Integrand size = 21, antiderivative size = 109 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {8 (a+a \sin (c+d x))^{4+m}}{a^4 d (4+m)}-\frac {12 (a+a \sin (c+d x))^{5+m}}{a^5 d (5+m)}+\frac {6 (a+a \sin (c+d x))^{6+m}}{a^6 d (6+m)}-\frac {(a+a \sin (c+d x))^{7+m}}{a^7 d (7+m)} \] Output:
8*(a+a*sin(d*x+c))^(4+m)/a^4/d/(4+m)-12*(a+a*sin(d*x+c))^(5+m)/a^5/d/(5+m) +6*(a+a*sin(d*x+c))^(6+m)/a^6/d/(6+m)-(a+a*sin(d*x+c))^(7+m)/a^7/d/(7+m)
Time = 0.72 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {(a (1+\sin (c+d x)))^{4+m} \left (\frac {8 a^3}{4+m}-\frac {12 a^3 (1+\sin (c+d x))}{5+m}+\frac {6 a^3 (1+\sin (c+d x))^2}{6+m}-\frac {(a+a \sin (c+d x))^3}{7+m}\right )}{a^7 d} \] Input:
Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^m,x]
Output:
((a*(1 + Sin[c + d*x]))^(4 + m)*((8*a^3)/(4 + m) - (12*a^3*(1 + Sin[c + d* x]))/(5 + m) + (6*a^3*(1 + Sin[c + d*x])^2)/(6 + m) - (a + a*Sin[c + d*x]) ^3/(7 + m)))/(a^7*d)
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3146, 53, 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.
| \(\displaystyle \int \cos ^7(c+d x) (a \sin (c+d x)+a)^m \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^7 (a \sin (c+d x)+a)^mdx\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {\int (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^{m+3}d(a \sin (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\int \left (8 a^3 (\sin (c+d x) a+a)^{m+3}-12 a^2 (\sin (c+d x) a+a)^{m+4}+6 a (\sin (c+d x) a+a)^{m+5}-(\sin (c+d x) a+a)^{m+6}\right )d(a \sin (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {8 a^3 (a \sin (c+d x)+a)^{m+4}}{m+4}-\frac {12 a^2 (a \sin (c+d x)+a)^{m+5}}{m+5}+\frac {6 a (a \sin (c+d x)+a)^{m+6}}{m+6}-\frac {(a \sin (c+d x)+a)^{m+7}}{m+7}}{a^7 d}\) |
Input:
Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^m,x]
Output:
((8*a^3*(a + a*Sin[c + d*x])^(4 + m))/(4 + m) - (12*a^2*(a + a*Sin[c + d*x ])^(5 + m))/(5 + m) + (6*a*(a + a*Sin[c + d*x])^(6 + m))/(6 + m) - (a + a* Sin[c + d*x])^(7 + m)/(7 + m))/(a^7*d)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
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 + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Time = 7.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.94
| method | result | size |
| parallelrisch | \(\frac {9 \left (\frac {8192}{3}+\left (\frac {1960}{3}+31 m^{2}+m^{3}+\frac {1070}{3} m \right ) \sin \left (3 d x +3 c \right )+\frac {\left (392+41 m^{2}+\frac {5}{3} m^{3}+\frac {706}{3} m \right ) \sin \left (5 d x +5 c \right )}{3}+\frac {\left (5+m \right ) \left (4+m \right ) \left (6+m \right ) \sin \left (7 d x +7 c \right )}{9}+2 \left (\frac {109}{3} m^{2}+\frac {5}{3} m^{3}+268 m \right ) \cos \left (2 d x +2 c \right )+\frac {4 m \left (m^{2}+17 m +44\right ) \cos \left (4 d x +4 c \right )}{3}+2 \left (m^{2}+\frac {20}{9} m +\frac {1}{9} m^{3}\right ) \cos \left (6 d x +6 c \right )+\left (19 m^{2}+\frac {9800}{3}+\frac {5}{9} m^{3}+\frac {2578}{9} m \right ) \sin \left (d x +c \right )+\frac {4336 m}{9}+\frac {20 m^{3}}{9}+52 m^{2}\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{m}}{64 \left (5+m \right ) \left (4+m \right ) \left (7+m \right ) \left (6+m \right ) d}\) | \(212\) |
| derivativedivides | \(\frac {\left (m^{3}+21 m^{2}+152 m +384\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}+\frac {\left (m^{3}+27 m^{2}+254 m +840\right ) \sin \left (d x +c \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {\sin \left (d x +c \right )^{7} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (7+m \right )}-\frac {3 \left (m^{3}+23 m^{2}+162 m +280\right ) \sin \left (d x +c \right )^{3} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {m \sin \left (d x +c \right )^{6} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{2}+13 m +42\right )}+\frac {3 \left (m^{2}+15 m +42\right ) \sin \left (d x +c \right )^{5} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{3}+18 m^{2}+107 m +210\right )}+\frac {3 m \left (m^{2}+13 m +32\right ) \sin \left (d x +c \right )^{4} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {3 m \left (m^{2}+17 m +76\right ) \sin \left (d x +c \right )^{2} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}\) | \(402\) |
| default | \(\frac {\left (m^{3}+21 m^{2}+152 m +384\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}+\frac {\left (m^{3}+27 m^{2}+254 m +840\right ) \sin \left (d x +c \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {\sin \left (d x +c \right )^{7} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (7+m \right )}-\frac {3 \left (m^{3}+23 m^{2}+162 m +280\right ) \sin \left (d x +c \right )^{3} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {m \sin \left (d x +c \right )^{6} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{2}+13 m +42\right )}+\frac {3 \left (m^{2}+15 m +42\right ) \sin \left (d x +c \right )^{5} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{3}+18 m^{2}+107 m +210\right )}+\frac {3 m \left (m^{2}+13 m +32\right ) \sin \left (d x +c \right )^{4} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {3 m \left (m^{2}+17 m +76\right ) \sin \left (d x +c \right )^{2} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (d x +c \right )\right )\right )}}{d \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}\) | \(402\) |
Input:
int(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x,method=_RETURNVERBOSE)
Output:
9/64*(8192/3+(1960/3+31*m^2+m^3+1070/3*m)*sin(3*d*x+3*c)+1/3*(392+41*m^2+5 /3*m^3+706/3*m)*sin(5*d*x+5*c)+1/9*(5+m)*(4+m)*(6+m)*sin(7*d*x+7*c)+2*(109 /3*m^2+5/3*m^3+268*m)*cos(2*d*x+2*c)+4/3*m*(m^2+17*m+44)*cos(4*d*x+4*c)+2* (m^2+20/9*m+1/9*m^3)*cos(6*d*x+6*c)+(19*m^2+9800/3+5/9*m^3+2578/9*m)*sin(d *x+c)+4336/9*m+20/9*m^3+52*m^2)*(a*(1+sin(d*x+c)))^m/(5+m)/(4+m)/(7+m)/(6+ m)/d
Time = 0.11 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {{\left ({\left (m^{3} + 9 \, m^{2} + 20 \, m\right )} \cos \left (d x + c\right )^{6} + 12 \, {\left (m^{2} + 3 \, m\right )} \cos \left (d x + c\right )^{4} + 96 \, m \cos \left (d x + c\right )^{2} + {\left ({\left (m^{3} + 15 \, m^{2} + 74 \, m + 120\right )} \cos \left (d x + c\right )^{6} + 12 \, {\left (m^{2} + 7 \, m + 12\right )} \cos \left (d x + c\right )^{4} + 96 \, {\left (m + 2\right )} \cos \left (d x + c\right )^{2} + 384\right )} \sin \left (d x + c\right ) + 384\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{4} + 22 \, d m^{3} + 179 \, d m^{2} + 638 \, d m + 840 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x, algorithm="fricas")
Output:
((m^3 + 9*m^2 + 20*m)*cos(d*x + c)^6 + 12*(m^2 + 3*m)*cos(d*x + c)^4 + 96* m*cos(d*x + c)^2 + ((m^3 + 15*m^2 + 74*m + 120)*cos(d*x + c)^6 + 12*(m^2 + 7*m + 12)*cos(d*x + c)^4 + 96*(m + 2)*cos(d*x + c)^2 + 384)*sin(d*x + c) + 384)*(a*sin(d*x + c) + a)^m/(d*m^4 + 22*d*m^3 + 179*d*m^2 + 638*d*m + 84 0*d)
Timed out. \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**7*(a+a*sin(d*x+c))**m,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (109) = 218\).
Time = 0.04 (sec) , antiderivative size = 520, normalized size of antiderivative = 4.77 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {\frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} a^{m} \sin \left (d x + c\right )^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} a^{m} \sin \left (d x + c\right )^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{m} \sin \left (d x + c\right )^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 360 \, {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 720 \, a^{m} m \sin \left (d x + c\right ) + 720 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} - \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (d x + c\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 24 \, a^{m} m \sin \left (d x + c\right ) + 24 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (d x + c\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 2 \, a^{m} m \sin \left (d x + c\right ) + 2 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} - \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m + 1}}{a {\left (m + 1\right )}}}{d} \] Input:
integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x, algorithm="maxima")
Output:
-(((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*a^m*sin(d* x + c)^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*a^m*sin(d*x + c)^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^m*sin(d*x + c)^5 + 3 0*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(d*x + c)^4 - 120*(m^3 + 3*m^2 + 2*m )*a^m*sin(d*x + c)^3 + 360*(m^2 + m)*a^m*sin(d*x + c)^2 - 720*a^m*m*sin(d* x + c) + 720*a^m)*(sin(d*x + c) + 1)^m/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040) - 3*((m^4 + 10*m^3 + 35*m^2 + 50* m + 24)*a^m*sin(d*x + c)^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(d*x + c) ^4 - 4*(m^3 + 3*m^2 + 2*m)*a^m*sin(d*x + c)^3 + 12*(m^2 + m)*a^m*sin(d*x + c)^2 - 24*a^m*m*sin(d*x + c) + 24*a^m)*(sin(d*x + c) + 1)^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + 3*((m^2 + 3*m + 2)*a^m*sin(d*x + c)^3 + (m^2 + m)*a^m*sin(d*x + c)^2 - 2*a^m*m*sin(d*x + c) + 2*a^m)*(sin(d*x + c) + 1)^m/(m^3 + 6*m^2 + 11*m + 6) - (a*sin(d*x + c) + a)^(m + 1)/(a*(m + 1)))/d
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (109) = 218\).
Time = 0.13 (sec) , antiderivative size = 517, normalized size of antiderivative = 4.74 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{3} - 6 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{3} + 12 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{3} - 8 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{3} + 15 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} - 96 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{2} + 204 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{2} - 144 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{2} + 74 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m - 498 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m + 1128 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m - 856 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m + 120 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} - 840 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a + 2016 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} - 1680 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3}}{{\left (a^{6} m^{4} + 22 \, a^{6} m^{3} + 179 \, a^{6} m^{2} + 638 \, a^{6} m + 840 \, a^{6}\right )} a d} \] Input:
integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x, algorithm="giac")
Output:
-((a*sin(d*x + c) + a)^7*(a*sin(d*x + c) + a)^m*m^3 - 6*(a*sin(d*x + c) + a)^6*(a*sin(d*x + c) + a)^m*a*m^3 + 12*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*a^2*m^3 - 8*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a^3*m ^3 + 15*(a*sin(d*x + c) + a)^7*(a*sin(d*x + c) + a)^m*m^2 - 96*(a*sin(d*x + c) + a)^6*(a*sin(d*x + c) + a)^m*a*m^2 + 204*(a*sin(d*x + c) + a)^5*(a*s in(d*x + c) + a)^m*a^2*m^2 - 144*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a^3*m^2 + 74*(a*sin(d*x + c) + a)^7*(a*sin(d*x + c) + a)^m*m - 498*(a *sin(d*x + c) + a)^6*(a*sin(d*x + c) + a)^m*a*m + 1128*(a*sin(d*x + c) + a )^5*(a*sin(d*x + c) + a)^m*a^2*m - 856*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a^3*m + 120*(a*sin(d*x + c) + a)^7*(a*sin(d*x + c) + a)^m - 840 *(a*sin(d*x + c) + a)^6*(a*sin(d*x + c) + a)^m*a + 2016*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*a^2 - 1680*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a^3)/((a^6*m^4 + 22*a^6*m^3 + 179*a^6*m^2 + 638*a^6*m + 840*a^6 )*a*d)
Time = 19.49 (sec) , antiderivative size = 555, normalized size of antiderivative = 5.09 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^7*(a + a*sin(c + d*x))^m,x)
Output:
exp(- c*7i - d*x*7i)*(a + a*sin(c + d*x))^m*((exp(c*7i + d*x*7i)*(m*8672i + m^2*936i + m^3*40i + 49152i))/(128*d*(m*638i + m^2*179i + m^3*22i + m^4* 1i + 840i)) + (exp(c*7i + d*x*7i)*cos(2*c + 2*d*x)*(m*4824i + m^2*654i + m ^3*30i))/(64*d*(m*638i + m^2*179i + m^3*22i + m^4*1i + 840i)) + (exp(c*7i + d*x*7i)*sin(5*c + 5*d*x)*(706*m + 123*m^2 + 5*m^3 + 1176)*1i)/(64*d*(m*6 38i + m^2*179i + m^3*22i + m^4*1i + 840i)) + (exp(c*7i + d*x*7i)*sin(3*c + 3*d*x)*(3210*m + 279*m^2 + 9*m^3 + 5880)*1i)/(64*d*(m*638i + m^2*179i + m ^3*22i + m^4*1i + 840i)) + (exp(c*7i + d*x*7i)*sin(7*c + 7*d*x)*(74*m + 15 *m^2 + m^3 + 120)*1i)/(64*d*(m*638i + m^2*179i + m^3*22i + m^4*1i + 840i)) + (exp(c*7i + d*x*7i)*sin(c + d*x)*(2578*m + 171*m^2 + 5*m^3 + 29400)*1i) /(64*d*(m*638i + m^2*179i + m^3*22i + m^4*1i + 840i)) + (m*exp(c*7i + d*x* 7i)*cos(6*c + 6*d*x)*(m*9i + m^2*1i + 20i))/(32*d*(m*638i + m^2*179i + m^3 *22i + m^4*1i + 840i)) + (3*m*exp(c*7i + d*x*7i)*cos(4*c + 4*d*x)*(m*17i + m^2*1i + 44i))/(16*d*(m*638i + m^2*179i + m^3*22i + m^4*1i + 840i)))
\[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx=\int \cos \left (d x +c \right )^{7} \left (\sin \left (d x +c \right ) a +a \right )^{m}d x \] Input:
int(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x)
Output:
int(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x)