Optimal. Leaf size=211 \[ \frac {b \left (1152 a^2+3912 a b+2279 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{1536 d}-\frac {b \left (1920 a^2+2232 a b+793 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{1024 d}+\frac {x \left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right )}{1024}+\frac {3 b^2 (40 a+139 b) \sinh (c+d x) \cosh ^7(c+d x)}{320 d}-\frac {b^2 (3000 a+3481 b) \sinh (c+d x) \cosh ^5(c+d x)}{1920 d}+\frac {b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{12 d}-\frac {61 b^3 \sinh (c+d x) \cosh ^9(c+d x)}{120 d} \]
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Rubi [A] time = 0.39, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3209, 1157, 1814, 385, 206} \[ \frac {b \left (1152 a^2+3912 a b+2279 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{1536 d}-\frac {b \left (1920 a^2+2232 a b+793 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{1024 d}+\frac {x \left (1152 a^2 b+1024 a^3+840 a b^2+231 b^3\right )}{1024}+\frac {3 b^2 (40 a+139 b) \sinh (c+d x) \cosh ^7(c+d x)}{320 d}-\frac {b^2 (3000 a+3481 b) \sinh (c+d x) \cosh ^5(c+d x)}{1920 d}+\frac {b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{12 d}-\frac {61 b^3 \sinh (c+d x) \cosh ^9(c+d x)}{120 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 1157
Rule 1814
Rule 3209
Rubi steps
\begin {align*} \int \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-2 a x^2+(a+b) x^4\right )^3}{\left (1-x^2\right )^7} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}-\frac {\operatorname {Subst}\left (\int \frac {-12 a^3+b^3+12 \left (5 a^3+b^3\right ) x^2-12 \left (10 a^3+3 a^2 b-b^3\right ) x^4+12 \left (10 a^3+9 a^2 b+b^3\right ) x^6-12 (5 a-b) (a+b)^2 x^8+12 (a+b)^3 x^{10}}{\left (1-x^2\right )^6} \, dx,x,\tanh (c+d x)\right )}{12 d}\\ &=-\frac {61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac {b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}+\frac {\operatorname {Subst}\left (\int \frac {3 \left (40 a^3+17 b^3\right )-480 \left (a^3-b^3\right ) x^2+360 \left (2 a^3+a^2 b+b^3\right ) x^4-240 (2 a-b) (a+b)^2 x^6+120 (a+b)^3 x^8}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{120 d}\\ &=\frac {3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac {61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac {b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}-\frac {\operatorname {Subst}\left (\int \frac {-3 \left (320 a^3-120 a b^2-281 b^3\right )+2880 \left (a^3+a b^2+2 b^3\right ) x^2-2880 (a-b) (a+b)^2 x^4+960 (a+b)^3 x^6}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{960 d}\\ &=-\frac {b^2 (3000 a+3481 b) \cosh ^5(c+d x) \sinh (c+d x)}{1920 d}+\frac {3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac {61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac {b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}+\frac {\operatorname {Subst}\left (\int \frac {15 \left (384 a^3+456 a b^2+359 b^3\right )-11520 (a-2 b) (a+b)^2 x^2+5760 (a+b)^3 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{5760 d}\\ &=\frac {b \left (1152 a^2+3912 a b+2279 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{1536 d}-\frac {b^2 (3000 a+3481 b) \cosh ^5(c+d x) \sinh (c+d x)}{1920 d}+\frac {3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac {61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac {b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}-\frac {\operatorname {Subst}\left (\int \frac {-45 \left (512 a^3-384 a^2 b-696 a b^2-281 b^3\right )+23040 (a+b)^3 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{23040 d}\\ &=-\frac {b \left (1920 a^2+2232 a b+793 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{1024 d}+\frac {b \left (1152 a^2+3912 a b+2279 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{1536 d}-\frac {b^2 (3000 a+3481 b) \cosh ^5(c+d x) \sinh (c+d x)}{1920 d}+\frac {3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac {61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac {b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}+\frac {\left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{1024 d}\\ &=\frac {\left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) x}{1024}-\frac {b \left (1920 a^2+2232 a b+793 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{1024 d}+\frac {b \left (1152 a^2+3912 a b+2279 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{1536 d}-\frac {b^2 (3000 a+3481 b) \cosh ^5(c+d x) \sinh (c+d x)}{1920 d}+\frac {3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac {61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac {b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 156, normalized size = 0.74 \[ \frac {-720 b \left (128 a^2+112 a b+33 b^2\right ) \sinh (2 (c+d x))+45 b \left (256 a^2+448 a b+165 b^2\right ) \sinh (4 (c+d x))+120 \left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) (c+d x)-40 b^2 (96 a+55 b) \sinh (6 (c+d x))+45 b^2 (8 a+11 b) \sinh (8 (c+d x))-72 b^3 \sinh (10 (c+d x))+5 b^3 \sinh (12 (c+d x))}{122880 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 461, normalized size = 2.18 \[ \frac {15 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{11} + 5 \, {\left (55 \, b^{3} \cosh \left (d x + c\right )^{3} - 36 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{9} + 90 \, {\left (11 \, b^{3} \cosh \left (d x + c\right )^{5} - 24 \, b^{3} \cosh \left (d x + c\right )^{3} + {\left (8 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 6 \, {\left (165 \, b^{3} \cosh \left (d x + c\right )^{7} - 756 \, b^{3} \cosh \left (d x + c\right )^{5} + 105 \, {\left (8 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 10 \, {\left (96 \, a b^{2} + 55 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (55 \, b^{3} \cosh \left (d x + c\right )^{9} - 432 \, b^{3} \cosh \left (d x + c\right )^{7} + 126 \, {\left (8 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 40 \, {\left (96 \, a b^{2} + 55 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 9 \, {\left (256 \, a^{2} b + 448 \, a b^{2} + 165 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 30 \, {\left (1024 \, a^{3} + 1152 \, a^{2} b + 840 \, a b^{2} + 231 \, b^{3}\right )} d x + 15 \, {\left (b^{3} \cosh \left (d x + c\right )^{11} - 12 \, b^{3} \cosh \left (d x + c\right )^{9} + 6 \, {\left (8 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 4 \, {\left (96 \, a b^{2} + 55 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 3 \, {\left (256 \, a^{2} b + 448 \, a b^{2} + 165 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 24 \, {\left (128 \, a^{2} b + 112 \, a b^{2} + 33 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{30720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 327, normalized size = 1.55 \[ \frac {b^{3} e^{\left (12 \, d x + 12 \, c\right )}}{49152 \, d} - \frac {3 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )}}{10240 \, d} + \frac {3 \, b^{3} e^{\left (-10 \, d x - 10 \, c\right )}}{10240 \, d} - \frac {b^{3} e^{\left (-12 \, d x - 12 \, c\right )}}{49152 \, d} + \frac {1}{1024} \, {\left (1024 \, a^{3} + 1152 \, a^{2} b + 840 \, a b^{2} + 231 \, b^{3}\right )} x + \frac {3 \, {\left (8 \, a b^{2} + 11 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )}}{16384 \, d} - \frac {{\left (96 \, a b^{2} + 55 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{6144 \, d} + \frac {3 \, {\left (256 \, a^{2} b + 448 \, a b^{2} + 165 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{16384 \, d} - \frac {3 \, {\left (128 \, a^{2} b + 112 \, a b^{2} + 33 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{1024 \, d} + \frac {3 \, {\left (128 \, a^{2} b + 112 \, a b^{2} + 33 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{1024 \, d} - \frac {3 \, {\left (256 \, a^{2} b + 448 \, a b^{2} + 165 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{16384 \, d} + \frac {{\left (96 \, a b^{2} + 55 \, b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{6144 \, d} - \frac {3 \, {\left (8 \, a b^{2} + 11 \, b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{16384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 193, normalized size = 0.91 \[ \frac {b^{3} \left (\left (\frac {\left (\sinh ^{11}\left (d x +c \right )\right )}{12}-\frac {11 \left (\sinh ^{9}\left (d x +c \right )\right )}{120}+\frac {33 \left (\sinh ^{7}\left (d x +c \right )\right )}{320}-\frac {77 \left (\sinh ^{5}\left (d x +c \right )\right )}{640}+\frac {77 \left (\sinh ^{3}\left (d x +c \right )\right )}{512}-\frac {231 \sinh \left (d x +c \right )}{1024}\right ) \cosh \left (d x +c \right )+\frac {231 d x}{1024}+\frac {231 c}{1024}\right )+3 a \,b^{2} \left (\left (\frac {\left (\sinh ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sinh ^{5}\left (d x +c \right )\right )}{48}+\frac {35 \left (\sinh ^{3}\left (d x +c \right )\right )}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )+3 a^{2} b \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 344, normalized size = 1.63 \[ \frac {3}{64} \, a^{2} b {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + a^{3} x - \frac {1}{245760} \, b^{3} {\left (\frac {{\left (72 \, e^{\left (-2 \, d x - 2 \, c\right )} - 495 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2200 \, e^{\left (-6 \, d x - 6 \, c\right )} - 7425 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23760 \, e^{\left (-10 \, d x - 10 \, c\right )} - 5\right )} e^{\left (12 \, d x + 12 \, c\right )}}{d} - \frac {55440 \, {\left (d x + c\right )}}{d} - \frac {23760 \, e^{\left (-2 \, d x - 2 \, c\right )} - 7425 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2200 \, e^{\left (-6 \, d x - 6 \, c\right )} - 495 \, e^{\left (-8 \, d x - 8 \, c\right )} + 72 \, e^{\left (-10 \, d x - 10 \, c\right )} - 5 \, e^{\left (-12 \, d x - 12 \, c\right )}}{d}\right )} - \frac {1}{2048} \, a b^{2} {\left (\frac {{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {1680 \, {\left (d x + c\right )}}{d} - \frac {672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 210, normalized size = 1.00 \[ \frac {\frac {7425\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{8}-2970\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-275\,b^3\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {495\,b^3\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}-9\,b^3\,\mathrm {sinh}\left (10\,c+10\,d\,x\right )+\frac {5\,b^3\,\mathrm {sinh}\left (12\,c+12\,d\,x\right )}{8}-10080\,a\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-11520\,a^2\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+2520\,a\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+1440\,a^2\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-480\,a\,b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+45\,a\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )+15360\,a^3\,d\,x+3465\,b^3\,d\,x+12600\,a\,b^2\,d\,x+17280\,a^2\,b\,d\,x}{15360\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 49.88, size = 666, normalized size = 3.16 \[ \begin {cases} a^{3} x + \frac {9 a^{2} b x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {9 a^{2} b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {9 a^{2} b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {15 a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {9 a^{2} b \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {105 a b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {105 a b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {315 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {105 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {105 a b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {279 a b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} - \frac {511 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {385 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{128 d} - \frac {105 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {231 b^{3} x \sinh ^{12}{\left (c + d x \right )}}{1024} - \frac {693 b^{3} x \sinh ^{10}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{512} + \frac {3465 b^{3} x \sinh ^{8}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{1024} - \frac {1155 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{256} + \frac {3465 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{8}{\left (c + d x \right )}}{1024} - \frac {693 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{10}{\left (c + d x \right )}}{512} + \frac {231 b^{3} x \cosh ^{12}{\left (c + d x \right )}}{1024} + \frac {793 b^{3} \sinh ^{11}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{1024 d} - \frac {7337 b^{3} \sinh ^{9}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3072 d} + \frac {9273 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{2560 d} - \frac {7623 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{2560 d} + \frac {1309 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{1024 d} - \frac {231 b^{3} \sinh {\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{1024 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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