Optimal. Leaf size=220 \[ \frac {(a+b)^3 \cosh (c+d x)}{d}-\frac {2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac {(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}-\frac {4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac {b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac {2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}+\frac {b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac {8 b^3 \cosh ^{15}(c+d x)}{15 d}+\frac {b^3 \cosh ^{17}(c+d x)}{17 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 220, 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 = {3294, 1167}
\begin {gather*} \frac {b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac {4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac {(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac {2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}-\frac {2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^{17}(c+d x)}{17 d}-\frac {8 b^3 \cosh ^{15}(c+d x)}{15 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1167
Rule 3294
Rubi steps
\begin {align*} \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (1-x^2\right )^2 \left (a+b-2 b x^2+b x^4\right )^3 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left ((a+b)^3-2 (a+b)^2 (a+4 b) x^2+(a+b) \left (a^2+17 a b+28 b^2\right ) x^4-4 b \left (3 a^2+15 a b+14 b^2\right ) x^6+b \left (3 a^2+45 a b+70 b^2\right ) x^8-2 b^2 (9 a+28 b) x^{10}+b^2 (3 a+28 b) x^{12}-8 b^3 x^{14}+b^3 x^{16}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {(a+b)^3 \cosh (c+d x)}{d}-\frac {2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac {(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}-\frac {4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac {b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac {2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}+\frac {b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac {8 b^3 \cosh ^{15}(c+d x)}{15 d}+\frac {b^3 \cosh ^{17}(c+d x)}{17 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.61, size = 288, normalized size = 1.31 \begin {gather*} \frac {1531530 \left (20480 a^3+48384 a^2 b+41184 a b^2+12155 b^3\right ) \cosh (c+d x)-2042040 \left (2560 a^3+8064 a^2 b+7722 a b^2+2431 b^3\right ) \cosh (3 (c+d x))+627314688 a^3 \cosh (5 (c+d x))+4234374144 a^2 b \cosh (5 (c+d x))+5256210960 a b^2 \cosh (5 (c+d x))+1895421528 b^3 \cosh (5 (c+d x))-756138240 a^2 b \cosh (7 (c+d x))-1501774560 a b^2 \cosh (7 (c+d x))-676936260 b^3 \cosh (7 (c+d x))+65345280 a^2 b \cosh (9 (c+d x))+318558240 a b^2 \cosh (9 (c+d x))+202502300 b^3 \cosh (9 (c+d x))-43439760 a b^2 \cosh (11 (c+d x))-47338200 b^3 \cosh (11 (c+d x))+2827440 a b^2 \cosh (13 (c+d x))+8011080 b^3 \cosh (13 (c+d x))-867867 b^3 \cosh (15 (c+d x))+45045 b^3 \cosh (17 (c+d x))}{50185175040 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.35, size = 259, normalized size = 1.18
method | result | size |
default | \(\frac {\left (-\frac {85}{8192} b^{3}-\frac {39}{4096} a \,b^{2}\right ) \cosh \left (11 d x +11 c \right )}{11 d}+\frac {\left (\frac {17}{8192} b^{3}+\frac {3}{4096} a \,b^{2}\right ) \cosh \left (13 d x +13 c \right )}{13 d}+\frac {\left (-\frac {1547}{16384} b^{3}-\frac {429}{2048} a \,b^{2}-\frac {27}{256} a^{2} b \right ) \cosh \left (7 d x +7 c \right )}{7 d}+\frac {\left (\frac {595}{16384} b^{3}+\frac {117}{2048} a \,b^{2}+\frac {3}{256} a^{2} b \right ) \cosh \left (9 d x +9 c \right )}{9 d}+\frac {\left (-\frac {2431}{8192} b^{3}-\frac {3861}{4096} a \,b^{2}-\frac {63}{64} a^{2} b -\frac {5}{16} a^{3}\right ) \cosh \left (3 d x +3 c \right )}{3 d}+\frac {\left (\frac {1547}{8192} b^{3}+\frac {2145}{4096} a \,b^{2}+\frac {27}{64} a^{2} b +\frac {1}{16} a^{3}\right ) \cosh \left (5 d x +5 c \right )}{5 d}+\frac {\left (\frac {12155}{32768} b^{3}+\frac {1287}{1024} a \,b^{2}+\frac {189}{128} a^{2} b +\frac {5}{8} a^{3}\right ) \cosh \left (d x +c \right )}{d}-\frac {17 b^{3} \cosh \left (15 d x +15 c \right )}{983040 d}+\frac {b^{3} \cosh \left (17 d x +17 c \right )}{1114112 d}\) | \(259\) |
risch | \(\frac {429 a \,b^{2} {\mathrm e}^{5 d x +5 c}}{8192 d}-\frac {1287 a \,b^{2} {\mathrm e}^{3 d x +3 c}}{8192 d}+\frac {b^{3} {\mathrm e}^{-17 d x -17 c}}{2228224 d}+\frac {189 b \,{\mathrm e}^{d x +c} a^{2}}{256 d}+\frac {595 b^{3} {\mathrm e}^{9 d x +9 c}}{294912 d}+\frac {{\mathrm e}^{5 d x +5 c} a^{3}}{160 d}+\frac {1547 \,{\mathrm e}^{5 d x +5 c} b^{3}}{81920 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} a^{3}}{96 d}-\frac {2431 \,{\mathrm e}^{3 d x +3 c} b^{3}}{49152 d}+\frac {17 b^{3} {\mathrm e}^{13 d x +13 c}}{212992 d}-\frac {85 b^{3} {\mathrm e}^{11 d x +11 c}}{180224 d}+\frac {5 a^{3} {\mathrm e}^{d x +c}}{16 d}+\frac {595 b^{3} {\mathrm e}^{-9 d x -9 c}}{294912 d}-\frac {85 b^{3} {\mathrm e}^{-11 d x -11 c}}{180224 d}+\frac {17 b^{3} {\mathrm e}^{-13 d x -13 c}}{212992 d}+\frac {1287 a \,{\mathrm e}^{-d x -c} b^{2}}{2048 d}-\frac {1287 a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{8192 d}+\frac {b^{3} {\mathrm e}^{17 d x +17 c}}{2228224 d}-\frac {17 b^{3} {\mathrm e}^{15 d x +15 c}}{1966080 d}+\frac {12155 b^{3} {\mathrm e}^{d x +c}}{65536 d}+\frac {12155 b^{3} {\mathrm e}^{-d x -c}}{65536 d}-\frac {2431 b^{3} {\mathrm e}^{-3 d x -3 c}}{49152 d}-\frac {221 b^{3} {\mathrm e}^{7 d x +7 c}}{32768 d}+\frac {429 a \,b^{2} {\mathrm e}^{-5 d x -5 c}}{8192 d}-\frac {17 b^{3} {\mathrm e}^{-15 d x -15 c}}{1966080 d}+\frac {5 \,{\mathrm e}^{-d x -c} a^{3}}{16 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} a^{3}}{96 d}+\frac {{\mathrm e}^{-5 d x -5 c} a^{3}}{160 d}+\frac {1547 \,{\mathrm e}^{-5 d x -5 c} b^{3}}{81920 d}-\frac {221 b^{3} {\mathrm e}^{-7 d x -7 c}}{32768 d}+\frac {1287 a \,{\mathrm e}^{d x +c} b^{2}}{2048 d}+\frac {189 \,{\mathrm e}^{-d x -c} a^{2} b}{256 d}-\frac {21 \,{\mathrm e}^{-3 d x -3 c} a^{2} b}{128 d}+\frac {27 \,{\mathrm e}^{-5 d x -5 c} a^{2} b}{640 d}+\frac {3 b^{2} {\mathrm e}^{13 d x +13 c} a}{106496 d}-\frac {39 b^{2} {\mathrm e}^{11 d x +11 c} a}{90112 d}+\frac {b \,{\mathrm e}^{9 d x +9 c} a^{2}}{1536 d}+\frac {13 b^{2} {\mathrm e}^{9 d x +9 c} a}{4096 d}-\frac {27 b \,{\mathrm e}^{7 d x +7 c} a^{2}}{3584 d}-\frac {429 b^{2} {\mathrm e}^{7 d x +7 c} a}{28672 d}+\frac {27 \,{\mathrm e}^{5 d x +5 c} a^{2} b}{640 d}-\frac {27 b \,{\mathrm e}^{-7 d x -7 c} a^{2}}{3584 d}-\frac {429 b^{2} {\mathrm e}^{-7 d x -7 c} a}{28672 d}+\frac {b \,{\mathrm e}^{-9 d x -9 c} a^{2}}{1536 d}+\frac {13 b^{2} {\mathrm e}^{-9 d x -9 c} a}{4096 d}-\frac {39 b^{2} {\mathrm e}^{-11 d x -11 c} a}{90112 d}+\frac {3 b^{2} {\mathrm e}^{-13 d x -13 c} a}{106496 d}-\frac {21 \,{\mathrm e}^{3 d x +3 c} a^{2} b}{128 d}\) | \(830\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 600 vs.
\(2 (204) = 408\).
time = 0.27, size = 600, normalized size = 2.73 \begin {gather*} -\frac {1}{14338621440} \, b^{3} {\left (\frac {{\left (123981 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1144440 \, e^{\left (-4 \, d x - 4 \, c\right )} + 6762600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 28928900 \, e^{\left (-8 \, d x - 8 \, c\right )} + 96705180 \, e^{\left (-10 \, d x - 10 \, c\right )} - 270774504 \, e^{\left (-12 \, d x - 12 \, c\right )} + 709171320 \, e^{\left (-14 \, d x - 14 \, c\right )} - 2659392450 \, e^{\left (-16 \, d x - 16 \, c\right )} - 6435\right )} e^{\left (17 \, d x + 17 \, c\right )}}{d} - \frac {2659392450 \, e^{\left (-d x - c\right )} - 709171320 \, e^{\left (-3 \, d x - 3 \, c\right )} + 270774504 \, e^{\left (-5 \, d x - 5 \, c\right )} - 96705180 \, e^{\left (-7 \, d x - 7 \, c\right )} + 28928900 \, e^{\left (-9 \, d x - 9 \, c\right )} - 6762600 \, e^{\left (-11 \, d x - 11 \, c\right )} + 1144440 \, e^{\left (-13 \, d x - 13 \, c\right )} - 123981 \, e^{\left (-15 \, d x - 15 \, c\right )} + 6435 \, e^{\left (-17 \, d x - 17 \, c\right )}}{d}\right )} - \frac {1}{8200192} \, a b^{2} {\left (\frac {{\left (3549 \, e^{\left (-2 \, d x - 2 \, c\right )} - 26026 \, e^{\left (-4 \, d x - 4 \, c\right )} + 122694 \, e^{\left (-6 \, d x - 6 \, c\right )} - 429429 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1288287 \, e^{\left (-10 \, d x - 10 \, c\right )} - 5153148 \, e^{\left (-12 \, d x - 12 \, c\right )} - 231\right )} e^{\left (13 \, d x + 13 \, c\right )}}{d} - \frac {5153148 \, e^{\left (-d x - c\right )} - 1288287 \, e^{\left (-3 \, d x - 3 \, c\right )} + 429429 \, e^{\left (-5 \, d x - 5 \, c\right )} - 122694 \, e^{\left (-7 \, d x - 7 \, c\right )} + 26026 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3549 \, e^{\left (-11 \, d x - 11 \, c\right )} + 231 \, e^{\left (-13 \, d x - 13 \, c\right )}}{d}\right )} - \frac {1}{53760} \, a^{2} b {\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}{480} \, a^{3} {\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1030 vs.
\(2 (204) = 408\).
time = 0.37, size = 1030, normalized size = 4.68 \begin {gather*} \frac {45045 \, b^{3} \cosh \left (d x + c\right )^{17} + 765765 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{16} - 867867 \, b^{3} \cosh \left (d x + c\right )^{15} + 765765 \, {\left (40 \, b^{3} \cosh \left (d x + c\right )^{3} - 17 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{14} + 471240 \, {\left (6 \, a b^{2} + 17 \, b^{3}\right )} \cosh \left (d x + c\right )^{13} + 255255 \, {\left (1092 \, b^{3} \cosh \left (d x + c\right )^{5} - 1547 \, b^{3} \cosh \left (d x + c\right )^{3} + 24 \, {\left (6 \, a b^{2} + 17 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{12} - 556920 \, {\left (78 \, a b^{2} + 85 \, b^{3}\right )} \cosh \left (d x + c\right )^{11} + 153153 \, {\left (5720 \, b^{3} \cosh \left (d x + c\right )^{7} - 17017 \, b^{3} \cosh \left (d x + c\right )^{5} + 880 \, {\left (6 \, a b^{2} + 17 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 40 \, {\left (78 \, a b^{2} + 85 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{10} + 340340 \, {\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} + 765765 \, {\left (1430 \, b^{3} \cosh \left (d x + c\right )^{9} - 7293 \, b^{3} \cosh \left (d x + c\right )^{7} + 792 \, {\left (6 \, a b^{2} + 17 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 120 \, {\left (78 \, a b^{2} + 85 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \, {\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} - 437580 \, {\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 255255 \, {\left (2184 \, b^{3} \cosh \left (d x + c\right )^{11} - 17017 \, b^{3} \cosh \left (d x + c\right )^{9} + 3168 \, {\left (6 \, a b^{2} + 17 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 1008 \, {\left (78 \, a b^{2} + 85 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 112 \, {\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 12 \, {\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 1225224 \, {\left (512 \, a^{3} + 3456 \, a^{2} b + 4290 \, a b^{2} + 1547 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 765765 \, {\left (140 \, b^{3} \cosh \left (d x + c\right )^{13} - 1547 \, b^{3} \cosh \left (d x + c\right )^{11} + 440 \, {\left (6 \, a b^{2} + 17 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} - 240 \, {\left (78 \, a b^{2} + 85 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 56 \, {\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 20 \, {\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 8 \, {\left (512 \, a^{3} + 3456 \, a^{2} b + 4290 \, a b^{2} + 1547 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 2042040 \, {\left (2560 \, a^{3} + 8064 \, a^{2} b + 7722 \, a b^{2} + 2431 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 765765 \, {\left (8 \, b^{3} \cosh \left (d x + c\right )^{15} - 119 \, b^{3} \cosh \left (d x + c\right )^{13} + 48 \, {\left (6 \, a b^{2} + 17 \, b^{3}\right )} \cosh \left (d x + c\right )^{11} - 40 \, {\left (78 \, a b^{2} + 85 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} + 16 \, {\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 12 \, {\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 16 \, {\left (512 \, a^{3} + 3456 \, a^{2} b + 4290 \, a b^{2} + 1547 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 8 \, {\left (2560 \, a^{3} + 8064 \, a^{2} b + 7722 \, a b^{2} + 2431 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 1531530 \, {\left (20480 \, a^{3} + 48384 \, a^{2} b + 41184 \, a b^{2} + 12155 \, b^{3}\right )} \cosh \left (d x + c\right )}{50185175040 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 592 vs.
\(2 (204) = 408\).
time = 17.34, size = 592, normalized size = 2.69 \begin {gather*} \begin {cases} \frac {a^{3} \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {8 a^{3} \cosh ^{5}{\left (c + d x \right )}}{15 d} + \frac {3 a^{2} b \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a^{2} b \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {48 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {192 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 a^{2} b \cosh ^{9}{\left (c + d x \right )}}{105 d} + \frac {3 a b^{2} \sinh ^{12}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {12 a b^{2} \sinh ^{10}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {24 a b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{d} - \frac {192 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac {128 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{7 d} - \frac {512 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{77 d} + \frac {1024 a b^{2} \cosh ^{13}{\left (c + d x \right )}}{1001 d} + \frac {b^{3} \sinh ^{16}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {16 b^{3} \sinh ^{14}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {224 b^{3} \sinh ^{12}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{15 d} - \frac {128 b^{3} \sinh ^{10}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{5 d} + \frac {256 b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{9 d} - \frac {2048 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{99 d} + \frac {4096 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{13}{\left (c + d x \right )}}{429 d} - \frac {16384 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{15}{\left (c + d x \right )}}{6435 d} + \frac {32768 b^{3} \cosh ^{17}{\left (c + d x \right )}}{109395 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{3} \sinh ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 520 vs.
\(2 (204) = 408\).
time = 0.54, size = 520, normalized size = 2.36 \begin {gather*} \frac {b^{3} e^{\left (17 \, d x + 17 \, c\right )}}{2228224 \, d} - \frac {17 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )}}{1966080 \, d} - \frac {17 \, b^{3} e^{\left (-15 \, d x - 15 \, c\right )}}{1966080 \, d} + \frac {b^{3} e^{\left (-17 \, d x - 17 \, c\right )}}{2228224 \, d} + \frac {{\left (6 \, a b^{2} + 17 \, b^{3}\right )} e^{\left (13 \, d x + 13 \, c\right )}}{212992 \, d} - \frac {{\left (78 \, a b^{2} + 85 \, b^{3}\right )} e^{\left (11 \, d x + 11 \, c\right )}}{180224 \, d} + \frac {{\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} e^{\left (9 \, d x + 9 \, c\right )}}{294912 \, d} - \frac {{\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{229376 \, d} + \frac {{\left (512 \, a^{3} + 3456 \, a^{2} b + 4290 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{81920 \, d} - \frac {{\left (2560 \, a^{3} + 8064 \, a^{2} b + 7722 \, a b^{2} + 2431 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{49152 \, d} + \frac {{\left (20480 \, a^{3} + 48384 \, a^{2} b + 41184 \, a b^{2} + 12155 \, b^{3}\right )} e^{\left (d x + c\right )}}{65536 \, d} + \frac {{\left (20480 \, a^{3} + 48384 \, a^{2} b + 41184 \, a b^{2} + 12155 \, b^{3}\right )} e^{\left (-d x - c\right )}}{65536 \, d} - \frac {{\left (2560 \, a^{3} + 8064 \, a^{2} b + 7722 \, a b^{2} + 2431 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{49152 \, d} + \frac {{\left (512 \, a^{3} + 3456 \, a^{2} b + 4290 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{81920 \, d} - \frac {{\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{229376 \, d} + \frac {{\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} e^{\left (-9 \, d x - 9 \, c\right )}}{294912 \, d} - \frac {{\left (78 \, a b^{2} + 85 \, b^{3}\right )} e^{\left (-11 \, d x - 11 \, c\right )}}{180224 \, d} + \frac {{\left (6 \, a b^{2} + 17 \, b^{3}\right )} e^{\left (-13 \, d x - 13 \, c\right )}}{212992 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.76, size = 319, normalized size = 1.45 \begin {gather*} \frac {\frac {a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {2\,a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+a^3\,\mathrm {cosh}\left (c+d\,x\right )+\frac {a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{3}-\frac {12\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {18\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-4\,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 )}^{13}}{13}-\frac {18\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+5\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9-\frac {60\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+9\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5-6\,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 )}^{17}}{17}-\frac {8\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{15}}{15}+\frac {28\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{13}}{13}-\frac {56\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+\frac {70\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-8\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^7+\frac {28\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {8\,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.
[In]
[Out]
________________________________________________________________________________________