\(\int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx\) [624]

Optimal result

Integrand size = 21, antiderivative size = 254 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=-\frac {\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{1+m}}{b^7 d (1+m)}+\frac {6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{2+m}}{b^7 d (2+m)}-\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+b \sin (c+d x))^{3+m}}{b^7 d (3+m)}+\frac {4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{4+m}}{b^7 d (4+m)}-\frac {3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{5+m}}{b^7 d (5+m)}+\frac {6 a (a+b \sin (c+d x))^{6+m}}{b^7 d (6+m)}-\frac {(a+b \sin (c+d x))^{7+m}}{b^7 d (7+m)} \] Output:

-(a^2-b^2)^3*(a+b*sin(d*x+c))^(1+m)/b^7/d/(1+m)+6*a*(a^2-b^2)^2*(a+b*sin(d 
*x+c))^(2+m)/b^7/d/(2+m)-3*(5*a^4-6*a^2*b^2+b^4)*(a+b*sin(d*x+c))^(3+m)/b^ 
7/d/(3+m)+4*a*(5*a^2-3*b^2)*(a+b*sin(d*x+c))^(4+m)/b^7/d/(4+m)-3*(5*a^2-b^ 
2)*(a+b*sin(d*x+c))^(5+m)/b^7/d/(5+m)+6*a*(a+b*sin(d*x+c))^(6+m)/b^7/d/(6+ 
m)-(a+b*sin(d*x+c))^(7+m)/b^7/d/(7+m)
 

Mathematica [A] (verified)

Time = 6.14 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.81 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\frac {\frac {b^6 \cos ^6(c+d x) (a+b \sin (c+d x))^{1+m}}{7+m}+\frac {6 \left (\left (-a^2+b^2\right ) \left (\frac {b^4 \cos ^4(c+d x) (a+b \sin (c+d x))^{1+m}}{5+m}+\frac {4 \left (\left (-a^2+b^2\right ) \left (-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{1+m}}{1+m}+\frac {2 a (a+b \sin (c+d x))^{2+m}}{2+m}-\frac {(a+b \sin (c+d x))^{3+m}}{3+m}\right )+a \left (-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{2+m}}{2+m}+\frac {2 a (a+b \sin (c+d x))^{3+m}}{3+m}-\frac {(a+b \sin (c+d x))^{4+m}}{4+m}\right )\right )}{5+m}\right )+a \left (\frac {b^4 \cos ^4(c+d x) (a+b \sin (c+d x))^{2+m}}{6+m}+\frac {4 \left (\left (-a^2+b^2\right ) \left (-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{2+m}}{2+m}+\frac {2 a (a+b \sin (c+d x))^{3+m}}{3+m}-\frac {(a+b \sin (c+d x))^{4+m}}{4+m}\right )+a \left (-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{3+m}}{3+m}+\frac {2 a (a+b \sin (c+d x))^{4+m}}{4+m}-\frac {(a+b \sin (c+d x))^{5+m}}{5+m}\right )\right )}{6+m}\right )\right )}{7+m}}{b^7 d} \] Input:

Integrate[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^m,x]
 

Output:

((b^6*Cos[c + d*x]^6*(a + b*Sin[c + d*x])^(1 + m))/(7 + m) + (6*((-a^2 + b 
^2)*((b^4*Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(1 + m))/(5 + m) + (4*((-a^2 
 + b^2)*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(1 + m))/(1 + m)) + (2*a*(a + 
 b*Sin[c + d*x])^(2 + m))/(2 + m) - (a + b*Sin[c + d*x])^(3 + m)/(3 + m)) 
+ a*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(2 + m))/(2 + m)) + (2*a*(a + b*S 
in[c + d*x])^(3 + m))/(3 + m) - (a + b*Sin[c + d*x])^(4 + m)/(4 + m))))/(5 
 + m)) + a*((b^4*Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(2 + m))/(6 + m) + (4 
*((-a^2 + b^2)*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(2 + m))/(2 + m)) + (2 
*a*(a + b*Sin[c + d*x])^(3 + m))/(3 + m) - (a + b*Sin[c + d*x])^(4 + m)/(4 
 + m)) + a*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(3 + m))/(3 + m)) + (2*a*( 
a + b*Sin[c + d*x])^(4 + m))/(4 + m) - (a + b*Sin[c + d*x])^(5 + m)/(5 + m 
))))/(6 + m))))/(7 + m))/(b^7*d)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3147, 476, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^7 (a+b \sin (c+d x))^mdx\)

\(\Big \downarrow \) 3147

\(\displaystyle \frac {\int (a+b \sin (c+d x))^m \left (b^2-b^2 \sin ^2(c+d x)\right )^3d(b \sin (c+d x))}{b^7 d}\)

\(\Big \downarrow \) 476

\(\displaystyle \frac {\int \left (-\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^m+6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{m+1}-3 \left (5 a^4-6 b^2 a^2+b^4\right ) (a+b \sin (c+d x))^{m+2}+4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{m+3}-3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{m+4}+6 a (a+b \sin (c+d x))^{m+5}-(a+b \sin (c+d x))^{m+6}\right )d(b \sin (c+d x))}{b^7 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{m+1}}{m+1}+\frac {6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{m+2}}{m+2}+\frac {4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{m+4}}{m+4}-\frac {3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{m+5}}{m+5}-\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+b \sin (c+d x))^{m+3}}{m+3}+\frac {6 a (a+b \sin (c+d x))^{m+6}}{m+6}-\frac {(a+b \sin (c+d x))^{m+7}}{m+7}}{b^7 d}\)

Input:

Int[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^m,x]
 

Output:

(-(((a^2 - b^2)^3*(a + b*Sin[c + d*x])^(1 + m))/(1 + m)) + (6*a*(a^2 - b^2 
)^2*(a + b*Sin[c + d*x])^(2 + m))/(2 + m) - (3*(5*a^4 - 6*a^2*b^2 + b^4)*( 
a + b*Sin[c + d*x])^(3 + m))/(3 + m) + (4*a*(5*a^2 - 3*b^2)*(a + b*Sin[c + 
 d*x])^(4 + m))/(4 + m) - (3*(5*a^2 - b^2)*(a + b*Sin[c + d*x])^(5 + m))/( 
5 + m) + (6*a*(a + b*Sin[c + d*x])^(6 + m))/(6 + m) - (a + b*Sin[c + d*x]) 
^(7 + m)/(7 + m))/(b^7*d)
 

Defintions of rubi rules used

rule 476

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, 
 x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

Maple [A] (verified)

Time = 18.76 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.97

method	result	size
parallelrisch	\(-\frac {720 \left (a +b \sin \left (d x +c \right )\right )^{m} \left (\frac {\left (1+m \right ) \left (2+m \right ) \left (-\frac {3 \left (4+m \right ) \left (m^{2}+16 m +\frac {245}{3}\right ) \left (6+m \right ) b^{4}}{640}-\frac {3 a^{2} m \left (m^{2}+23 m +92\right ) b^{2}}{80}+a^{4} m \right ) b^{3} \sin \left (3 d x +3 c \right )}{24}-\frac {\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 ) a \left (1+m \right ) m \,b^{2} \cos \left (2 d x +2 c \right )}{4}-\frac {\left (\left (\frac {5}{24} m^{2}+\frac {79}{24} m +\frac {49}{4}\right ) b^{2}+a^{2} m \right ) \left (3+m \right ) \left (4+m \right ) \left (1+m \right ) \left (2+m \right ) b^{5} \sin \left (5 d x +5 c \right )}{1920}+\frac {\left (3+m \right ) a \left (1+m \right ) \left (2+m \right ) m \left (\left (-\frac {1}{20} m^{2}-\frac {17}{20} m -\frac {16}{5}\right ) b^{2}+a^{2}\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 (4+m \right ) \left (2+m \right ) \left (m^{3}+\frac {93}{5} m^{2}+\frac {691}{5} m +735\right ) \left (6+m \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 )+a \left (\left (-7-\frac {1}{2304} m^{6}-\frac {49}{3840} m^{5}-\frac {1829}{11520} m^{4}-\frac {875}{768} m^{3}-\frac {27271}{5760} m^{2}-\frac {1169}{120} m \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 )\right )}{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 {\sin \left (d x +c \right )^{7} {\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 ) \sin \left (d x +c \right )^{5} {\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 ) \sin \left (d x +c \right )^{3} {\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 \sin \left (d x +c \right )^{6} {\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 \sin \left (d x +c \right )^{4} {\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 b^{2} a^{2}+840 b^{4}\right ) a m \sin \left (d x +c \right )^{2} {\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 {\sin \left (d x +c \right )^{7} {\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 ) \sin \left (d x +c \right )^{5} {\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 ) \sin \left (d x +c \right )^{3} {\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 \sin \left (d x +c \right )^{6} {\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 \sin \left (d x +c \right )^{4} {\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 b^{2} a^{2}+840 b^{4}\right ) a m \sin \left (d x +c \right )^{2} {\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 )}\)	\(877\)

Input:

int(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x,method=_RETURNVERBOSE)
 

Output:

-720*(a+b*sin(d*x+c))^m*(1/24*(1+m)*(2+m)*(-3/640*(4+m)*(m^2+16*m+245/3)*( 
6+m)*b^4-3/80*a^2*m*(m^2+23*m+92)*b^2+a^4*m)*b^3*sin(3*d*x+3*c)-1/4*((1/38 
4*m^4+67/960*m^3+1411/1920*m^2+637/192*m+417/80)*b^4-1/60*a^2*(m^2+53*m+22 
2)*b^2+a^4)*a*(1+m)*m*b^2*cos(2*d*x+2*c)-1/1920*((5/24*m^2+79/24*m+49/4)*b 
^2+a^2*m)*(3+m)*(4+m)*(1+m)*(2+m)*b^5*sin(5*d*x+5*c)+1/192*(3+m)*a*(1+m)*( 
2+m)*m*((-1/20*m^2-17/20*m-16/5)*b^2+a^2)*b^4*cos(4*d*x+4*c)-1/46080*b^7*( 
6+m)*(5+m)*(4+m)*(3+m)*(2+m)*(1+m)*sin(7*d*x+7*c)-1/23040*a*b^6*m*(5+m)*(4 
+m)*(3+m)*(2+m)*(1+m)*cos(6*d*x+6*c)-(1/9216*(4+m)*(2+m)*(m^3+93/5*m^2+691 
/5*m+735)*(6+m)*b^6+1/960*a^2*m*(m^4+34*m^3+611*m^2+3530*m+5832)*b^4+1/40* 
a^4*m*(m^2-37*m-158)*b^2+a^6*m)*b*sin(d*x+c)+a*((-7-1/2304*m^6-49/3840*m^5 
-1829/11520*m^4-875/768*m^3-27271/5760*m^2-1169/120*m)*b^6-1/960*a^2*(2+m) 
*(m^3+68*m^2-413*m-3360)*b^4+3/20*a^4*(m^2-7*m-28)*b^2+a^6))/b^7/(7+m)/(6+ 
m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (254) = 508\).

Time = 0.21 (sec) , antiderivative size = 814, normalized size of antiderivative = 3.20 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x, algorithm="fricas")
 

Output:

-(720*a^7 - 3024*a^5*b^2 + 5040*a^3*b^4 - 5040*a*b^6 - (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)*cos(d 
*x + c)^6 - 6*(2*a*b^6*m^5 - (5*a^3*b^4 - 23*a*b^6)*m^4 - 2*(15*a^3*b^4 - 
44*a*b^6)*m^3 - (55*a^3*b^4 - 133*a*b^6)*m^2 - 6*(5*a^3*b^4 - 11*a*b^6)*m) 
*cos(d*x + c)^4 - 192*(a^3*b^4 + a*b^6)*m^3 + 288*(a^5*b^2 - 2*a^3*b^4 - 7 
*a*b^6)*m^2 - 24*((a^3*b^4 + 3*a*b^6)*m^4 - 6*(a^3*b^4 - 5*a*b^6)*m^3 + (1 
5*a^5*b^2 - 55*a^3*b^4 + 84*a*b^6)*m^2 + 3*(5*a^5*b^2 - 16*a^3*b^4 + 19*a* 
b^6)*m)*cos(d*x + c)^2 - 192*(3*a^5*b^2 - 13*a^3*b^4 + 32*a*b^6)*m - (2304 
*b^7 + (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)*cos(d*x + c)^6 + 6*(144*b^7 + (a^2*b^5 + b^7)*m^5 + 
2*(5*a^2*b^5 + 8*b^7)*m^4 + 5*(7*a^2*b^5 + 19*b^7)*m^3 + 10*(5*a^2*b^5 + 2 
6*b^7)*m^2 + 12*(2*a^2*b^5 + 27*b^7)*m)*cos(d*x + c)^4 + 48*(a^4*b^3 + 6*a 
^2*b^5 + b^7)*m^3 - 576*(a^4*b^3 - 4*a^2*b^5 - b^7)*m^2 + 24*(48*b^7 + (3* 
a^2*b^5 + b^7)*m^4 - (5*a^4*b^3 - 24*a^2*b^5 - 13*b^7)*m^3 - (15*a^4*b^3 - 
 51*a^2*b^5 - 56*b^7)*m^2 - 2*(5*a^4*b^3 - 15*a^2*b^5 - 46*b^7)*m)*cos(d*x 
 + c)^2 + 48*(15*a^6*b - 58*a^4*b^3 + 87*a^2*b^5 + 44*b^7)*m)*sin(d*x + c) 
)*(b*sin(d*x + c) + a)^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)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**7*(a+b*sin(d*x+c))**m,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (254) = 508\).

Time = 0.05 (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} \] Input:

integrate(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x, algorithm="maxima")
 

Output:

((b*sin(d*x + c) + a)^(m + 1)/(b*(m + 1)) - 3*((m^2 + 3*m + 2)*b^3*sin(d*x 
 + c)^3 + (m^2 + m)*a*b^2*sin(d*x + c)^2 - 2*a^2*b*m*sin(d*x + c) + 2*a^3) 
*(b*sin(d*x + c) + a)^m/((m^3 + 6*m^2 + 11*m + 6)*b^3) + 3*((m^4 + 10*m^3 
+ 35*m^2 + 50*m + 24)*b^5*sin(d*x + c)^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a* 
b^4*sin(d*x + c)^4 - 4*(m^3 + 3*m^2 + 2*m)*a^2*b^3*sin(d*x + c)^3 + 12*(m^ 
2 + m)*a^3*b^2*sin(d*x + c)^2 - 24*a^4*b*m*sin(d*x + c) + 24*a^5)*(b*sin(d 
*x + c) + a)^m/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^5) - ((m 
^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*b^7*sin(d*x + c 
)^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*a*b^6*sin(d*x + 
c)^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^2*b^5*sin(d*x + c)^5 + 
30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^3*b^4*sin(d*x + c)^4 - 120*(m^3 + 3*m^2 
+ 2*m)*a^4*b^3*sin(d*x + c)^3 + 360*(m^2 + m)*a^5*b^2*sin(d*x + c)^2 - 720 
*a^6*b*m*sin(d*x + c) + 720*a^7)*(b*sin(d*x + c) + a)^m/((m^7 + 28*m^6 + 3 
22*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*b^7))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3579 vs. \(2 (254) = 508\).

Time = 0.17 (sec) , antiderivative size = 3579, normalized size of antiderivative = 14.09 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x, algorithm="giac")
 

Output:

-((b*sin(d*x + c) + a)^7*(b*sin(d*x + c) + a)^m*m^6 - 6*(b*sin(d*x + c) + 
a)^6*(b*sin(d*x + c) + a)^m*a*m^6 + 15*(b*sin(d*x + c) + a)^5*(b*sin(d*x + 
 c) + a)^m*a^2*m^6 - 20*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a^3* 
m^6 + 15*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^4*m^6 - 6*(b*sin( 
d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a^5*m^6 + (b*sin(d*x + c) + a)*(b*s 
in(d*x + c) + a)^m*a^6*m^6 - 3*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a) 
^m*b^2*m^6 + 12*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a*b^2*m^6 - 
18*(b*sin(d*x + c) + a)^3*(b*sin(d*x + c) + a)^m*a^2*b^2*m^6 + 12*(b*sin(d 
*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a^3*b^2*m^6 - 3*(b*sin(d*x + c) + a) 
*(b*sin(d*x + c) + a)^m*a^4*b^2*m^6 + 3*(b*sin(d*x + c) + a)^3*(b*sin(d*x 
+ c) + a)^m*b^4*m^6 - 6*(b*sin(d*x + c) + a)^2*(b*sin(d*x + c) + a)^m*a*b^ 
4*m^6 + 3*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a)^m*a^2*b^4*m^6 - (b*sin 
(d*x + c) + a)*(b*sin(d*x + c) + a)^m*b^6*m^6 + 21*(b*sin(d*x + c) + a)^7* 
(b*sin(d*x + c) + a)^m*m^5 - 132*(b*sin(d*x + c) + a)^6*(b*sin(d*x + c) + 
a)^m*a*m^5 + 345*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*a^2*m^5 - 4 
80*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a^3*m^5 + 375*(b*sin(d*x 
+ c) + a)^3*(b*sin(d*x + c) + a)^m*a^4*m^5 - 156*(b*sin(d*x + c) + a)^2*(b 
*sin(d*x + c) + a)^m*a^5*m^5 + 27*(b*sin(d*x + c) + a)*(b*sin(d*x + c) + a 
)^m*a^6*m^5 - 69*(b*sin(d*x + c) + a)^5*(b*sin(d*x + c) + a)^m*b^2*m^5 + 2 
88*(b*sin(d*x + c) + a)^4*(b*sin(d*x + c) + a)^m*a*b^2*m^5 - 450*(b*sin...
 

Mupad [B] (verification not implemented)

Time = 27.15 (sec) , antiderivative size = 1196, normalized size of antiderivative = 4.71 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\text {Too large to display} \] Input:

int(cos(c + d*x)^7*(a + b*sin(c + d*x))^m,x)
 

Output:

((a + b*sin(c + d*x))^m*(a*b^6*645120i - a^7*92160i - a^3*b^4*645120i + a^ 
5*b^2*387072i - a^3*b^4*m*401856i + a^5*b^2*m*96768i + a*b^6*m^2*436336i + 
 a*b^6*m^3*105000i + a*b^6*m^4*14632i + a*b^6*m^5*1176i + a*b^6*m^6*40i - 
a^3*b^4*m^2*26592i - a^5*b^2*m^2*13824i + a^3*b^4*m^3*6720i + a^3*b^4*m^4* 
96i + a*b^6*m*897792i))/(128*b^7*d*(m*13068i + m^2*13132i + m^3*6769i + m^ 
4*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i)) + (sin(7*c + 7*d*x)*(a + b 
*sin(c + d*x))^m*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 7 
20)*1i)/(64*d*(m*13068i + m^2*13132i + m^3*6769i + m^4*1960i + m^5*322i + 
m^6*28i + m^7*1i + 5040i)) + (sin(c + d*x)*(a + b*sin(c + d*x))^m*(194868* 
b^7*m + 176400*b^7 + 78968*b^7*m^2 + 16299*b^7*m^3 + 2027*b^7*m^4 + 153*b^ 
7*m^5 + 5*b^7*m^6 + 279936*a^2*b^5*m - 182016*a^4*b^3*m + 169440*a^2*b^5*m 
^2 - 42624*a^4*b^3*m^2 + 29328*a^2*b^5*m^3 + 1152*a^4*b^3*m^3 + 1632*a^2*b 
^5*m^4 + 48*a^2*b^5*m^5 + 46080*a^6*b*m)*1i)/(64*b^7*d*(m*13068i + m^2*131 
32i + m^3*6769i + m^4*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i)) + (sin 
(3*c + 3*d*x)*(a + b*sin(c + d*x))^m*(3*m + m^2 + 2)*(3602*b^4*m - 640*a^4 
*m + 5880*b^4 + 797*b^4*m^2 + 78*b^4*m^3 + 3*b^4*m^4 + 2208*a^2*b^2*m + 55 
2*a^2*b^2*m^2 + 24*a^2*b^2*m^3)*3i)/(64*b^4*d*(m*13068i + m^2*13132i + m^3 
*6769i + m^4*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i)) + (sin(5*c + 5* 
d*x)*(a + b*sin(c + d*x))^m*(24*a^2*m + 79*b^2*m + 294*b^2 + 5*b^2*m^2)*(5 
0*m + 35*m^2 + 10*m^3 + m^4 + 24)*1i)/(64*b^2*d*(m*13068i + m^2*13132i ...
 

Reduce [F]

\[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx=\int \cos \left (d x +c \right )^{7} \left (\sin \left (d x +c \right ) b +a \right )^{m}d x \] Input:

int(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x)
 

Output:

int(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x)