Optimal. Leaf size=255 \[ \frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \sinh (c+d x) \cosh ^5(c+d x)}{3840 d}-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{3072 d}+\frac {\left (1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{2048 d}-\frac {x \left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right )}{2048}+\frac {b^2 (504 a+2593 b) \sinh (c+d x) \cosh ^9(c+d x)}{1680 d}-\frac {b^2 (6888 a+11821 b) \sinh (c+d x) \cosh ^7(c+d x)}{4480 d}+\frac {b^3 \sinh (c+d x) \cosh ^{13}(c+d x)}{14 d}-\frac {85 b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{168 d} \]
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Rubi [A] time = 0.56, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3217, 1257, 1814, 1157, 385, 206} \[ \frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \sinh (c+d x) \cosh ^5(c+d x)}{3840 d}-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{3072 d}+\frac {\left (4224 a^2 b+1024 a^3+4632 a b^2+1619 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{2048 d}-\frac {x \left (1920 a^2 b+1024 a^3+1512 a b^2+429 b^3\right )}{2048}+\frac {b^2 (504 a+2593 b) \sinh (c+d x) \cosh ^9(c+d x)}{1680 d}-\frac {b^2 (6888 a+11821 b) \sinh (c+d x) \cosh ^7(c+d x)}{4480 d}+\frac {b^3 \sinh (c+d x) \cosh ^{13}(c+d x)}{14 d}-\frac {85 b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{168 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 1157
Rule 1257
Rule 1814
Rule 3217
Rubi steps
\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a-2 a x^2+(a+b) x^4\right )^3}{\left (1-x^2\right )^8} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}+\frac {\operatorname {Subst}\left (\int \frac {-b^3+14 \left (a^3-b^3\right ) x^2-14 \left (5 a^3+b^3\right ) x^4+14 \left (10 a^3+3 a^2 b-b^3\right ) x^6-14 \left (10 a^3+9 a^2 b+b^3\right ) x^8+14 (5 a-b) (a+b)^2 x^{10}-14 (a+b)^3 x^{12}}{\left (1-x^2\right )^7} \, dx,x,\tanh (c+d x)\right )}{14 d}\\ &=-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac {\operatorname {Subst}\left (\int \frac {-73 b^3-168 \left (a^3+5 b^3\right ) x^2+672 \left (a^3-b^3\right ) x^4-504 \left (2 a^3+a^2 b+b^3\right ) x^6+336 (2 a-b) (a+b)^2 x^8-168 (a+b)^3 x^{10}}{\left (1-x^2\right )^6} \, dx,x,\tanh (c+d x)\right )}{168 d}\\ &=\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}+\frac {\operatorname {Subst}\left (\int \frac {-9 b^2 (56 a+207 b)+1680 \left (a^3-3 a b^2-10 b^3\right ) x^2-5040 \left (a^3+a b^2+2 b^3\right ) x^4+5040 (a-b) (a+b)^2 x^6-1680 (a+b)^3 x^8}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{1680 d}\\ &=-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac {\operatorname {Subst}\left (\int \frac {-231 b^2 (72 a+89 b)-13440 \left (a^3+9 a b^2+10 b^3\right ) x^2+26880 (a-2 b) (a+b)^2 x^4-13440 (a+b)^3 x^6}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{13440 d}\\ &=\frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}+\frac {\operatorname {Subst}\left (\int \frac {-105 b \left (384 a^2+1512 a b+941 b^2\right )+80640 (a-5 b) (a+b)^2 x^2-80640 (a+b)^3 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{80640 d}\\ &=-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac {\operatorname {Subst}\left (\int \frac {-315 b \left (1152 a^2+1560 a b+595 b^2\right )-322560 (a+b)^3 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{322560 d}\\ &=\frac {\left (1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{2048 d}-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac {\left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2048 d}\\ &=-\frac {\left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) x}{2048}+\frac {\left (1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{2048 d}-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 189, normalized size = 0.74 \[ \frac {-105 b \left (2304 a^2+2880 a b+1001 b^2\right ) \sinh (4 (c+d x))+35 b \left (768 a^2+2160 a b+1001 b^2\right ) \sinh (6 (c+d x))-840 \left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) (c+d x)+105 \left (4096 a^3+11520 a^2 b+10080 a b^2+3003 b^3\right ) \sinh (2 (c+d x))-105 b^2 (120 a+91 b) \sinh (8 (c+d x))+21 b^2 (48 a+91 b) \sinh (10 (c+d x))-245 b^3 \sinh (12 (c+d x))+15 b^3 \sinh (14 (c+d x))}{1720320 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 627, normalized size = 2.46 \[ \frac {105 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{13} + 210 \, {\left (13 \, b^{3} \cosh \left (d x + c\right )^{3} - 7 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{11} + 35 \, {\left (429 \, b^{3} \cosh \left (d x + c\right )^{5} - 770 \, b^{3} \cosh \left (d x + c\right )^{3} + 3 \, {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{9} + 60 \, {\left (429 \, b^{3} \cosh \left (d x + c\right )^{7} - 1617 \, b^{3} \cosh \left (d x + c\right )^{5} + 21 \, {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 7 \, {\left (120 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 21 \, {\left (715 \, b^{3} \cosh \left (d x + c\right )^{9} - 4620 \, b^{3} \cosh \left (d x + c\right )^{7} + 126 \, {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 140 \, {\left (120 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 5 \, {\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 70 \, {\left (39 \, b^{3} \cosh \left (d x + c\right )^{11} - 385 \, b^{3} \cosh \left (d x + c\right )^{9} + 18 \, {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 42 \, {\left (120 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (2304 \, a^{2} b + 2880 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 420 \, {\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} d x + 105 \, {\left (b^{3} \cosh \left (d x + c\right )^{13} - 14 \, b^{3} \cosh \left (d x + c\right )^{11} + {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} - 4 \, {\left (120 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + {\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 2 \, {\left (2304 \, a^{2} b + 2880 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (4096 \, a^{3} + 11520 \, a^{2} b + 10080 \, a b^{2} + 3003 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{860160 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 401, normalized size = 1.57 \[ \frac {b^{3} e^{\left (14 \, d x + 14 \, c\right )}}{229376 \, d} - \frac {7 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )}}{98304 \, d} + \frac {7 \, b^{3} e^{\left (-12 \, d x - 12 \, c\right )}}{98304 \, d} - \frac {b^{3} e^{\left (-14 \, d x - 14 \, c\right )}}{229376 \, d} - \frac {1}{2048} \, {\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} x + \frac {{\left (48 \, a b^{2} + 91 \, b^{3}\right )} e^{\left (10 \, d x + 10 \, c\right )}}{163840 \, d} - \frac {{\left (120 \, a b^{2} + 91 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )}}{32768 \, d} + \frac {{\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{98304 \, d} - \frac {{\left (2304 \, a^{2} b + 2880 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{32768 \, d} + \frac {{\left (4096 \, a^{3} + 11520 \, a^{2} b + 10080 \, a b^{2} + 3003 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{32768 \, d} - \frac {{\left (4096 \, a^{3} + 11520 \, a^{2} b + 10080 \, a b^{2} + 3003 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32768 \, d} + \frac {{\left (2304 \, a^{2} b + 2880 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{32768 \, d} - \frac {{\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{98304 \, d} + \frac {{\left (120 \, a b^{2} + 91 \, b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{32768 \, d} - \frac {{\left (48 \, a b^{2} + 91 \, b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{163840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 240, normalized size = 0.94 \[ \frac {b^{3} \left (\left (\frac {\left (\sinh ^{13}\left (d x +c \right )\right )}{14}-\frac {13 \left (\sinh ^{11}\left (d x +c \right )\right )}{168}+\frac {143 \left (\sinh ^{9}\left (d x +c \right )\right )}{1680}-\frac {429 \left (\sinh ^{7}\left (d x +c \right )\right )}{4480}+\frac {143 \left (\sinh ^{5}\left (d x +c \right )\right )}{1280}-\frac {143 \left (\sinh ^{3}\left (d x +c \right )\right )}{1024}+\frac {429 \sinh \left (d x +c \right )}{2048}\right ) \cosh \left (d x +c \right )-\frac {429 d x}{2048}-\frac {429 c}{2048}\right )+3 a \,b^{2} \left (\left (\frac {\left (\sinh ^{9}\left (d x +c \right )\right )}{10}-\frac {9 \left (\sinh ^{7}\left (d x +c \right )\right )}{80}+\frac {21 \left (\sinh ^{5}\left (d x +c \right )\right )}{160}-\frac {21 \left (\sinh ^{3}\left (d x +c \right )\right )}{128}+\frac {63 \sinh \left (d x +c \right )}{256}\right ) \cosh \left (d x +c \right )-\frac {63 d x}{256}-\frac {63 c}{256}\right )+3 a^{2} b \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+a^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 442, normalized size = 1.73 \[ -\frac {1}{8} \, a^{3} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{3440640} \, b^{3} {\left (\frac {{\left (245 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1911 \, e^{\left (-4 \, d x - 4 \, c\right )} + 9555 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35035 \, e^{\left (-8 \, d x - 8 \, c\right )} + 105105 \, e^{\left (-10 \, d x - 10 \, c\right )} - 315315 \, e^{\left (-12 \, d x - 12 \, c\right )} - 15\right )} e^{\left (14 \, d x + 14 \, c\right )}}{d} + \frac {720720 \, {\left (d x + c\right )}}{d} + \frac {315315 \, e^{\left (-2 \, d x - 2 \, c\right )} - 105105 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35035 \, e^{\left (-6 \, d x - 6 \, c\right )} - 9555 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1911 \, e^{\left (-10 \, d x - 10 \, c\right )} - 245 \, e^{\left (-12 \, d x - 12 \, c\right )} + 15 \, e^{\left (-14 \, d x - 14 \, c\right )}}{d}\right )} - \frac {3}{20480} \, a b^{2} {\left (\frac {{\left (25 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 \, e^{\left (-4 \, d x - 4 \, c\right )} + 600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 2100 \, e^{\left (-8 \, d x - 8 \, c\right )} - 2\right )} e^{\left (10 \, d x + 10 \, c\right )}}{d} + \frac {5040 \, {\left (d x + c\right )}}{d} + \frac {2100 \, e^{\left (-2 \, d x - 2 \, c\right )} - 600 \, e^{\left (-4 \, d x - 4 \, c\right )} + 150 \, e^{\left (-6 \, d x - 6 \, c\right )} - 25 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d}\right )} - \frac {1}{128} \, a^{2} b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 393, normalized size = 1.54 \[ \frac {{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (768\,a^2\,b+2160\,a\,b^2+1001\,b^3\right )}{98304\,d}-\frac {{\mathrm {e}}^{-6\,c-6\,d\,x}\,\left (768\,a^2\,b+2160\,a\,b^2+1001\,b^3\right )}{98304\,d}-x\,\left (\frac {a^3}{2}+\frac {15\,a^2\,b}{16}+\frac {189\,a\,b^2}{256}+\frac {429\,b^3}{2048}\right )+\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}\,\left (2304\,a^2\,b+2880\,a\,b^2+1001\,b^3\right )}{32768\,d}-\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2304\,a^2\,b+2880\,a\,b^2+1001\,b^3\right )}{32768\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (4096\,a^3+11520\,a^2\,b+10080\,a\,b^2+3003\,b^3\right )}{32768\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (4096\,a^3+11520\,a^2\,b+10080\,a\,b^2+3003\,b^3\right )}{32768\,d}+\frac {7\,b^3\,{\mathrm {e}}^{-12\,c-12\,d\,x}}{98304\,d}-\frac {7\,b^3\,{\mathrm {e}}^{12\,c+12\,d\,x}}{98304\,d}-\frac {b^3\,{\mathrm {e}}^{-14\,c-14\,d\,x}}{229376\,d}+\frac {b^3\,{\mathrm {e}}^{14\,c+14\,d\,x}}{229376\,d}-\frac {b^2\,{\mathrm {e}}^{-10\,c-10\,d\,x}\,\left (48\,a+91\,b\right )}{163840\,d}+\frac {b^2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (48\,a+91\,b\right )}{163840\,d}+\frac {b^2\,{\mathrm {e}}^{-8\,c-8\,d\,x}\,\left (120\,a+91\,b\right )}{32768\,d}-\frac {b^2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (120\,a+91\,b\right )}{32768\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 104.16, size = 877, normalized size = 3.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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