^{3} {\mathrm e}^{-5 d x -5 c}}{640 d}-\frac {9 b^{3} {\mathrm e}^{-7 d x -7 c}}{3584 d}+\frac {15 \,{\mathrm e}^{-d x -c} a^{2} b}{16 d}-\frac {105 \,{\mathrm e}^{-d x -c} a \,b^{2}}{128 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} a^{2} b}{32 d}+\frac {15 \,{\mathrm e}^{d x +c} a^{2} b}{16 d}-\frac {105 \,{\mathrm e}^{d x +c} a \,b^{2}}{128 d}+\frac {21 \,{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{128 d}+\frac {3 b \,{\mathrm e}^{-5 d x -5 c} a^{2}}{160 d}-\frac {21 b^{2} {\mathrm e}^{-5 d x -5 c} a}{640 d}+\frac {3 b^{2} {\mathrm e}^{-7 d x -7 c} a}{896 d}+\frac {3 b^{2} {\mathrm e}^{7 d x +7 c} a}{896 d}+\frac {3 b \,{\mathrm e}^{5 d x +5 c} a^{2}}{160 d}-\frac {21 b^{2} {\mathrm e}^{5 d x +5 c} a}{640 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} a^{2} b}{32 d}+\frac {21 \,{\mathrm e}^{3 d x +3 c} a \,b^{2}}{128 d}\) | \(480\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 376 vs.
\(2 (107) = 214\).
time = 0.27, size = 376, normalized size = 3.27 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 373 vs.
\(2 (107) = 214\).
time = 0.41, size = 373, normalized size = 3.24 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 330 vs.
\(2 (100) = 200\).
time = 1.43, size = 330, normalized size = 2.87 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 296 vs.
\(2 (107) = 214\).
time = 0.45, size = 296, normalized size = 2.57 \begin {gather*} \frac {b^{3} e^{\left (9 \, d x + 9 \, c\right )}}{4608 \, d} + \frac {b^{3} e^{\left (-9 \, d x - 9 \, c\right )}}{4608 \, d} + \frac {3 \, {\left (4 \, a b^{2} - 3 \, b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{3584 \, d} + \frac {3 \, {\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} - \frac {3 \, {\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} e^{\left (d x + c\right )}}{256 \, d} - \frac {3 \, {\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} e^{\left (-d x - c\right )}}{256 \, d} + \frac {{\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} + \frac {3 \, {\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} + \frac {3 \, {\left (4 \, a b^{2} - 3 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{3584 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 185, normalized size = 1.61 \begin {gather*} \frac {\frac {a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}-a^3\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-2\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a^2\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}-\frac {9\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}+3\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3-3\,a\,b^2\,\mathrm {cosh}\left (c+d\,x\right )+\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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