Integrand size = 27, antiderivative size = 167 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {a \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {3 a \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {3 a \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {3 a \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {3 a \sin ^{6+n}(c+d x)}{d (6+n)}-\frac {a \sin ^{7+n}(c+d x)}{d (7+n)}-\frac {a \sin ^{8+n}(c+d x)}{d (8+n)} \] Output:
a*sin(d*x+c)^(1+n)/d/(1+n)+a*sin(d*x+c)^(2+n)/d/(2+n)-3*a*sin(d*x+c)^(3+n) /d/(3+n)-3*a*sin(d*x+c)^(4+n)/d/(4+n)+3*a*sin(d*x+c)^(5+n)/d/(5+n)+3*a*sin (d*x+c)^(6+n)/d/(6+n)-a*sin(d*x+c)^(7+n)/d/(7+n)-a*sin(d*x+c)^(8+n)/d/(8+n )
Time = 2.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.92 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^{1+n}(c+d x) (1+\sin (c+d x)) \left (\frac {1}{1+n}+\frac {\sin (c+d x)}{2+n}-\frac {3 \sin ^2(c+d x)}{3+n}-\frac {3 \sin ^3(c+d x)}{4+n}+\frac {3 \sin ^4(c+d x)}{5+n}+\frac {3 \sin ^5(c+d x)}{6+n}-\frac {\sin ^6(c+d x)}{7+n}-\frac {\sin ^7(c+d x)}{8+n}\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \] Input:
Integrate[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]
Output:
(a*Sin[c + d*x]^(1 + n)*(1 + Sin[c + d*x])*((1 + n)^(-1) + Sin[c + d*x]/(2 + n) - (3*Sin[c + d*x]^2)/(3 + n) - (3*Sin[c + d*x]^3)/(4 + n) + (3*Sin[c + d*x]^4)/(5 + n) + (3*Sin[c + d*x]^5)/(6 + n) - Sin[c + d*x]^6/(7 + n) - Sin[c + d*x]^7/(8 + n)))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)
Time = 0.39 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, 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.
| \(\displaystyle \int \cos ^7(c+d x) (a \sin (c+d x)+a) \sin ^n(c+d x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^7 (a \sin (c+d x)+a) \sin (c+d x)^ndx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \sin ^n(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^4d(a \sin (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (a^7 \sin ^n(c+d x)+a^7 \sin ^{n+1}(c+d x)-3 a^7 \sin ^{n+2}(c+d x)-3 a^7 \sin ^{n+3}(c+d x)+3 a^7 \sin ^{n+4}(c+d x)+3 a^7 \sin ^{n+5}(c+d x)-a^7 \sin ^{n+6}(c+d x)-a^7 \sin ^{n+7}(c+d x)\right )d(a \sin (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^8 \sin ^{n+1}(c+d x)}{n+1}+\frac {a^8 \sin ^{n+2}(c+d x)}{n+2}-\frac {3 a^8 \sin ^{n+3}(c+d x)}{n+3}-\frac {3 a^8 \sin ^{n+4}(c+d x)}{n+4}+\frac {3 a^8 \sin ^{n+5}(c+d x)}{n+5}+\frac {3 a^8 \sin ^{n+6}(c+d x)}{n+6}-\frac {a^8 \sin ^{n+7}(c+d x)}{n+7}-\frac {a^8 \sin ^{n+8}(c+d x)}{n+8}}{a^7 d}\) |
Input:
Int[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]
Output:
((a^8*Sin[c + d*x]^(1 + n))/(1 + n) + (a^8*Sin[c + d*x]^(2 + n))/(2 + n) - (3*a^8*Sin[c + d*x]^(3 + n))/(3 + n) - (3*a^8*Sin[c + d*x]^(4 + n))/(4 + n) + (3*a^8*Sin[c + d*x]^(5 + n))/(5 + n) + (3*a^8*Sin[c + d*x]^(6 + n))/( 6 + n) - (a^8*Sin[c + d*x]^(7 + n))/(7 + n) - (a^8*Sin[c + d*x]^(8 + n))/( 8 + 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]
Time = 12.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.38
| method | result | size |
| derivativedivides | \(\frac {a \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a \sin \left (d x +c \right )^{2} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}-\frac {3 a \sin \left (d x +c \right )^{3} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}-\frac {3 a \sin \left (d x +c \right )^{4} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}+\frac {3 a \sin \left (d x +c \right )^{5} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {3 a \sin \left (d x +c \right )^{6} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}-\frac {a \sin \left (d x +c \right )^{7} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (7+n \right )}-\frac {a \sin \left (d x +c \right )^{8} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (8+n \right )}\) | \(230\) |
| default | \(\frac {a \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a \sin \left (d x +c \right )^{2} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}-\frac {3 a \sin \left (d x +c \right )^{3} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}-\frac {3 a \sin \left (d x +c \right )^{4} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}+\frac {3 a \sin \left (d x +c \right )^{5} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {3 a \sin \left (d x +c \right )^{6} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}-\frac {a \sin \left (d x +c \right )^{7} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (7+n \right )}-\frac {a \sin \left (d x +c \right )^{8} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (8+n \right )}\) | \(230\) |
| parallelrisch | \(\frac {5 \sin \left (d x +c \right )^{n} \left (\frac {2 \left (n -4\right ) \left (7+n \right ) \left (5+n \right ) \left (3+n \right ) \left (1+n \right ) \left (n^{2}+22 n +168\right ) \cos \left (2 d x +2 c \right )}{5}-\frac {2 \left (7+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \left (n^{2}+22 n +168\right ) \cos \left (4 d x +4 c \right )}{5}-\frac {2 \left (n +12\right ) \left (7+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (6 d x +6 c \right )}{5}-\frac {\left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (8 d x +8 c \right )}{10}+\frac {9 \left (2+n \right ) \left (1+n \right ) \left (8+n \right ) \left (4+n \right ) \left (n^{2}+16 n +\frac {245}{3}\right ) \left (6+n \right ) \sin \left (3 d x +3 c \right )}{5}+\left (2+n \right ) \left (1+n \right ) \left (8+n \right ) \left (3+n \right ) \left (4+n \right ) \left (n +\frac {49}{5}\right ) \left (6+n \right ) \sin \left (5 d x +5 c \right )+\frac {\left (8+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (7 d x +7 c \right )}{5}+\left (2+n \right ) \left (8+n \right ) \left (n^{3}+\frac {93}{5} n^{2}+\frac {691}{5} n +735\right ) \left (4+n \right ) \left (6+n \right ) \sin \left (d x +c \right )+\frac {\left (1+n \right ) \left (n^{3}+\frac {108}{5} n^{2}+\frac {892}{5} n +\frac {4464}{5}\right ) \left (3+n \right ) \left (5+n \right ) \left (7+n \right )}{2}\right ) a}{64 \left (2+n \right ) \left (1+n \right ) \left (4+n \right ) \left (3+n \right ) \left (8+n \right ) \left (6+n \right ) \left (7+n \right ) d \left (5+n \right )}\) | \(344\) |
Input:
int(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
a/d/(1+n)*sin(d*x+c)*exp(n*ln(sin(d*x+c)))+a/d/(2+n)*sin(d*x+c)^2*exp(n*ln (sin(d*x+c)))-3*a/d/(3+n)*sin(d*x+c)^3*exp(n*ln(sin(d*x+c)))-3*a/d/(4+n)*s in(d*x+c)^4*exp(n*ln(sin(d*x+c)))+3*a/d/(5+n)*sin(d*x+c)^5*exp(n*ln(sin(d* x+c)))+3*a/d/(6+n)*sin(d*x+c)^6*exp(n*ln(sin(d*x+c)))-a/d/(7+n)*sin(d*x+c) ^7*exp(n*ln(sin(d*x+c)))-a/d/(8+n)*sin(d*x+c)^8*exp(n*ln(sin(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (167) = 334\).
Time = 0.16 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.66 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {{\left ({\left (a n^{7} + 28 \, a n^{6} + 322 \, a n^{5} + 1960 \, a n^{4} + 6769 \, a n^{3} + 13132 \, a n^{2} + 13068 \, a n + 5040 \, a\right )} \cos \left (d x + c\right )^{8} - {\left (a n^{7} + 22 \, a n^{6} + 190 \, a n^{5} + 820 \, a n^{4} + 1849 \, a n^{3} + 2038 \, a n^{2} + 840 \, a n\right )} \cos \left (d x + c\right )^{6} - 48 \, a n^{4} - 6 \, {\left (a n^{6} + 18 \, a n^{5} + 118 \, a n^{4} + 348 \, a n^{3} + 457 \, a n^{2} + 210 \, a n\right )} \cos \left (d x + c\right )^{4} - 768 \, a n^{3} - 4128 \, a n^{2} - 24 \, {\left (a n^{5} + 16 \, a n^{4} + 86 \, a n^{3} + 176 \, a n^{2} + 105 \, a n\right )} \cos \left (d x + c\right )^{2} - 8448 \, a n - {\left ({\left (a n^{7} + 29 \, a n^{6} + 343 \, a n^{5} + 2135 \, a n^{4} + 7504 \, a n^{3} + 14756 \, a n^{2} + 14832 \, a n + 5760 \, a\right )} \cos \left (d x + c\right )^{6} + 48 \, a n^{4} + 6 \, {\left (a n^{6} + 24 \, a n^{5} + 223 \, a n^{4} + 1020 \, a n^{3} + 2404 \, a n^{2} + 2736 \, a n + 1152 \, a\right )} \cos \left (d x + c\right )^{4} + 960 \, a n^{3} + 6720 \, a n^{2} + 24 \, {\left (a n^{5} + 21 \, a n^{4} + 160 \, a n^{3} + 540 \, a n^{2} + 784 \, a n + 384 \, a\right )} \cos \left (d x + c\right )^{2} + 19200 \, a n + 18432 \, a\right )} \sin \left (d x + c\right ) - 5040 \, a\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 36 \, d n^{7} + 546 \, d n^{6} + 4536 \, d n^{5} + 22449 \, d n^{4} + 67284 \, d n^{3} + 118124 \, d n^{2} + 109584 \, d n + 40320 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-((a*n^7 + 28*a*n^6 + 322*a*n^5 + 1960*a*n^4 + 6769*a*n^3 + 13132*a*n^2 + 13068*a*n + 5040*a)*cos(d*x + c)^8 - (a*n^7 + 22*a*n^6 + 190*a*n^5 + 820*a *n^4 + 1849*a*n^3 + 2038*a*n^2 + 840*a*n)*cos(d*x + c)^6 - 48*a*n^4 - 6*(a *n^6 + 18*a*n^5 + 118*a*n^4 + 348*a*n^3 + 457*a*n^2 + 210*a*n)*cos(d*x + c )^4 - 768*a*n^3 - 4128*a*n^2 - 24*(a*n^5 + 16*a*n^4 + 86*a*n^3 + 176*a*n^2 + 105*a*n)*cos(d*x + c)^2 - 8448*a*n - ((a*n^7 + 29*a*n^6 + 343*a*n^5 + 2 135*a*n^4 + 7504*a*n^3 + 14756*a*n^2 + 14832*a*n + 5760*a)*cos(d*x + c)^6 + 48*a*n^4 + 6*(a*n^6 + 24*a*n^5 + 223*a*n^4 + 1020*a*n^3 + 2404*a*n^2 + 2 736*a*n + 1152*a)*cos(d*x + c)^4 + 960*a*n^3 + 6720*a*n^2 + 24*(a*n^5 + 21 *a*n^4 + 160*a*n^3 + 540*a*n^2 + 784*a*n + 384*a)*cos(d*x + c)^2 + 19200*a *n + 18432*a)*sin(d*x + c) - 5040*a)*sin(d*x + c)^n/(d*n^8 + 36*d*n^7 + 54 6*d*n^6 + 4536*d*n^5 + 22449*d*n^4 + 67284*d*n^3 + 118124*d*n^2 + 109584*d *n + 40320*d)
Leaf count of result is larger than twice the leaf count of optimal. 19968 vs. \(2 (141) = 282\).
Time = 28.86 (sec) , antiderivative size = 19968, normalized size of antiderivative = 119.57 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**7*sin(d*x+c)**n*(a+a*sin(d*x+c)),x)
Output:
Piecewise((x*(a*sin(c) + a)*sin(c)**n*cos(c)**7, Eq(d, 0)), (-a*log(sin(c + d*x))/d + 16*a/(35*d*sin(c + d*x)) - a*cos(c + d*x)**2/(2*d*sin(c + d*x) **2) - 8*a*cos(c + d*x)**2/(35*d*sin(c + d*x)**3) + a*cos(c + d*x)**4/(4*d *sin(c + d*x)**4) + 6*a*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - a*cos(c + d*x)**6/(6*d*sin(c + d*x)**6) - a*cos(c + d*x)**6/(7*d*sin(c + d*x)**7), Eq(n, -8)), (-a*log(sin(c + d*x))/d - 16*a*sin(c + d*x)/(5*d) - 8*a*cos(c + d*x)**2/(5*d*sin(c + d*x)) - a*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) + 2 *a*cos(c + d*x)**4/(5*d*sin(c + d*x)**3) + a*cos(c + d*x)**4/(4*d*sin(c + d*x)**4) - a*cos(c + d*x)**6/(5*d*sin(c + d*x)**5) - a*cos(c + d*x)**6/(6* d*sin(c + d*x)**6), Eq(n, -7)), (-960*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c /2 + d*x/2)**9/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 32 0*d*tan(c/2 + d*x/2)**5) - 1920*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d *x/2)**7/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*ta n(c/2 + d*x/2)**5) - 960*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)** 5/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) + 960*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**9/(320*d*tan(c /2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) + 1920*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7/(320*d*tan(c/2 + d*x/2)** 9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) + 960*a*log(tan (c/2 + d*x/2))*tan(c/2 + d*x/2)**5/(320*d*tan(c/2 + d*x/2)**9 + 640*d*t...
Time = 0.03 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {a \sin \left (d x + c\right )^{n + 8}}{n + 8} + \frac {a \sin \left (d x + c\right )^{n + 7}}{n + 7} - \frac {3 \, a \sin \left (d x + c\right )^{n + 6}}{n + 6} - \frac {3 \, a \sin \left (d x + c\right )^{n + 5}}{n + 5} + \frac {3 \, a \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {3 \, a \sin \left (d x + c\right )^{n + 3}}{n + 3} - \frac {a \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-(a*sin(d*x + c)^(n + 8)/(n + 8) + a*sin(d*x + c)^(n + 7)/(n + 7) - 3*a*si n(d*x + c)^(n + 6)/(n + 6) - 3*a*sin(d*x + c)^(n + 5)/(n + 5) + 3*a*sin(d* x + c)^(n + 4)/(n + 4) + 3*a*sin(d*x + c)^(n + 3)/(n + 3) - a*sin(d*x + c) ^(n + 2)/(n + 2) - a*sin(d*x + c)^(n + 1)/(n + 1))/d
Time = 0.15 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.28 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {a \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7}}{n + 7} + \frac {a e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{7}}{n + 8} - \frac {3 \, a \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5}}{n + 5} - \frac {3 \, a e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{5}}{n + 6} + \frac {3 \, a \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac {3 \, a e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{3}}{n + 4} - \frac {a \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-(a*sin(d*x + c)^n*sin(d*x + c)^7/(n + 7) + a*e^(n*log(sin(d*x + c)) + log (sin(d*x + c)))*sin(d*x + c)^7/(n + 8) - 3*a*sin(d*x + c)^n*sin(d*x + c)^5 /(n + 5) - 3*a*e^(n*log(sin(d*x + c)) + log(sin(d*x + c)))*sin(d*x + c)^5/ (n + 6) + 3*a*sin(d*x + c)^n*sin(d*x + c)^3/(n + 3) + 3*a*e^(n*log(sin(d*x + c)) + log(sin(d*x + c)))*sin(d*x + c)^3/(n + 4) - a*sin(d*x + c)^(n + 2 )/(n + 2) - a*sin(d*x + c)^(n + 1)/(n + 1))/d
Time = 40.29 (sec) , antiderivative size = 901, normalized size of antiderivative = 5.40 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^7*sin(c + d*x)^n*(a + a*sin(c + d*x)),x)
Output:
(a*sin(c + d*x)^n*(879324*n + 552236*n^2 + 167669*n^3 + 28904*n^4 + 3050*n ^5 + 188*n^6 + 5*n^7 + 468720))/(128*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (a*sin(c + d*x )^n*cos(2*c + 2*d*x)*(109872*n + 41822*n^2 + 599*n^3 - 2332*n^4 - 454*n^5 - 34*n^6 - n^7 + 70560))/(32*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449* n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (a*sin(c + d*x)^n*sin( 7*c + 7*d*x)*(n*14832i + n^2*14756i + n^3*7504i + n^4*2135i + n^5*343i + n ^6*29i + n^7*1i + 5760i)*1i)/(64*d*(109584*n + 118124*n^2 + 67284*n^3 + 22 449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (a*sin(c + d*x)^n* sin(5*c + 5*d*x)*(n*139824i + n^2*131476i + n^3*62000i + n^4*16027i + n^5* 2291i + n^6*169i + n^7*5i + 56448i)*1i)/(64*d*(109584*n + 118124*n^2 + 672 84*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (a*sin( c + d*x)^n*sin(3*c + 3*d*x)*(n*210512i + n^2*171084i + n^3*67472i + n^4*14 445i + n^5*1733i + n^6*111i + n^7*3i + 94080i)*3i)/(64*d*(109584*n + 11812 4*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320) ) - (a*sin(c + d*x)^n*cos(8*c + 8*d*x)*(13068*n + 13132*n^2 + 6769*n^3 + 1 960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040))/(128*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (a*s in(c + d*x)^n*cos(6*c + 6*d*x)*(25296*n + 24226*n^2 + 11689*n^3 + 3100*n^4 + 454*n^5 + 34*n^6 + n^7 + 10080))/(32*d*(109584*n + 118124*n^2 + 6728...
\[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )d x +\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{7}d x \right ) \] Input:
int(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c)),x)
Output:
a*(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))