Integrand size = 29, antiderivative size = 184 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {2 a^2 \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {2 a^2 \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {6 a^2 \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {6 a^2 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac {2 a^2 \sin ^{7+n}(c+d x)}{d (7+n)}-\frac {2 a^2 \sin ^{8+n}(c+d x)}{d (8+n)}-\frac {a^2 \sin ^{9+n}(c+d x)}{d (9+n)} \] Output:
a^2*sin(d*x+c)^(1+n)/d/(1+n)+2*a^2*sin(d*x+c)^(2+n)/d/(2+n)-2*a^2*sin(d*x+ c)^(3+n)/d/(3+n)-6*a^2*sin(d*x+c)^(4+n)/d/(4+n)+6*a^2*sin(d*x+c)^(6+n)/d/( 6+n)+2*a^2*sin(d*x+c)^(7+n)/d/(7+n)-2*a^2*sin(d*x+c)^(8+n)/d/(8+n)-a^2*sin (d*x+c)^(9+n)/d/(9+n)
Time = 0.85 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.68 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \sin ^{1+n}(c+d x) \left (\frac {1}{1+n}+\frac {2 \sin (c+d x)}{2+n}-\frac {2 \sin ^2(c+d x)}{3+n}-\frac {6 \sin ^3(c+d x)}{4+n}+\frac {6 \sin ^5(c+d x)}{6+n}+\frac {2 \sin ^6(c+d x)}{7+n}-\frac {2 \sin ^7(c+d x)}{8+n}-\frac {\sin ^8(c+d x)}{9+n}\right )}{d} \] Input:
Integrate[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]
Output:
(a^2*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (2*Sin[c + d*x])/(2 + n) - (2*Si n[c + d*x]^2)/(3 + n) - (6*Sin[c + d*x]^3)/(4 + n) + (6*Sin[c + d*x]^5)/(6 + n) + (2*Sin[c + d*x]^6)/(7 + n) - (2*Sin[c + d*x]^7)/(8 + n) - Sin[c + d*x]^8/(9 + n)))/d
Time = 0.44 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3042, 3315, 99, 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.
| \(\displaystyle \int \cos ^7(c+d x) (a \sin (c+d x)+a)^2 \sin ^n(c+d x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^7 (a \sin (c+d x)+a)^2 \sin (c+d x)^ndx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \sin ^n(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^5d(a \sin (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (a^8 \sin ^n(c+d x)+2 a^8 \sin ^{n+1}(c+d x)-2 a^8 \sin ^{n+2}(c+d x)-6 a^8 \sin ^{n+3}(c+d x)+6 a^8 \sin ^{n+5}(c+d x)+2 a^8 \sin ^{n+6}(c+d x)-2 a^8 \sin ^{n+7}(c+d x)-a^8 \sin ^{n+8}(c+d x)\right )d(a \sin (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^9 \sin ^{n+1}(c+d x)}{n+1}+\frac {2 a^9 \sin ^{n+2}(c+d x)}{n+2}-\frac {2 a^9 \sin ^{n+3}(c+d x)}{n+3}-\frac {6 a^9 \sin ^{n+4}(c+d x)}{n+4}+\frac {6 a^9 \sin ^{n+6}(c+d x)}{n+6}+\frac {2 a^9 \sin ^{n+7}(c+d x)}{n+7}-\frac {2 a^9 \sin ^{n+8}(c+d x)}{n+8}-\frac {a^9 \sin ^{n+9}(c+d x)}{n+9}}{a^7 d}\) |
Input:
Int[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]
Output:
((a^9*Sin[c + d*x]^(1 + n))/(1 + n) + (2*a^9*Sin[c + d*x]^(2 + n))/(2 + n) - (2*a^9*Sin[c + d*x]^(3 + n))/(3 + n) - (6*a^9*Sin[c + d*x]^(4 + n))/(4 + n) + (6*a^9*Sin[c + d*x]^(6 + n))/(6 + n) + (2*a^9*Sin[c + d*x]^(7 + n)) /(7 + n) - (2*a^9*Sin[c + d*x]^(8 + n))/(8 + n) - (a^9*Sin[c + d*x]^(9 + n ))/(9 + n))/(a^7*d)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(378\) vs. \(2(184)=368\).
Time = 17.98 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.06
| method | result | size |
| parallelrisch | \(\frac {13 \left (\frac {8 \left (n -4\right ) \left (9+n \right ) \left (7+n \right ) \left (3+n \right ) \left (1+n \right ) \left (n^{2}+22 n +168\right ) \cos \left (2 d x +2 c \right )}{13}-\frac {8 \left (9+n \right ) \left (7+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \left (n^{2}+22 n +168\right ) \cos \left (4 d x +4 c \right )}{13}-\frac {8 \left (n +12\right ) \left (9+n \right ) \left (7+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (6 d x +6 c \right )}{13}-\frac {2 \left (9+n \right ) \left (7+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (8 d x +8 c \right )}{13}+\frac {22 \left (2+n \right ) \left (1+n \right ) \left (8+n \right ) \left (n^{2}+\frac {218}{11} n +\frac {1323}{11}\right ) \left (4+n \right ) \left (6+n \right ) \sin \left (3 d x +3 c \right )}{13}+\frac {10 \left (2+n \right ) \left (1+n \right ) \left (8+n \right ) \left (3+n \right ) \left (4+n \right ) \left (n +\frac {63}{5}\right ) \left (6+n \right ) \sin \left (5 d x +5 c \right )}{13}+\frac {\left (n -9\right ) \left (8+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (7 d x +7 c \right )}{26}-\frac {\left (8+n \right ) \left (7+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (9 d x +9 c \right )}{26}+\left (2+n \right ) \left (8+n \right ) \left (4+n \right ) \left (n^{3}+\frac {295}{13} n^{2}+\frac {2547}{13} n +\frac {14553}{13}\right ) \left (6+n \right ) \sin \left (d x +c \right )+\frac {10 \left (9+n \right ) \left (1+n \right ) \left (n^{3}+\frac {108}{5} n^{2}+\frac {892}{5} n +\frac {4464}{5}\right ) \left (3+n \right ) \left (7+n \right )}{13}\right ) \sin \left (d x +c \right )^{n} a^{2}}{128 \left (4+n \right ) \left (2+n \right ) \left (9+n \right ) \left (3+n \right ) \left (7+n \right ) \left (1+n \right ) \left (8+n \right ) d \left (6+n \right )}\) | \(379\) |
Input:
int(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
13/128*(8/13*(n-4)*(9+n)*(7+n)*(3+n)*(1+n)*(n^2+22*n+168)*cos(2*d*x+2*c)-8 /13*(9+n)*(7+n)*(3+n)*(2+n)*(1+n)*(n^2+22*n+168)*cos(4*d*x+4*c)-8/13*(n+12 )*(9+n)*(7+n)*(4+n)*(3+n)*(2+n)*(1+n)*cos(6*d*x+6*c)-2/13*(9+n)*(7+n)*(6+n )*(4+n)*(3+n)*(2+n)*(1+n)*cos(8*d*x+8*c)+22/13*(2+n)*(1+n)*(8+n)*(n^2+218/ 11*n+1323/11)*(4+n)*(6+n)*sin(3*d*x+3*c)+10/13*(2+n)*(1+n)*(8+n)*(3+n)*(4+ n)*(n+63/5)*(6+n)*sin(5*d*x+5*c)+1/26*(n-9)*(8+n)*(6+n)*(4+n)*(3+n)*(2+n)* (1+n)*sin(7*d*x+7*c)-1/26*(8+n)*(7+n)*(6+n)*(4+n)*(3+n)*(2+n)*(1+n)*sin(9* d*x+9*c)+(2+n)*(8+n)*(4+n)*(n^3+295/13*n^2+2547/13*n+14553/13)*(6+n)*sin(d *x+c)+10/13*(9+n)*(1+n)*(n^3+108/5*n^2+892/5*n+4464/5)*(3+n)*(7+n))*sin(d* x+c)^n*a^2/(4+n)/(2+n)/(9+n)/(3+n)/(7+n)/(1+n)/(8+n)/d/(6+n)
Leaf count of result is larger than twice the leaf count of optimal. 628 vs. \(2 (184) = 368\).
Time = 0.15 (sec) , antiderivative size = 628, normalized size of antiderivative = 3.41 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {{\left (2 \, {\left (a^{2} n^{7} + 32 \, a^{2} n^{6} + 414 \, a^{2} n^{5} + 2788 \, a^{2} n^{4} + 10469 \, a^{2} n^{3} + 21708 \, a^{2} n^{2} + 22716 \, a^{2} n + 9072 \, a^{2}\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (a^{2} n^{7} + 26 \, a^{2} n^{6} + 258 \, a^{2} n^{5} + 1240 \, a^{2} n^{4} + 3029 \, a^{2} n^{3} + 3534 \, a^{2} n^{2} + 1512 \, a^{2} n\right )} \cos \left (d x + c\right )^{6} - 96 \, a^{2} n^{4} - 1920 \, a^{2} n^{3} - 12 \, {\left (a^{2} n^{6} + 22 \, a^{2} n^{5} + 170 \, a^{2} n^{4} + 560 \, a^{2} n^{3} + 789 \, a^{2} n^{2} + 378 \, a^{2} n\right )} \cos \left (d x + c\right )^{4} - 12480 \, a^{2} n^{2} - 28800 \, a^{2} n - 48 \, {\left (a^{2} n^{5} + 20 \, a^{2} n^{4} + 130 \, a^{2} n^{3} + 300 \, a^{2} n^{2} + 189 \, a^{2} n\right )} \cos \left (d x + c\right )^{2} - 18144 \, a^{2} + {\left ({\left (a^{2} n^{7} + 31 \, a^{2} n^{6} + 391 \, a^{2} n^{5} + 2581 \, a^{2} n^{4} + 9544 \, a^{2} n^{3} + 19564 \, a^{2} n^{2} + 20304 \, a^{2} n + 8064 \, a^{2}\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (a^{2} n^{7} + 29 \, a^{2} n^{6} + 343 \, a^{2} n^{5} + 2135 \, a^{2} n^{4} + 7504 \, a^{2} n^{3} + 14756 \, a^{2} n^{2} + 14832 \, a^{2} n + 5760 \, a^{2}\right )} \cos \left (d x + c\right )^{6} - 96 \, a^{2} n^{4} - 1920 \, a^{2} n^{3} - 12 \, {\left (a^{2} n^{6} + 24 \, a^{2} n^{5} + 223 \, a^{2} n^{4} + 1020 \, a^{2} n^{3} + 2404 \, a^{2} n^{2} + 2736 \, a^{2} n + 1152 \, a^{2}\right )} \cos \left (d x + c\right )^{4} - 13440 \, a^{2} n^{2} - 38400 \, a^{2} n - 48 \, {\left (a^{2} n^{5} + 21 \, a^{2} n^{4} + 160 \, a^{2} n^{3} + 540 \, a^{2} n^{2} + 784 \, a^{2} n + 384 \, a^{2}\right )} \cos \left (d x + c\right )^{2} - 36864 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 40 \, d n^{7} + 670 \, d n^{6} + 6100 \, d n^{5} + 32773 \, d n^{4} + 105460 \, d n^{3} + 196380 \, d n^{2} + 190800 \, d n + 72576 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-(2*(a^2*n^7 + 32*a^2*n^6 + 414*a^2*n^5 + 2788*a^2*n^4 + 10469*a^2*n^3 + 2 1708*a^2*n^2 + 22716*a^2*n + 9072*a^2)*cos(d*x + c)^8 - 2*(a^2*n^7 + 26*a^ 2*n^6 + 258*a^2*n^5 + 1240*a^2*n^4 + 3029*a^2*n^3 + 3534*a^2*n^2 + 1512*a^ 2*n)*cos(d*x + c)^6 - 96*a^2*n^4 - 1920*a^2*n^3 - 12*(a^2*n^6 + 22*a^2*n^5 + 170*a^2*n^4 + 560*a^2*n^3 + 789*a^2*n^2 + 378*a^2*n)*cos(d*x + c)^4 - 1 2480*a^2*n^2 - 28800*a^2*n - 48*(a^2*n^5 + 20*a^2*n^4 + 130*a^2*n^3 + 300* a^2*n^2 + 189*a^2*n)*cos(d*x + c)^2 - 18144*a^2 + ((a^2*n^7 + 31*a^2*n^6 + 391*a^2*n^5 + 2581*a^2*n^4 + 9544*a^2*n^3 + 19564*a^2*n^2 + 20304*a^2*n + 8064*a^2)*cos(d*x + c)^8 - 2*(a^2*n^7 + 29*a^2*n^6 + 343*a^2*n^5 + 2135*a ^2*n^4 + 7504*a^2*n^3 + 14756*a^2*n^2 + 14832*a^2*n + 5760*a^2)*cos(d*x + c)^6 - 96*a^2*n^4 - 1920*a^2*n^3 - 12*(a^2*n^6 + 24*a^2*n^5 + 223*a^2*n^4 + 1020*a^2*n^3 + 2404*a^2*n^2 + 2736*a^2*n + 1152*a^2)*cos(d*x + c)^4 - 13 440*a^2*n^2 - 38400*a^2*n - 48*(a^2*n^5 + 21*a^2*n^4 + 160*a^2*n^3 + 540*a ^2*n^2 + 784*a^2*n + 384*a^2)*cos(d*x + c)^2 - 36864*a^2)*sin(d*x + c))*si n(d*x + c)^n/(d*n^8 + 40*d*n^7 + 670*d*n^6 + 6100*d*n^5 + 32773*d*n^4 + 10 5460*d*n^3 + 196380*d*n^2 + 190800*d*n + 72576*d)
Leaf count of result is larger than twice the leaf count of optimal. 29818 vs. \(2 (158) = 316\).
Time = 49.89 (sec) , antiderivative size = 29818, normalized size of antiderivative = 162.05 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**7*sin(d*x+c)**n*(a+a*sin(d*x+c))**2,x)
Output:
Piecewise((x*(a*sin(c) + a)**2*sin(c)**n*cos(c)**7, Eq(d, 0)), (-a**2*log( sin(c + d*x))/d + 32*a**2/(35*d*sin(c + d*x)) - a**2*cos(c + d*x)**2/(2*d* sin(c + d*x)**2) + a**2/(8*d*sin(c + d*x)**2) - 16*a**2*cos(c + d*x)**2/(3 5*d*sin(c + d*x)**3) + a**2*cos(c + d*x)**4/(4*d*sin(c + d*x)**4) - a**2*c os(c + d*x)**2/(8*d*sin(c + d*x)**4) + 12*a**2*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - a**2*cos(c + d*x)**6/(6*d*sin(c + d*x)**6) + a**2*cos(c + d* x)**4/(8*d*sin(c + d*x)**6) - 2*a**2*cos(c + d*x)**6/(7*d*sin(c + d*x)**7) - a**2*cos(c + d*x)**6/(8*d*sin(c + d*x)**8), Eq(n, -9)), (-2*a**2*log(si n(c + d*x))/d - 16*a**2*sin(c + d*x)/(5*d) - 8*a**2*cos(c + d*x)**2/(5*d*s in(c + d*x)) + 16*a**2/(35*d*sin(c + d*x)) - a**2*cos(c + d*x)**2/(d*sin(c + d*x)**2) + 2*a**2*cos(c + d*x)**4/(5*d*sin(c + d*x)**3) - 8*a**2*cos(c + d*x)**2/(35*d*sin(c + d*x)**3) + a**2*cos(c + d*x)**4/(2*d*sin(c + d*x)* *4) - a**2*cos(c + d*x)**6/(5*d*sin(c + d*x)**5) + 6*a**2*cos(c + d*x)**4/ (35*d*sin(c + d*x)**5) - a**2*cos(c + d*x)**6/(3*d*sin(c + d*x)**6) - a**2 *cos(c + d*x)**6/(7*d*sin(c + d*x)**7), Eq(n, -8)), (-3840*a**2*log(tan(c/ 2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**10/(1920*d*tan(c/2 + d*x/2)**10 + 384 0*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) - 7680*a**2*log(tan( c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**8/(1920*d*tan(c/2 + d*x/2)**10 + 38 40*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) - 3840*a**2*log(tan (c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(1920*d*tan(c/2 + d*x/2)**10 ...
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {\frac {a^{2} \sin \left (d x + c\right )^{n + 9}}{n + 9} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 8}}{n + 8} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 7}}{n + 7} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{n + 6}}{n + 6} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 3}}{n + 3} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a^{2} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-(a^2*sin(d*x + c)^(n + 9)/(n + 9) + 2*a^2*sin(d*x + c)^(n + 8)/(n + 8) - 2*a^2*sin(d*x + c)^(n + 7)/(n + 7) - 6*a^2*sin(d*x + c)^(n + 6)/(n + 6) + 6*a^2*sin(d*x + c)^(n + 4)/(n + 4) + 2*a^2*sin(d*x + c)^(n + 3)/(n + 3) - 2*a^2*sin(d*x + c)^(n + 2)/(n + 2) - a^2*sin(d*x + c)^(n + 1)/(n + 1))/d
Time = 0.18 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.98 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {\frac {a^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7}}{n + 7} + \frac {a^{2} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 2 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{7}}{n + 9} + \frac {2 \, a^{2} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{7}}{n + 8} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5}}{n + 5} - \frac {3 \, a^{2} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 2 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{5}}{n + 7} - \frac {6 \, a^{2} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{5}}{n + 6} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac {3 \, a^{2} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 2 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{3}}{n + 5} + \frac {6 \, a^{2} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{3}}{n + 4} - \frac {a^{2} \sin \left (d x + c\right )^{n + 3}}{n + 3} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a^{2} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
-(a^2*sin(d*x + c)^n*sin(d*x + c)^7/(n + 7) + a^2*e^(n*log(sin(d*x + c)) + 2*log(sin(d*x + c)))*sin(d*x + c)^7/(n + 9) + 2*a^2*e^(n*log(sin(d*x + c) ) + log(sin(d*x + c)))*sin(d*x + c)^7/(n + 8) - 3*a^2*sin(d*x + c)^n*sin(d *x + c)^5/(n + 5) - 3*a^2*e^(n*log(sin(d*x + c)) + 2*log(sin(d*x + c)))*si n(d*x + c)^5/(n + 7) - 6*a^2*e^(n*log(sin(d*x + c)) + log(sin(d*x + c)))*s in(d*x + c)^5/(n + 6) + 3*a^2*sin(d*x + c)^n*sin(d*x + c)^3/(n + 3) + 3*a^ 2*e^(n*log(sin(d*x + c)) + 2*log(sin(d*x + c)))*sin(d*x + c)^3/(n + 5) + 6 *a^2*e^(n*log(sin(d*x + c)) + log(sin(d*x + c)))*sin(d*x + c)^3/(n + 4) - a^2*sin(d*x + c)^(n + 3)/(n + 3) - 2*a^2*sin(d*x + c)^(n + 2)/(n + 2) - a^ 2*sin(d*x + c)^(n + 1)/(n + 1))/d
Time = 41.35 (sec) , antiderivative size = 1142, normalized size of antiderivative = 6.21 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Too large to display} \] Input:
int(cos(c + d*x)^7*sin(c + d*x)^n*(a + a*sin(c + d*x))^2,x)
Output:
(a^2*sin(c + d*x)^n*(n*1507788i + n^2*868332i + n^3*238585i + n^4*37844i + n^5*3702i + n^6*208i + n^7*5i + 843696i))/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 725 76i)) - (a^2*sin(c + d*x)^n*sin(9*c + 9*d*x)*(20304*n + 19564*n^2 + 9544*n ^3 + 2581*n^4 + 391*n^5 + 31*n^6 + n^7 + 8064)*1i)/(256*d*(n*190800i + n^2 *196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8 *1i + 72576i)) + (a^2*sin(c + d*x)*sin(c + d*x)^n*(6799248*n + 3169500*n^2 + 770632*n^3 + 111993*n^4 + 10267*n^5 + 555*n^6 + 13*n^7 + 5588352)*1i)/( 128*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^ 6*670i + n^7*40i + n^8*1i + 72576i)) - (a^2*sin(c + d*x)^n*cos(8*c + 8*d*x )*(n*22716i + n^2*21708i + n^3*10469i + n^4*2788i + n^5*414i + n^6*32i + n ^7*1i + 9072i))/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i)) - (a^2*sin(c + d*x)^n *cos(6*c + 6*d*x)*(n*43920i + n^2*39882i + n^3*17909i + n^4*4336i + n^5*57 0i + n^6*38i + n^7*1i + 18144i))/(16*d*(n*190800i + n^2*196380i + n^3*1054 60i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i)) - (a ^2*sin(c + d*x)^n*cos(4*c + 4*d*x)*(n*140868i + n^2*111816i + n^3*41669i + n^4*7996i + n^5*822i + n^6*44i + n^7*1i + 63504i))/(16*d*(n*190800i + n^2 *196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8 *1i + 72576i)) + (a^2*sin(c + d*x)^n*cos(2*c + 2*d*x)*(n^3*2549i - n^2*...
\[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}d x +2 \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )d x \right )+\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{7}d x \right ) \] Input:
int(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x)
Output:
a**2*(int(sin(c + d*x)**n*cos(c + d*x)**7*sin(c + d*x)**2,x) + 2*int(sin(c + d*x)**n*cos(c + d*x)**7*sin(c + d*x),x) + int(sin(c + d*x)**n*cos(c + d *x)**7,x))