Optimal. Leaf size=192 \[ -\frac {5}{8} a b x-\frac {a^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}+\frac {5 a b \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac {5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d} \]
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Rubi [A]
time = 0.13, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3299, 2713,
2715, 8} \begin {gather*} \frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {a^2 \cosh (c+d x)}{d}+\frac {a b \sinh ^5(c+d x) \cosh (c+d x)}{3 d}-\frac {5 a b \sinh ^3(c+d x) \cosh (c+d x)}{12 d}+\frac {5 a b \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac {5 a b x}{8}+\frac {b^2 \cosh ^9(c+d x)}{9 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^2 \cosh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 3299
Rubi steps
\begin {align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=i \int \left (-i a^2 \sinh ^3(c+d x)-2 i a b \sinh ^6(c+d x)-i b^2 \sinh ^9(c+d x)\right ) \, dx\\ &=a^2 \int \sinh ^3(c+d x) \, dx+(2 a b) \int \sinh ^6(c+d x) \, dx+b^2 \int \sinh ^9(c+d x) \, dx\\ &=\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}-\frac {1}{3} (5 a b) \int \sinh ^4(c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac {b^2 \text {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}\\ &=-\frac {a^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}-\frac {5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}+\frac {1}{4} (5 a b) \int \sinh ^2(c+d x) \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}+\frac {5 a b \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac {5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}-\frac {1}{8} (5 a b) \int 1 \, dx\\ &=-\frac {5}{8} a b x-\frac {a^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}+\frac {5 a b \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac {5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 125, normalized size = 0.65 \begin {gather*} \frac {-1890 \left (32 a^2-21 b^2\right ) \cosh (c+d x)+420 \left (16 a^2-21 b^2\right ) \cosh (3 (c+d x))+b (2268 b \cosh (5 (c+d x))-405 b \cosh (7 (c+d x))+35 b \cosh (9 (c+d x))-840 a (60 c+60 d x-45 \sinh (2 (c+d x))+9 \sinh (4 (c+d x))-\sinh (6 (c+d x))))}{80640 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.42, size = 152, normalized size = 0.79
method | result | size |
default | \(\frac {\left (-\frac {21 b^{2}}{64}+\frac {a^{2}}{4}\right ) \cosh \left (3 d x +3 c \right )}{3 d}+\frac {\left (\frac {63 b^{2}}{128}-\frac {3 a^{2}}{4}\right ) \cosh \left (d x +c \right )}{d}-\frac {5 a b x}{8}+\frac {9 b^{2} \cosh \left (5 d x +5 c \right )}{320 d}-\frac {9 b^{2} \cosh \left (7 d x +7 c \right )}{1792 d}+\frac {b^{2} \cosh \left (9 d x +9 c \right )}{2304 d}+\frac {15 a b \sinh \left (2 d x +2 c \right )}{32 d}-\frac {3 a b \sinh \left (4 d x +4 c \right )}{32 d}+\frac {a b \sinh \left (6 d x +6 c \right )}{96 d}\) | \(152\) |
risch | \(-\frac {5 a b x}{8}+\frac {b^{2} {\mathrm e}^{9 d x +9 c}}{4608 d}-\frac {9 b^{2} {\mathrm e}^{7 d x +7 c}}{3584 d}+\frac {a b \,{\mathrm e}^{6 d x +6 c}}{192 d}+\frac {9 b^{2} {\mathrm e}^{5 d x +5 c}}{640 d}-\frac {3 \,{\mathrm e}^{4 d x +4 c} a b}{64 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2}}{24 d}-\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{2}}{128 d}+\frac {15 \,{\mathrm e}^{2 d x +2 c} a b}{64 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{2}}{8 d}+\frac {63 \,{\mathrm e}^{d x +c} b^{2}}{256 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2}}{8 d}+\frac {63 \,{\mathrm e}^{-d x -c} b^{2}}{256 d}-\frac {15 \,{\mathrm e}^{-2 d x -2 c} a b}{64 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2}}{24 d}-\frac {7 \,{\mathrm e}^{-3 d x -3 c} b^{2}}{128 d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} a b}{64 d}+\frac {9 b^{2} {\mathrm e}^{-5 d x -5 c}}{640 d}-\frac {a b \,{\mathrm e}^{-6 d x -6 c}}{192 d}-\frac {9 b^{2} {\mathrm e}^{-7 d x -7 c}}{3584 d}+\frac {b^{2} {\mathrm e}^{-9 d x -9 c}}{4608 d}\) | \(335\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 272, normalized size = 1.42 \begin {gather*} -\frac {1}{161280} \, b^{2} {\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}{192} \, a b {\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}{24} \, a^{2} {\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. 355 vs.
\(2 (174) = 348\).
time = 0.44, size = 355, normalized size = 1.85 \begin {gather*} \frac {35 \, b^{2} \cosh \left (d x + c\right )^{9} + 315 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} - 405 \, b^{2} \cosh \left (d x + c\right )^{7} + 5040 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2268 \, b^{2} \cosh \left (d x + c\right )^{5} + 105 \, {\left (28 \, b^{2} \cosh \left (d x + c\right )^{3} - 27 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 315 \, {\left (14 \, b^{2} \cosh \left (d x + c\right )^{5} - 45 \, b^{2} \cosh \left (d x + c\right )^{3} + 36 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 50400 \, a b d x + 420 \, {\left (16 \, a^{2} - 21 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3360 \, {\left (5 \, a b \cosh \left (d x + c\right )^{3} - 9 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 315 \, {\left (4 \, b^{2} \cosh \left (d x + c\right )^{7} - 27 \, b^{2} \cosh \left (d x + c\right )^{5} + 72 \, b^{2} \cosh \left (d x + c\right )^{3} + 4 \, {\left (16 \, a^{2} - 21 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 1890 \, {\left (32 \, a^{2} - 21 \, b^{2}\right )} \cosh \left (d x + c\right ) + 5040 \, {\left (a b \cosh \left (d x + c\right )^{5} - 6 \, a b \cosh \left (d x + c\right )^{3} + 15 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{80640 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.40, size = 325, normalized size = 1.69 \begin {gather*} \begin {cases} \frac {a^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a^{2} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 a b x \sinh ^{6}{\left (c + d x \right )}}{8} - \frac {15 a b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {15 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {5 a b x \cosh ^{6}{\left (c + d x \right )}}{8} + \frac {11 a b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {5 a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 a b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{8 d} + \frac {b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {64 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 b^{2} \cosh ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{2} \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 [A]
time = 0.45, size = 301, normalized size = 1.57 \begin {gather*} -\frac {5}{8} \, a b x + \frac {b^{2} e^{\left (9 \, d x + 9 \, c\right )}}{4608 \, d} - \frac {9 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{3584 \, d} + \frac {a b e^{\left (6 \, d x + 6 \, c\right )}}{192 \, d} + \frac {9 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} - \frac {3 \, a b e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} + \frac {15 \, a b e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d} - \frac {15 \, a b e^{\left (-2 \, d x - 2 \, c\right )}}{64 \, d} + \frac {3 \, a b e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} + \frac {9 \, b^{2} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} - \frac {a b e^{\left (-6 \, d x - 6 \, c\right )}}{192 \, d} - \frac {9 \, b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{3584 \, d} + \frac {b^{2} e^{\left (-9 \, d x - 9 \, c\right )}}{4608 \, d} + \frac {{\left (16 \, a^{2} - 21 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} - \frac {3 \, {\left (32 \, a^{2} - 21 \, b^{2}\right )} e^{\left (d x + c\right )}}{256 \, d} - \frac {3 \, {\left (32 \, a^{2} - 21 \, b^{2}\right )} e^{\left (-d x - c\right )}}{256 \, d} + \frac {{\left (16 \, a^{2} - 21 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.93, size = 149, normalized size = 0.78 \begin {gather*} \frac {\frac {a^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}-a^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {\mathrm {sinh}\left (c+d\,x\right )\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{3}-\frac {13\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{12}+\frac {11\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )}{8}-\frac {5\,d\,x\,a\,b}{8}+\frac {b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-\frac {4\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {6\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {4\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^2\,\mathrm {cosh}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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