Integrand size = 29, antiderivative size = 467 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}+\frac {2 a^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^9 d}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d} \]
[Out]
Time = 1.25 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2975, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{140 a b^4 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{192 b^5 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{240 a^2 b^3 d}+\frac {2 a^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^9 d}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{128 b^7 d}-\frac {x \left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right )}{128 b^9}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d} \]
[In]
[Out]
Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2975
Rule 3102
Rule 3128
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin ^5(c+d x) \left (40 \left (24 a^4-49 a^2 b^2+28 b^4\right )-4 a b \left (5 a^2-14 b^2\right ) \sin (c+d x)-28 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1120 a^2 b^2} \\ & = \frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin ^4(c+d x) \left (-140 a \left (40 a^4-85 a^2 b^2+48 b^4\right )+20 a^2 b \left (8 a^2+7 b^2\right ) \sin (c+d x)+240 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6720 a^2 b^3} \\ & = \frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin ^3(c+d x) \left (960 a^2 \left (28 a^4-60 a^2 b^2+35 b^4\right )-20 a^3 b \left (56 a^2-95 b^2\right ) \sin (c+d x)-700 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{33600 a^2 b^4} \\ & = \frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin ^2(c+d x) \left (-2100 a^3 \left (48 a^4-104 a^2 b^2+59 b^4\right )+60 a^2 b \left (112 a^4-200 a^2 b^2+175 b^4\right ) \sin (c+d x)+3840 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{134400 a^2 b^5} \\ & = -\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin (c+d x) \left (7680 a^4 \left (35 a^4-77 a^2 b^2+45 b^4\right )-60 a^3 b \left (560 a^4-1064 a^2 b^2+435 b^4\right ) \sin (c+d x)-6300 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{403200 a^2 b^6} \\ & = \frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {-6300 a^3 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right )+60 a^2 b \left (2240 a^6-4592 a^4 b^2+2280 a^2 b^4+525 b^6\right ) \sin (c+d x)+7680 a^3 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{806400 a^2 b^7} \\ & = -\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {-6300 a^3 b \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right )-6300 a^2 \left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{806400 a^2 b^8} \\ & = -\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\left (a^3 \left (a^2-b^2\right )^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^9} \\ & = -\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\left (2 a^3 \left (a^2-b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^9 d} \\ & = -\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}-\frac {\left (4 a^3 \left (a^2-b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^9 d} \\ & = -\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}+\frac {2 a^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^9 d}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d} \\ \end{align*}
Time = 2.27 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-107520 a^8 c+268800 a^6 b^2 c-201600 a^4 b^4 c+33600 a^2 b^6 c+4200 b^8 c-107520 a^8 d x+268800 a^6 b^2 d x-201600 a^4 b^4 d x+33600 a^2 b^6 d x+4200 b^8 d x+215040 a^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-1680 a b \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x)+560 \left (16 a^5 b^3-28 a^3 b^5+9 a b^7\right ) \cos (3 (c+d x))-1344 a^3 b^5 \cos (5 (c+d x))+1680 a b^7 \cos (5 (c+d x))+240 a b^7 \cos (7 (c+d x))+26880 a^6 b^2 \sin (2 (c+d x))-53760 a^4 b^4 \sin (2 (c+d x))+25200 a^2 b^6 \sin (2 (c+d x))+1680 b^8 \sin (2 (c+d x))-3360 a^4 b^4 \sin (4 (c+d x))+5040 a^2 b^6 \sin (4 (c+d x))-840 b^8 \sin (4 (c+d x))+560 a^2 b^6 \sin (6 (c+d x))-560 b^8 \sin (6 (c+d x))-105 b^8 \sin (8 (c+d x))}{107520 b^9 d} \]
[In]
[Out]
Time = 1.56 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.71
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) a^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9} \sqrt {a^{2}-b^{2}}}-\frac {2 \left (\frac {a^{7} b -\frac {7 a^{5} b^{3}}{3}+\frac {23 a^{3} b^{5}}{15}-\frac {a \,b^{7}}{7}+\left (-\frac {1}{2} a^{6} b^{2}+\frac {9}{8} a^{4} b^{4}-\frac {11}{16} a^{2} b^{6}+\frac {5}{128} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7 a^{7} b -\frac {47}{3} a^{5} b^{3}+\frac {139}{15} a^{3} b^{5}-\frac {1}{7} a \,b^{7}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{6} b^{2}+\frac {37}{8} a^{4} b^{4}-\frac {61}{48} a^{2} b^{6}-\frac {397}{384} b^{8}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{7} b -\frac {139}{3} a^{5} b^{3}+\frac {419}{15} a^{3} b^{5}-3 a \,b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {9}{2} a^{6} b^{2}+\frac {57}{8} a^{4} b^{4}-\frac {113}{48} a^{2} b^{6}+\frac {895}{384} b^{8}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (35 a^{7} b -\frac {235}{3} a^{5} b^{3}+\frac {743}{15} a^{3} b^{5}-3 a \,b^{7}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{6} b^{2}+\frac {29}{8} a^{4} b^{4}-\frac {85}{48} a^{2} b^{6}-\frac {1765}{384} b^{8}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (35 a^{7} b -\frac {245}{3} a^{5} b^{3}+\frac {161}{3} a^{3} b^{5}-5 a \,b^{7}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{6} b^{2}-\frac {29}{8} a^{4} b^{4}+\frac {85}{48} a^{2} b^{6}+\frac {1765}{384} b^{8}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{7} b -\frac {157}{3} a^{5} b^{3}+\frac {109}{3} a^{3} b^{5}-5 a \,b^{7}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {9}{2} a^{6} b^{2}-\frac {57}{8} a^{4} b^{4}+\frac {113}{48} a^{2} b^{6}-\frac {895}{384} b^{8}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (7 a^{7} b -19 a^{5} b^{3}+15 a^{3} b^{5}-a \,b^{7}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{6} b^{2}-\frac {37}{8} a^{4} b^{4}+\frac {61}{48} a^{2} b^{6}+\frac {397}{384} b^{8}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{7} b -3 a^{5} b^{3}+3 a^{3} b^{5}-a \,b^{7}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{6} b^{2}-\frac {9}{8} a^{4} b^{4}+\frac {11}{16} a^{2} b^{6}-\frac {5}{128} b^{8}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {\left (128 a^{8}-320 a^{6} b^{2}+240 a^{4} b^{4}-40 a^{2} b^{6}-5 b^{8}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}\right )}{b^{9}}}{d}\) | \(797\) |
default | \(\frac {\frac {2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) a^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9} \sqrt {a^{2}-b^{2}}}-\frac {2 \left (\frac {a^{7} b -\frac {7 a^{5} b^{3}}{3}+\frac {23 a^{3} b^{5}}{15}-\frac {a \,b^{7}}{7}+\left (-\frac {1}{2} a^{6} b^{2}+\frac {9}{8} a^{4} b^{4}-\frac {11}{16} a^{2} b^{6}+\frac {5}{128} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7 a^{7} b -\frac {47}{3} a^{5} b^{3}+\frac {139}{15} a^{3} b^{5}-\frac {1}{7} a \,b^{7}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{6} b^{2}+\frac {37}{8} a^{4} b^{4}-\frac {61}{48} a^{2} b^{6}-\frac {397}{384} b^{8}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{7} b -\frac {139}{3} a^{5} b^{3}+\frac {419}{15} a^{3} b^{5}-3 a \,b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {9}{2} a^{6} b^{2}+\frac {57}{8} a^{4} b^{4}-\frac {113}{48} a^{2} b^{6}+\frac {895}{384} b^{8}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (35 a^{7} b -\frac {235}{3} a^{5} b^{3}+\frac {743}{15} a^{3} b^{5}-3 a \,b^{7}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{6} b^{2}+\frac {29}{8} a^{4} b^{4}-\frac {85}{48} a^{2} b^{6}-\frac {1765}{384} b^{8}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (35 a^{7} b -\frac {245}{3} a^{5} b^{3}+\frac {161}{3} a^{3} b^{5}-5 a \,b^{7}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{6} b^{2}-\frac {29}{8} a^{4} b^{4}+\frac {85}{48} a^{2} b^{6}+\frac {1765}{384} b^{8}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{7} b -\frac {157}{3} a^{5} b^{3}+\frac {109}{3} a^{3} b^{5}-5 a \,b^{7}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {9}{2} a^{6} b^{2}-\frac {57}{8} a^{4} b^{4}+\frac {113}{48} a^{2} b^{6}-\frac {895}{384} b^{8}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (7 a^{7} b -19 a^{5} b^{3}+15 a^{3} b^{5}-a \,b^{7}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{6} b^{2}-\frac {37}{8} a^{4} b^{4}+\frac {61}{48} a^{2} b^{6}+\frac {397}{384} b^{8}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{7} b -3 a^{5} b^{3}+3 a^{3} b^{5}-a \,b^{7}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{6} b^{2}-\frac {9}{8} a^{4} b^{4}+\frac {11}{16} a^{2} b^{6}-\frac {5}{128} b^{8}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {\left (128 a^{8}-320 a^{6} b^{2}+240 a^{4} b^{4}-40 a^{2} b^{6}-5 b^{8}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}\right )}{b^{9}}}{d}\) | \(797\) |
risch | \(\frac {i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}+\frac {i \sqrt {a^{2}-b^{2}}\, a^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{9}}-\frac {2 i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{7}}-\frac {i \sqrt {a^{2}-b^{2}}\, a^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{9}}+\frac {2 i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{7}}-\frac {i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}+\frac {5 x}{128 b}+\frac {5 a \,{\mathrm e}^{-i \left (d x +c \right )}}{128 b^{2} d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{2}}{64 b^{3} d}+\frac {\sin \left (6 d x +6 c \right ) a^{2}}{192 b^{3} d}-\frac {a^{3} \cos \left (5 d x +5 c \right )}{80 d \,b^{4}}+\frac {a \cos \left (5 d x +5 c \right )}{64 d \,b^{2}}-\frac {\sin \left (4 d x +4 c \right ) a^{4}}{32 b^{5} d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{2}}{64 b^{3} d}+\frac {a^{5} \cos \left (3 d x +3 c \right )}{12 d \,b^{6}}-\frac {7 a^{3} \cos \left (3 d x +3 c \right )}{48 d \,b^{4}}+\frac {3 a \cos \left (3 d x +3 c \right )}{64 d \,b^{2}}+\frac {\sin \left (2 d x +2 c \right ) a^{6}}{4 b^{7} d}-\frac {\sin \left (2 d x +2 c \right ) a^{4}}{2 b^{5} d}-\frac {\sin \left (6 d x +6 c \right )}{192 b d}-\frac {\sin \left (4 d x +4 c \right )}{128 b d}+\frac {a \cos \left (7 d x +7 c \right )}{448 b^{2} d}+\frac {5 a \,{\mathrm e}^{i \left (d x +c \right )}}{128 d \,b^{2}}-\frac {x \,a^{8}}{b^{9}}+\frac {5 x \,a^{6}}{2 b^{7}}-\frac {15 x \,a^{4}}{8 b^{5}}-\frac {\sin \left (8 d x +8 c \right )}{1024 b d}+\frac {5 x \,a^{2}}{16 b^{3}}+\frac {\sin \left (2 d x +2 c \right )}{64 b d}-\frac {a^{7} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{8} d}+\frac {9 a^{5} {\mathrm e}^{i \left (d x +c \right )}}{8 b^{6} d}-\frac {11 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 b^{4} d}-\frac {a^{7} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{8} d}+\frac {9 a^{5} {\mathrm e}^{-i \left (d x +c \right )}}{8 b^{6} d}-\frac {11 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 b^{4} d}\) | \(826\) |
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Time = 0.52 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.51 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\left [\frac {1920 \, a b^{7} \cos \left (d x + c\right )^{7} - 2688 \, a^{3} b^{5} \cos \left (d x + c\right )^{5} + 4480 \, {\left (a^{5} b^{3} - a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (128 \, a^{8} - 320 \, a^{6} b^{2} + 240 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - 5 \, b^{8}\right )} d x + 6720 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 13440 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right ) - 35 \, {\left (48 \, b^{8} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (48 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - 5 \, b^{8}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (64 \, a^{6} b^{2} - 112 \, a^{4} b^{4} + 40 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, b^{9} d}, \frac {1920 \, a b^{7} \cos \left (d x + c\right )^{7} - 2688 \, a^{3} b^{5} \cos \left (d x + c\right )^{5} + 4480 \, {\left (a^{5} b^{3} - a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (128 \, a^{8} - 320 \, a^{6} b^{2} + 240 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - 5 \, b^{8}\right )} d x - 13440 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 13440 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right ) - 35 \, {\left (48 \, b^{8} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (48 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - 5 \, b^{8}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (64 \, a^{6} b^{2} - 112 \, a^{4} b^{4} + 40 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, b^{9} d}\right ] \]
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Timed out. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (440) = 880\).
Time = 0.40 (sec) , antiderivative size = 1244, normalized size of antiderivative = 2.66 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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Time = 14.99 (sec) , antiderivative size = 4505, normalized size of antiderivative = 9.65 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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