Integrand size = 29, antiderivative size = 181 \[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac {a^3 \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {5 a^3 \sin ^{4+n}(c+d x)}{d (4+n)}-\frac {5 a^3 \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {a^3 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac {3 a^3 \sin ^{7+n}(c+d x)}{d (7+n)}+\frac {a^3 \sin ^{8+n}(c+d x)}{d (8+n)} \] Output:
a^3*sin(d*x+c)^(1+n)/d/(1+n)+3*a^3*sin(d*x+c)^(2+n)/d/(2+n)+a^3*sin(d*x+c) ^(3+n)/d/(3+n)-5*a^3*sin(d*x+c)^(4+n)/d/(4+n)-5*a^3*sin(d*x+c)^(5+n)/d/(5+ n)+a^3*sin(d*x+c)^(6+n)/d/(6+n)+3*a^3*sin(d*x+c)^(7+n)/d/(7+n)+a^3*sin(d*x +c)^(8+n)/d/(8+n)
Time = 0.69 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.68 \[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^{1+n}(c+d x) \left (\frac {1}{1+n}+\frac {3 \sin (c+d x)}{2+n}+\frac {\sin ^2(c+d x)}{3+n}-\frac {5 \sin ^3(c+d x)}{4+n}-\frac {5 \sin ^4(c+d x)}{5+n}+\frac {\sin ^5(c+d x)}{6+n}+\frac {3 \sin ^6(c+d x)}{7+n}+\frac {\sin ^7(c+d x)}{8+n}\right )}{d} \] Input:
Integrate[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) + Sin[c + d*x]^2/(3 + n) - (5*Sin[c + d*x]^3)/(4 + n) - (5*Sin[c + d*x]^4)/(5 + n ) + Sin[c + d*x]^5/(6 + n) + (3*Sin[c + d*x]^6)/(7 + n) + Sin[c + d*x]^7/( 8 + n)))/d
Time = 0.43 (sec) , antiderivative size = 164, 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.
| \(\displaystyle \int \cos ^5(c+d x) (a \sin (c+d x)+a)^3 \sin ^n(c+d x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^5 (a \sin (c+d x)+a)^3 \sin (c+d x)^ndx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \sin ^n(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^5d(a \sin (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (a^7 \sin ^n(c+d x)+3 a^7 \sin ^{n+1}(c+d x)+a^7 \sin ^{n+2}(c+d x)-5 a^7 \sin ^{n+3}(c+d x)-5 a^7 \sin ^{n+4}(c+d x)+a^7 \sin ^{n+5}(c+d x)+3 a^7 \sin ^{n+6}(c+d x)+a^7 \sin ^{n+7}(c+d x)\right )d(a \sin (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^8 \sin ^{n+1}(c+d x)}{n+1}+\frac {3 a^8 \sin ^{n+2}(c+d x)}{n+2}+\frac {a^8 \sin ^{n+3}(c+d x)}{n+3}-\frac {5 a^8 \sin ^{n+4}(c+d x)}{n+4}-\frac {5 a^8 \sin ^{n+5}(c+d x)}{n+5}+\frac {a^8 \sin ^{n+6}(c+d x)}{n+6}+\frac {3 a^8 \sin ^{n+7}(c+d x)}{n+7}+\frac {a^8 \sin ^{n+8}(c+d x)}{n+8}}{a^5 d}\) |
Input:
Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]
Output:
((a^8*Sin[c + d*x]^(1 + n))/(1 + n) + (3*a^8*Sin[c + d*x]^(2 + n))/(2 + n) + (a^8*Sin[c + d*x]^(3 + n))/(3 + n) - (5*a^8*Sin[c + d*x]^(4 + n))/(4 + n) - (5*a^8*Sin[c + d*x]^(5 + n))/(5 + n) + (a^8*Sin[c + d*x]^(6 + n))/(6 + n) + (3*a^8*Sin[c + d*x]^(7 + n))/(7 + n) + (a^8*Sin[c + d*x]^(8 + n))/( 8 + n))/(a^5*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 = 14.17 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.35
| method | result | size |
| derivativedivides | \(\frac {a^{3} \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a^{3} \sin \left (d x +c \right )^{3} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}+\frac {a^{3} \sin \left (d x +c \right )^{6} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}+\frac {a^{3} \sin \left (d x +c \right )^{8} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (8+n \right )}+\frac {3 a^{3} \sin \left (d x +c \right )^{2} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}-\frac {5 a^{3} \sin \left (d x +c \right )^{4} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}-\frac {5 a^{3} \sin \left (d x +c \right )^{5} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {3 a^{3} \sin \left (d x +c \right )^{7} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (7+n \right )}\) | \(244\) |
| default | \(\frac {a^{3} \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a^{3} \sin \left (d x +c \right )^{3} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}+\frac {a^{3} \sin \left (d x +c \right )^{6} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}+\frac {a^{3} \sin \left (d x +c \right )^{8} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (8+n \right )}+\frac {3 a^{3} \sin \left (d x +c \right )^{2} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}-\frac {5 a^{3} \sin \left (d x +c \right )^{4} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}-\frac {5 a^{3} \sin \left (d x +c \right )^{5} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {3 a^{3} \sin \left (d x +c \right )^{7} {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (7+n \right )}\) | \(244\) |
| parallelrisch | \(\frac {17 \sin \left (d x +c \right )^{n} \left (\frac {6 \left (5+n \right ) \left (1+n \right ) \left (7+n \right ) \left (n^{3}+\frac {38}{3} n^{2}-\frac {368}{3} n -1056\right ) \left (3+n \right ) \cos \left (2 d x +2 c \right )}{17}-\frac {14 \left (5+n \right ) \left (1+n \right ) \left (7+n \right ) \left (2+n \right ) \left (n^{2}+\frac {138}{7} n +\frac {600}{7}\right ) \left (3+n \right ) \cos \left (4 d x +4 c \right )}{17}-\frac {6 \left (5+n \right ) \left (n +\frac {20}{3}\right ) \left (4+n \right ) \left (1+n \right ) \left (7+n \right ) \left (2+n \right ) \left (3+n \right ) \cos \left (6 d x +6 c \right )}{17}+\frac {\left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (8 d x +8 c \right )}{34}+\frac {21 \left (8+n \right ) \left (4+n \right ) \left (1+n \right ) \left (n +\frac {85}{7}\right ) \left (n +\frac {7}{3}\right ) \left (6+n \right ) \left (2+n \right ) \sin \left (3 d x +3 c \right )}{17}+\frac {\left (n -35\right ) \left (8+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (5 d x +5 c \right )}{17}-\frac {3 \left (8+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (7 d x +7 c \right )}{17}+\left (8+n \right ) \left (4+n \right ) \left (n^{3}+\frac {329}{17} n^{2}+\frac {3015}{17} n +\frac {5775}{17}\right ) \left (6+n \right ) \left (2+n \right ) \sin \left (d x +c \right )+\frac {27 \left (5+n \right ) \left (1+n \right ) \left (7+n \right ) \left (n^{3}+\frac {596}{27} n^{2}+204 n +\frac {18064}{27}\right ) \left (3+n \right )}{34}\right ) a^{3}}{64 \left (7+n \right ) \left (8+n \right ) \left (2+n \right ) \left (1+n \right ) \left (6+n \right ) \left (4+n \right ) \left (5+n \right ) \left (3+n \right ) d}\) | \(347\) |
Input:
int(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
a^3/d/(1+n)*sin(d*x+c)*exp(n*ln(sin(d*x+c)))+a^3/d/(3+n)*sin(d*x+c)^3*exp( n*ln(sin(d*x+c)))+a^3/d/(6+n)*sin(d*x+c)^6*exp(n*ln(sin(d*x+c)))+a^3/d/(8+ n)*sin(d*x+c)^8*exp(n*ln(sin(d*x+c)))+3*a^3/d/(2+n)*sin(d*x+c)^2*exp(n*ln( sin(d*x+c)))-5*a^3/d/(4+n)*sin(d*x+c)^4*exp(n*ln(sin(d*x+c)))-5*a^3/d/(5+n )*sin(d*x+c)^5*exp(n*ln(sin(d*x+c)))+3*a^3/d/(7+n)*sin(d*x+c)^7*exp(n*ln(s in(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (181) = 362\).
Time = 0.13 (sec) , antiderivative size = 616, normalized size of antiderivative = 3.40 \[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {{\left ({\left (a^{3} n^{7} + 28 \, a^{3} n^{6} + 322 \, a^{3} n^{5} + 1960 \, a^{3} n^{4} + 6769 \, a^{3} n^{3} + 13132 \, a^{3} n^{2} + 13068 \, a^{3} n + 5040 \, a^{3}\right )} \cos \left (d x + c\right )^{8} + 32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} - {\left (5 \, a^{3} n^{7} + 142 \, a^{3} n^{6} + 1654 \, a^{3} n^{5} + 10180 \, a^{3} n^{4} + 35485 \, a^{3} n^{3} + 69358 \, a^{3} n^{2} + 69416 \, a^{3} n + 26880 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 23520 \, a^{3} n^{2} + 2 \, {\left (2 \, a^{3} n^{7} + 49 \, a^{3} n^{6} + 470 \, a^{3} n^{5} + 2230 \, a^{3} n^{4} + 5438 \, a^{3} n^{3} + 6361 \, a^{3} n^{2} + 2730 \, a^{3} n\right )} \cos \left (d x + c\right )^{4} + 39968 \, a^{3} n + 21840 \, a^{3} + 8 \, {\left (2 \, a^{3} n^{6} + 45 \, a^{3} n^{5} + 380 \, a^{3} n^{4} + 1470 \, a^{3} n^{3} + 2498 \, a^{3} n^{2} + 1365 \, a^{3} n\right )} \cos \left (d x + c\right )^{2} + {\left (32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} - 3 \, {\left (a^{3} n^{7} + 29 \, a^{3} n^{6} + 343 \, a^{3} n^{5} + 2135 \, a^{3} n^{4} + 7504 \, a^{3} n^{3} + 14756 \, a^{3} n^{2} + 14832 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 24000 \, a^{3} n^{2} + 2 \, {\left (2 \, a^{3} n^{7} + 53 \, a^{3} n^{6} + 566 \, a^{3} n^{5} + 3155 \, a^{3} n^{4} + 9908 \, a^{3} n^{3} + 17492 \, a^{3} n^{2} + 15984 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{4} + 44288 \, a^{3} n + 30720 \, a^{3} + 8 \, {\left (2 \, a^{3} n^{6} + 47 \, a^{3} n^{5} + 425 \, a^{3} n^{4} + 1880 \, a^{3} n^{3} + 4268 \, a^{3} n^{2} + 4688 \, a^{3} n + 1920 \, a^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 36 \, d n^{7} + 546 \, d n^{6} + 4536 \, d n^{5} + 22449 \, d n^{4} + 67284 \, d n^{3} + 118124 \, d n^{2} + 109584 \, d n + 40320 \, d} \] Input:
integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
((a^3*n^7 + 28*a^3*n^6 + 322*a^3*n^5 + 1960*a^3*n^4 + 6769*a^3*n^3 + 13132 *a^3*n^2 + 13068*a^3*n + 5040*a^3)*cos(d*x + c)^8 + 32*a^3*n^5 + 720*a^3*n ^4 - (5*a^3*n^7 + 142*a^3*n^6 + 1654*a^3*n^5 + 10180*a^3*n^4 + 35485*a^3*n ^3 + 69358*a^3*n^2 + 69416*a^3*n + 26880*a^3)*cos(d*x + c)^6 + 6080*a^3*n^ 3 + 23520*a^3*n^2 + 2*(2*a^3*n^7 + 49*a^3*n^6 + 470*a^3*n^5 + 2230*a^3*n^4 + 5438*a^3*n^3 + 6361*a^3*n^2 + 2730*a^3*n)*cos(d*x + c)^4 + 39968*a^3*n + 21840*a^3 + 8*(2*a^3*n^6 + 45*a^3*n^5 + 380*a^3*n^4 + 1470*a^3*n^3 + 249 8*a^3*n^2 + 1365*a^3*n)*cos(d*x + c)^2 + (32*a^3*n^5 + 720*a^3*n^4 - 3*(a^ 3*n^7 + 29*a^3*n^6 + 343*a^3*n^5 + 2135*a^3*n^4 + 7504*a^3*n^3 + 14756*a^3 *n^2 + 14832*a^3*n + 5760*a^3)*cos(d*x + c)^6 + 6080*a^3*n^3 + 24000*a^3*n ^2 + 2*(2*a^3*n^7 + 53*a^3*n^6 + 566*a^3*n^5 + 3155*a^3*n^4 + 9908*a^3*n^3 + 17492*a^3*n^2 + 15984*a^3*n + 5760*a^3)*cos(d*x + c)^4 + 44288*a^3*n + 30720*a^3 + 8*(2*a^3*n^6 + 47*a^3*n^5 + 425*a^3*n^4 + 1880*a^3*n^3 + 4268* a^3*n^2 + 4688*a^3*n + 1920*a^3)*cos(d*x + c)^2)*sin(d*x + c))*sin(d*x + c )^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)
Leaf count of result is larger than twice the leaf count of optimal. 23312 vs. \(2 (155) = 310\).
Time = 31.03 (sec) , antiderivative size = 23312, normalized size of antiderivative = 128.80 \[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**5*sin(d*x+c)**n*(a+a*sin(d*x+c))**3,x)
Output:
Piecewise((x*(a*sin(c) + a)**3*sin(c)**n*cos(c)**5, Eq(d, 0)), (a**3*log(s in(c + d*x))/d - 8*a**3/(5*d*sin(c + d*x)) + a**3*cos(c + d*x)**2/(2*d*sin (c + d*x)**2) - a**3/(2*d*sin(c + d*x)**2) + 4*a**3*cos(c + d*x)**2/(5*d*s in(c + d*x)**3) - 8*a**3/(105*d*sin(c + d*x)**3) - a**3*cos(c + d*x)**4/(4 *d*sin(c + d*x)**4) + a**3*cos(c + d*x)**2/(2*d*sin(c + d*x)**4) - 3*a**3* cos(c + d*x)**4/(5*d*sin(c + d*x)**5) + 4*a**3*cos(c + d*x)**2/(35*d*sin(c + d*x)**5) - a**3*cos(c + d*x)**4/(2*d*sin(c + d*x)**6) - a**3*cos(c + d* x)**4/(7*d*sin(c + d*x)**7), Eq(n, -8)), (3*a**3*log(sin(c + d*x))/d + 8*a **3*sin(c + d*x)/(3*d) + 4*a**3*cos(c + d*x)**2/(3*d*sin(c + d*x)) - 8*a** 3/(5*d*sin(c + d*x)) + 3*a**3*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) - a**3 /(6*d*sin(c + d*x)**2) - a**3*cos(c + d*x)**4/(3*d*sin(c + d*x)**3) + 4*a* *3*cos(c + d*x)**2/(5*d*sin(c + d*x)**3) - 3*a**3*cos(c + d*x)**4/(4*d*sin (c + d*x)**4) + a**3*cos(c + d*x)**2/(6*d*sin(c + d*x)**4) - 3*a**3*cos(c + d*x)**4/(5*d*sin(c + d*x)**5) - a**3*cos(c + d*x)**4/(6*d*sin(c + d*x)** 6), Eq(n, -7)), (-960*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)** 9/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) - 1920*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**7/ (960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) - 960*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**5/(96 0*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + ...
Time = 0.04 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.89 \[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {a^{3} \sin \left (d x + c\right )^{n + 8}}{n + 8} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 7}}{n + 7} + \frac {a^{3} \sin \left (d x + c\right )^{n + 6}}{n + 6} - \frac {5 \, a^{3} \sin \left (d x + c\right )^{n + 5}}{n + 5} - \frac {5 \, a^{3} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {a^{3} \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \] Input:
integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
(a^3*sin(d*x + c)^(n + 8)/(n + 8) + 3*a^3*sin(d*x + c)^(n + 7)/(n + 7) + a ^3*sin(d*x + c)^(n + 6)/(n + 6) - 5*a^3*sin(d*x + c)^(n + 5)/(n + 5) - 5*a ^3*sin(d*x + c)^(n + 4)/(n + 4) + a^3*sin(d*x + c)^(n + 3)/(n + 3) + 3*a^3 *sin(d*x + c)^(n + 2)/(n + 2) + a^3*sin(d*x + c)^(n + 1)/(n + 1))/d
Time = 0.17 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.97 \[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5}}{n + 5} + \frac {a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 3 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{5}}{n + 8} + \frac {3 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 2 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{5}}{n + 7} + \frac {3 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{5}}{n + 6} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} - \frac {2 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 3 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{3}}{n + 6} - \frac {6 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 2 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{3}}{n + 5} - \frac {6 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{3}}{n + 4} + \frac {a^{3} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \] Input:
integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
(a^3*sin(d*x + c)^n*sin(d*x + c)^5/(n + 5) + a^3*e^(n*log(sin(d*x + c)) + 3*log(sin(d*x + c)))*sin(d*x + c)^5/(n + 8) + 3*a^3*e^(n*log(sin(d*x + c)) + 2*log(sin(d*x + c)))*sin(d*x + c)^5/(n + 7) + 3*a^3*e^(n*log(sin(d*x + c)) + log(sin(d*x + c)))*sin(d*x + c)^5/(n + 6) - 2*a^3*sin(d*x + c)^n*sin (d*x + c)^3/(n + 3) - 2*a^3*e^(n*log(sin(d*x + c)) + 3*log(sin(d*x + c)))* sin(d*x + c)^3/(n + 6) - 6*a^3*e^(n*log(sin(d*x + c)) + 2*log(sin(d*x + c) ))*sin(d*x + c)^3/(n + 5) - 6*a^3*e^(n*log(sin(d*x + c)) + log(sin(d*x + c )))*sin(d*x + c)^3/(n + 4) + a^3*sin(d*x + c)^(n + 4)/(n + 4) + 3*a^3*sin( d*x + c)^(n + 3)/(n + 3) + 3*a^3*sin(d*x + c)^(n + 2)/(n + 2) + a^3*sin(d* x + c)^(n + 1)/(n + 1))/d
Time = 42.62 (sec) , antiderivative size = 923, normalized size of antiderivative = 5.10 \[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^5*sin(c + d*x)^n*(a + a*sin(c + d*x))^3,x)
Output:
(a^3*sin(c + d*x)^n*(3757604*n + 2585492*n^2 + 870443*n^3 + 162200*n^4 + 1 7366*n^5 + 1028*n^6 + 27*n^7 + 1896720))/(128*d*(109584*n + 118124*n^2 + 6 7284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (a^3* sin(c + d*x)^n*cos(8*c + 8*d*x)*(13068*n + 13132*n^2 + 6769*n^3 + 1960*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^3*sin(c + d*x)*sin(c + d*x)^n*(n*3467760i + n^2*2140836i + n^3*675728i + n^4*11893 5i + n^5*11975i + n^6*669i + n^7*17i + 2217600i)*1i)/(64*d*(109584*n + 118 124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 4032 0)) - (a^3*sin(c + d*x)^n*cos(6*c + 6*d*x)*(43280*n + 43094*n^2 + 21947*n^ 3 + 6260*n^4 + 1010*n^5 + 86*n^6 + 3*n^7 + 16800))/(32*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^3*sin(c + d*x)^n*cos(4*c + 4*d*x)*(303180*n + 273336*n^2 + 122023*n ^3 + 29520*n^4 + 3910*n^5 + 264*n^6 + 7*n^7 + 126000))/(32*d*(109584*n + 1 18124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40 320)) - (a^3*sin(c + d*x)^n*cos(2*c + 2*d*x)*(596208*n + 333226*n^2 + 7533 3*n^3 + 5260*n^4 - 498*n^5 - 86*n^6 - 3*n^7 + 332640))/(32*d*(109584*n + 1 18124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40 320)) + (a^3*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)*3i)/(64*d*(109...
\[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )^{3}d x +3 \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )^{2}d x \right )+3 \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )d x \right )+\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{5}d x \right ) \] Input:
int(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x)
Output:
a**3*(int(sin(c + d*x)**n*cos(c + d*x)**5*sin(c + d*x)**3,x) + 3*int(sin(c + d*x)**n*cos(c + d*x)**5*sin(c + d*x)**2,x) + 3*int(sin(c + d*x)**n*cos( c + d*x)**5*sin(c + d*x),x) + int(sin(c + d*x)**n*cos(c + d*x)**5,x))