Integrand size = 21, antiderivative size = 254 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=-\frac {\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{1+m}}{b^7 d (1+m)}+\frac {6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{2+m}}{b^7 d (2+m)}-\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+b \sin (c+d x))^{3+m}}{b^7 d (3+m)}+\frac {4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{4+m}}{b^7 d (4+m)}-\frac {3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{5+m}}{b^7 d (5+m)}+\frac {6 a (a+b \sin (c+d x))^{6+m}}{b^7 d (6+m)}-\frac {(a+b \sin (c+d x))^{7+m}}{b^7 d (7+m)} \]
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Time = 0.12 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2747, 711} \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=-\frac {\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{m+1}}{b^7 d (m+1)}+\frac {6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{m+2}}{b^7 d (m+2)}+\frac {4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{m+4}}{b^7 d (m+4)}-\frac {3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{m+5}}{b^7 d (m+5)}-\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+b \sin (c+d x))^{m+3}}{b^7 d (m+3)}+\frac {6 a (a+b \sin (c+d x))^{m+6}}{b^7 d (m+6)}-\frac {(a+b \sin (c+d x))^{m+7}}{b^7 d (m+7)} \]
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Rule 711
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^m \left (b^2-x^2\right )^3 \, dx,x,b \sin (c+d x)\right )}{b^7 d} \\ & = \frac {\text {Subst}\left (\int \left (-\left (a^2-b^2\right )^3 (a+x)^m+6 a \left (a^2-b^2\right )^2 (a+x)^{1+m}-3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+x)^{2+m}+4 a \left (5 a^2-3 b^2\right ) (a+x)^{3+m}-3 \left (5 a^2-b^2\right ) (a+x)^{4+m}+6 a (a+x)^{5+m}-(a+x)^{6+m}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d} \\ & = -\frac {\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{1+m}}{b^7 d (1+m)}+\frac {6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{2+m}}{b^7 d (2+m)}-\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+b \sin (c+d x))^{3+m}}{b^7 d (3+m)}+\frac {4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{4+m}}{b^7 d (4+m)}-\frac {3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{5+m}}{b^7 d (5+m)}+\frac {6 a (a+b \sin (c+d x))^{6+m}}{b^7 d (6+m)}-\frac {(a+b \sin (c+d x))^{7+m}}{b^7 d (7+m)} \\ \end{align*}
Time = 4.78 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.41 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\frac {(a+b \sin (c+d x))^{1+m} \left (b^6 \cos ^6(c+d x)+\frac {6 \left (-a^2+b^2\right ) \left (b^4 \cos ^4(c+d x)+4 \left (-a^2+b^2\right ) \left (\frac {-a^2+b^2}{1+m}+\frac {2 a (a+b \sin (c+d x))}{2+m}-\frac {(a+b \sin (c+d x))^2}{3+m}\right )+4 a (a+b \sin (c+d x)) \left (\frac {-a^2+b^2}{2+m}+\frac {2 a (a+b \sin (c+d x))}{3+m}-\frac {(a+b \sin (c+d x))^2}{4+m}\right )\right )}{5+m}+\frac {6 a (a+b \sin (c+d x)) \left (b^4 \cos ^4(c+d x)+4 \left (-a^2+b^2\right ) \left (\frac {-a^2+b^2}{2+m}+\frac {2 a (a+b \sin (c+d x))}{3+m}-\frac {(a+b \sin (c+d x))^2}{4+m}\right )+4 a (a+b \sin (c+d x)) \left (\frac {-a^2+b^2}{3+m}+\frac {2 a (a+b \sin (c+d x))}{4+m}-\frac {(a+b \sin (c+d x))^2}{5+m}\right )\right )}{6+m}\right )}{b^7 d (7+m)} \]
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Time = 18.53 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.97
method | result | size |
parallelrisch | \(-\frac {720 \left (\frac {\left (-\frac {3 \left (m^{2}+16 m +\frac {245}{3}\right ) \left (6+m \right ) \left (4+m \right ) b^{4}}{640}-\frac {3 a^{2} m \left (m^{2}+23 m +92\right ) b^{2}}{80}+a^{4} m \right ) \left (2+m \right ) \left (1+m \right ) b^{3} \sin \left (3 d x +3 c \right )}{24}-\frac {a m \left (1+m \right ) b^{2} \left (\left (\frac {1}{384} m^{4}+\frac {67}{960} m^{3}+\frac {1411}{1920} m^{2}+\frac {637}{192} m +\frac {417}{80}\right ) b^{4}-\frac {a^{2} \left (m^{2}+53 m +222\right ) b^{2}}{60}+a^{4}\right ) \cos \left (2 d x +2 c \right )}{4}-\frac {\left (2+m \right ) \left (\left (\frac {5}{24} m^{2}+\frac {79}{24} m +\frac {49}{4}\right ) b^{2}+a^{2} m \right ) \left (1+m \right ) \left (4+m \right ) \left (3+m \right ) b^{5} \sin \left (5 d x +5 c \right )}{1920}+\frac {\left (2+m \right ) \left (\left (-\frac {1}{20} m^{2}-\frac {17}{20} m -\frac {16}{5}\right ) b^{2}+a^{2}\right ) a m \left (1+m \right ) \left (3+m \right ) b^{4} \cos \left (4 d x +4 c \right )}{192}-\frac {b^{7} \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \sin \left (7 d x +7 c \right )}{46080}-\frac {a \,b^{6} m \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \cos \left (6 d x +6 c \right )}{23040}-\left (\frac {\left (2+m \right ) \left (6+m \right ) \left (4+m \right ) \left (m^{3}+\frac {93}{5} m^{2}+\frac {691}{5} m +735\right ) b^{6}}{9216}+\frac {a^{2} m \left (m^{4}+34 m^{3}+611 m^{2}+3530 m +5832\right ) b^{4}}{960}+\frac {a^{4} m \left (m^{2}-37 m -158\right ) b^{2}}{40}+a^{6} m \right ) b \sin \left (d x +c \right )+\left (\left (-7-\frac {875}{768} m^{3}-\frac {27271}{5760} m^{2}-\frac {1169}{120} m -\frac {1}{2304} m^{6}-\frac {49}{3840} m^{5}-\frac {1829}{11520} m^{4}\right ) b^{6}-\frac {a^{2} \left (2+m \right ) \left (m^{3}+68 m^{2}-413 m -3360\right ) b^{4}}{960}+\frac {3 a^{4} \left (m^{2}-7 m -28\right ) b^{2}}{20}+a^{6}\right ) a \right ) \left (a +b \sin \left (d x +c \right )\right )^{m}}{b^{7} \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) d}\) | \(500\) |
derivativedivides | \(\frac {\left (b^{6} m^{6}+6 a^{2} b^{4} m^{5}+27 b^{6} m^{5}+132 a^{2} b^{4} m^{4}+295 b^{6} m^{4}-72 a^{4} b^{2} m^{3}+1074 a^{2} b^{4} m^{3}+1665 b^{6} m^{3}-936 a^{4} b^{2} m^{2}+3828 a^{2} b^{4} m^{2}+5104 b^{6} m^{2}+720 a^{6} m -3024 a^{4} b^{2} m +5040 a^{2} b^{4} m +8028 b^{6} m +5040 b^{6}\right ) \sin \left (d x +c \right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{6} \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right ) d}-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{d \left (7+m \right )}-\frac {a \left (-b^{6} m^{6}-27 b^{6} m^{5}+6 a^{2} b^{4} m^{4}-295 b^{6} m^{4}+132 a^{2} b^{4} m^{3}-1665 b^{6} m^{3}-72 a^{4} b^{2} m^{2}+1074 a^{2} b^{4} m^{2}-5104 b^{6} m^{2}-936 a^{4} b^{2} m +3828 a^{2} b^{4} m -8028 b^{6} m +720 a^{6}-3024 a^{4} b^{2}+5040 a^{2} b^{4}-5040 b^{6}\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{7} d \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right )}+\frac {3 \left (b^{2} m^{2}+2 a^{2} m +13 b^{2} m +42 b^{2}\right ) \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{2} d \left (m^{3}+18 m^{2}+107 m +210\right )}+\frac {3 \left (-b^{4} m^{4}-4 a^{2} b^{2} m^{3}-22 b^{4} m^{3}-52 a^{2} b^{2} m^{2}-179 b^{4} m^{2}+40 a^{4} m -168 a^{2} b^{2} m -638 b^{4} m -840 b^{4}\right ) \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{4} d \left (m^{5}+25 m^{4}+245 m^{3}+1175 m^{2}+2754 m +2520\right )}-\frac {a m \left (\sin ^{6}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b d \left (m^{2}+13 m +42\right )}-\frac {3 \left (-b^{2} m^{2}-13 b^{2} m +10 a^{2}-42 b^{2}\right ) a m \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{3} d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {3 \left (b^{4} m^{4}+22 b^{4} m^{3}-12 a^{2} b^{2} m^{2}+179 b^{4} m^{2}-156 a^{2} b^{2} m +638 b^{4} m +120 a^{4}-504 a^{2} b^{2}+840 b^{4}\right ) a m \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{5} d \left (m^{6}+27 m^{5}+295 m^{4}+1665 m^{3}+5104 m^{2}+8028 m +5040\right )}\) | \(877\) |
default | \(\frac {\left (b^{6} m^{6}+6 a^{2} b^{4} m^{5}+27 b^{6} m^{5}+132 a^{2} b^{4} m^{4}+295 b^{6} m^{4}-72 a^{4} b^{2} m^{3}+1074 a^{2} b^{4} m^{3}+1665 b^{6} m^{3}-936 a^{4} b^{2} m^{2}+3828 a^{2} b^{4} m^{2}+5104 b^{6} m^{2}+720 a^{6} m -3024 a^{4} b^{2} m +5040 a^{2} b^{4} m +8028 b^{6} m +5040 b^{6}\right ) \sin \left (d x +c \right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{6} \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right ) d}-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{d \left (7+m \right )}-\frac {a \left (-b^{6} m^{6}-27 b^{6} m^{5}+6 a^{2} b^{4} m^{4}-295 b^{6} m^{4}+132 a^{2} b^{4} m^{3}-1665 b^{6} m^{3}-72 a^{4} b^{2} m^{2}+1074 a^{2} b^{4} m^{2}-5104 b^{6} m^{2}-936 a^{4} b^{2} m +3828 a^{2} b^{4} m -8028 b^{6} m +720 a^{6}-3024 a^{4} b^{2}+5040 a^{2} b^{4}-5040 b^{6}\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{7} d \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right )}+\frac {3 \left (b^{2} m^{2}+2 a^{2} m +13 b^{2} m +42 b^{2}\right ) \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{2} d \left (m^{3}+18 m^{2}+107 m +210\right )}+\frac {3 \left (-b^{4} m^{4}-4 a^{2} b^{2} m^{3}-22 b^{4} m^{3}-52 a^{2} b^{2} m^{2}-179 b^{4} m^{2}+40 a^{4} m -168 a^{2} b^{2} m -638 b^{4} m -840 b^{4}\right ) \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{4} d \left (m^{5}+25 m^{4}+245 m^{3}+1175 m^{2}+2754 m +2520\right )}-\frac {a m \left (\sin ^{6}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b d \left (m^{2}+13 m +42\right )}-\frac {3 \left (-b^{2} m^{2}-13 b^{2} m +10 a^{2}-42 b^{2}\right ) a m \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{3} d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {3 \left (b^{4} m^{4}+22 b^{4} m^{3}-12 a^{2} b^{2} m^{2}+179 b^{4} m^{2}-156 a^{2} b^{2} m +638 b^{4} m +120 a^{4}-504 a^{2} b^{2}+840 b^{4}\right ) a m \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{m \ln \left (a +b \sin \left (d x +c \right )\right )}}{b^{5} d \left (m^{6}+27 m^{5}+295 m^{4}+1665 m^{3}+5104 m^{2}+8028 m +5040\right )}\) | \(877\) |
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Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (254) = 508\).
Time = 0.41 (sec) , antiderivative size = 814, normalized size of antiderivative = 3.20 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=-\frac {{\left (720 \, a^{7} - 3024 \, a^{5} b^{2} + 5040 \, a^{3} b^{4} - 5040 \, a b^{6} - {\left (a b^{6} m^{6} + 15 \, a b^{6} m^{5} + 85 \, a b^{6} m^{4} + 225 \, a b^{6} m^{3} + 274 \, a b^{6} m^{2} + 120 \, a b^{6} m\right )} \cos \left (d x + c\right )^{6} - 6 \, {\left (2 \, a b^{6} m^{5} - {\left (5 \, a^{3} b^{4} - 23 \, a b^{6}\right )} m^{4} - 2 \, {\left (15 \, a^{3} b^{4} - 44 \, a b^{6}\right )} m^{3} - {\left (55 \, a^{3} b^{4} - 133 \, a b^{6}\right )} m^{2} - 6 \, {\left (5 \, a^{3} b^{4} - 11 \, a b^{6}\right )} m\right )} \cos \left (d x + c\right )^{4} - 192 \, {\left (a^{3} b^{4} + a b^{6}\right )} m^{3} + 288 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} - 7 \, a b^{6}\right )} m^{2} - 24 \, {\left ({\left (a^{3} b^{4} + 3 \, a b^{6}\right )} m^{4} - 6 \, {\left (a^{3} b^{4} - 5 \, a b^{6}\right )} m^{3} + {\left (15 \, a^{5} b^{2} - 55 \, a^{3} b^{4} + 84 \, a b^{6}\right )} m^{2} + 3 \, {\left (5 \, a^{5} b^{2} - 16 \, a^{3} b^{4} + 19 \, a b^{6}\right )} m\right )} \cos \left (d x + c\right )^{2} - 192 \, {\left (3 \, a^{5} b^{2} - 13 \, a^{3} b^{4} + 32 \, a b^{6}\right )} m - {\left (2304 \, b^{7} + {\left (b^{7} m^{6} + 21 \, b^{7} m^{5} + 175 \, b^{7} m^{4} + 735 \, b^{7} m^{3} + 1624 \, b^{7} m^{2} + 1764 \, b^{7} m + 720 \, b^{7}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (144 \, b^{7} + {\left (a^{2} b^{5} + b^{7}\right )} m^{5} + 2 \, {\left (5 \, a^{2} b^{5} + 8 \, b^{7}\right )} m^{4} + 5 \, {\left (7 \, a^{2} b^{5} + 19 \, b^{7}\right )} m^{3} + 10 \, {\left (5 \, a^{2} b^{5} + 26 \, b^{7}\right )} m^{2} + 12 \, {\left (2 \, a^{2} b^{5} + 27 \, b^{7}\right )} m\right )} \cos \left (d x + c\right )^{4} + 48 \, {\left (a^{4} b^{3} + 6 \, a^{2} b^{5} + b^{7}\right )} m^{3} - 576 \, {\left (a^{4} b^{3} - 4 \, a^{2} b^{5} - b^{7}\right )} m^{2} + 24 \, {\left (48 \, b^{7} + {\left (3 \, a^{2} b^{5} + b^{7}\right )} m^{4} - {\left (5 \, a^{4} b^{3} - 24 \, a^{2} b^{5} - 13 \, b^{7}\right )} m^{3} - {\left (15 \, a^{4} b^{3} - 51 \, a^{2} b^{5} - 56 \, b^{7}\right )} m^{2} - 2 \, {\left (5 \, a^{4} b^{3} - 15 \, a^{2} b^{5} - 46 \, b^{7}\right )} m\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (15 \, a^{6} b - 58 \, a^{4} b^{3} + 87 \, a^{2} b^{5} + 44 \, b^{7}\right )} m\right )} \sin \left (d x + c\right )\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{b^{7} d m^{7} + 28 \, b^{7} d m^{6} + 322 \, b^{7} d m^{5} + 1960 \, b^{7} d m^{4} + 6769 \, b^{7} d m^{3} + 13132 \, b^{7} d m^{2} + 13068 \, b^{7} d m + 5040 \, b^{7} d} \]
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Timed out. \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (254) = 508\).
Time = 0.21 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.20 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\frac {\frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m + 1}}{b {\left (m + 1\right )}} - \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} \sin \left (d x + c\right )^{3} + {\left (m^{2} + m\right )} a b^{2} \sin \left (d x + c\right )^{2} - 2 \, a^{2} b m \sin \left (d x + c\right ) + 2 \, a^{3}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{5} \sin \left (d x + c\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a b^{4} \sin \left (d x + c\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{2} b^{3} \sin \left (d x + c\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{3} b^{2} \sin \left (d x + c\right )^{2} - 24 \, a^{4} b m \sin \left (d x + c\right ) + 24 \, a^{5}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} b^{5}} - \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} b^{7} \sin \left (d x + c\right )^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} a b^{6} \sin \left (d x + c\right )^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{2} b^{5} \sin \left (d x + c\right )^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{3} b^{4} \sin \left (d x + c\right )^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{4} b^{3} \sin \left (d x + c\right )^{3} + 360 \, {\left (m^{2} + m\right )} a^{5} b^{2} \sin \left (d x + c\right )^{2} - 720 \, a^{6} b m \sin \left (d x + c\right ) + 720 \, a^{7}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} b^{7}}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3579 vs. \(2 (254) = 508\).
Time = 0.36 (sec) , antiderivative size = 3579, normalized size of antiderivative = 14.09 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\text {Too large to display} \]
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Time = 19.11 (sec) , antiderivative size = 1196, normalized size of antiderivative = 4.71 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\text {Too large to display} \]
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