Integrand size = 27, antiderivative size = 97 \[ \int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^{10}(c+d x)}{10 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {a \sin ^{11}(c+d x)}{11 d} \] Output:
-1/8*a*cos(d*x+c)^8/d+1/10*a*cos(d*x+c)^10/d+1/5*a*sin(d*x+c)^5/d-3/7*a*si n(d*x+c)^7/d+1/3*a*sin(d*x+c)^9/d-1/11*a*sin(d*x+c)^11/d
Time = 0.60 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21 \[ \int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (-16170 \cos (2 (c+d x))-4620 \cos (4 (c+d x))+1155 \cos (6 (c+d x))+1155 \cos (8 (c+d x))+231 \cos (10 (c+d x))+16170 \sin (c+d x)-2310 \sin (3 (c+d x))-2541 \sin (5 (c+d x))-165 \sin (7 (c+d x))+385 \sin (9 (c+d x))+105 \sin (11 (c+d x)))}{1182720 d} \] Input:
Integrate[Cos[c + d*x]^7*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
(a*(-16170*Cos[2*(c + d*x)] - 4620*Cos[4*(c + d*x)] + 1155*Cos[6*(c + d*x) ] + 1155*Cos[8*(c + d*x)] + 231*Cos[10*(c + d*x)] + 16170*Sin[c + d*x] - 2 310*Sin[3*(c + d*x)] - 2541*Sin[5*(c + d*x)] - 165*Sin[7*(c + d*x)] + 385* Sin[9*(c + d*x)] + 105*Sin[11*(c + d*x)]))/(1182720*d)
Time = 0.44 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3313, 3042, 3044, 244, 2009, 3045, 244, 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.
\(\displaystyle \int \sin ^3(c+d x) \cos ^7(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^7 (a \sin (c+d x)+a)dx\) |
\(\Big \downarrow \) 3313 |
\(\displaystyle a \int \cos ^7(c+d x) \sin ^4(c+d x)dx+a \int \cos ^7(c+d x) \sin ^3(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \cos (c+d x)^7 \sin (c+d x)^3dx+a \int \cos (c+d x)^7 \sin (c+d x)^4dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {a \int \sin ^4(c+d x) \left (1-\sin ^2(c+d x)\right )^3d\sin (c+d x)}{d}+a \int \cos (c+d x)^7 \sin (c+d x)^3dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {a \int \left (-\sin ^{10}(c+d x)+3 \sin ^8(c+d x)-3 \sin ^6(c+d x)+\sin ^4(c+d x)\right )d\sin (c+d x)}{d}+a \int \cos (c+d x)^7 \sin (c+d x)^3dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int \cos (c+d x)^7 \sin (c+d x)^3dx+\frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle \frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \int \cos ^7(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \int \left (\cos ^7(c+d x)-\cos ^9(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \left (\frac {1}{8} \cos ^8(c+d x)-\frac {1}{10} \cos ^{10}(c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^7*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
-((a*(Cos[c + d*x]^8/8 - Cos[c + d*x]^10/10))/d) + (a*(Sin[c + d*x]^5/5 - (3*Sin[c + d*x]^7)/7 + Sin[c + d*x]^9/3 - Sin[c + d*x]^11/11))/d
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[Cos[e + f*x]^ p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[Cos[e + f*x]^p*(d*Sin[e + f*x ])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2 ] && IntegerQ[n] && ((LtQ[p, 0] && NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] | | LtQ[p + 1, -n, 2*p + 1])
Time = 357.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\sin \left (d x +c \right )^{11}}{11}+\frac {\sin \left (d x +c \right )^{10}}{10}-\frac {\sin \left (d x +c \right )^{9}}{3}-\frac {3 \sin \left (d x +c \right )^{8}}{8}+\frac {3 \sin \left (d x +c \right )^{7}}{7}+\frac {\sin \left (d x +c \right )^{6}}{2}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{4}}{4}\right )}{d}\) | \(88\) |
default | \(-\frac {a \left (\frac {\sin \left (d x +c \right )^{11}}{11}+\frac {\sin \left (d x +c \right )^{10}}{10}-\frac {\sin \left (d x +c \right )^{9}}{3}-\frac {3 \sin \left (d x +c \right )^{8}}{8}+\frac {3 \sin \left (d x +c \right )^{7}}{7}+\frac {\sin \left (d x +c \right )^{6}}{2}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{4}}{4}\right )}{d}\) | \(88\) |
parallelrisch | \(-\frac {\left (\frac {11 \sin \left (5 d x +5 c \right )}{10}-\frac {\cos \left (8 d x +8 c \right )}{2}-\frac {\cos \left (6 d x +6 c \right )}{2}+\frac {\sin \left (7 d x +7 c \right )}{14}+7 \cos \left (2 d x +2 c \right )-7 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )-\frac {79}{10}+2 \cos \left (4 d x +4 c \right )-\frac {\sin \left (11 d x +11 c \right )}{22}-\frac {\cos \left (10 d x +10 c \right )}{10}-\frac {\sin \left (9 d x +9 c \right )}{6}\right ) a}{512 d}\) | \(125\) |
risch | \(\frac {7 a \sin \left (d x +c \right )}{512 d}+\frac {a \sin \left (11 d x +11 c \right )}{11264 d}+\frac {a \cos \left (10 d x +10 c \right )}{5120 d}+\frac {a \sin \left (9 d x +9 c \right )}{3072 d}+\frac {a \cos \left (8 d x +8 c \right )}{1024 d}-\frac {a \sin \left (7 d x +7 c \right )}{7168 d}+\frac {a \cos \left (6 d x +6 c \right )}{1024 d}-\frac {11 a \sin \left (5 d x +5 c \right )}{5120 d}-\frac {a \cos \left (4 d x +4 c \right )}{256 d}-\frac {a \sin \left (3 d x +3 c \right )}{512 d}-\frac {7 a \cos \left (2 d x +2 c \right )}{512 d}\) | \(164\) |
orering | \(\text {Expression too large to display}\) | \(2880\) |
Input:
int(cos(d*x+c)^7*sin(d*x+c)^3*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-a/d*(1/11*sin(d*x+c)^11+1/10*sin(d*x+c)^10-1/3*sin(d*x+c)^9-3/8*sin(d*x+c )^8+3/7*sin(d*x+c)^7+1/2*sin(d*x+c)^6-1/5*sin(d*x+c)^5-1/4*sin(d*x+c)^4)
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98 \[ \int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {924 \, a \cos \left (d x + c\right )^{10} - 1155 \, a \cos \left (d x + c\right )^{8} + 8 \, {\left (105 \, a \cos \left (d x + c\right )^{10} - 140 \, a \cos \left (d x + c\right )^{8} + 5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{9240 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/9240*(924*a*cos(d*x + c)^10 - 1155*a*cos(d*x + c)^8 + 8*(105*a*cos(d*x + c)^10 - 140*a*cos(d*x + c)^8 + 5*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 + 8*a*cos(d*x + c)^2 + 16*a)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (82) = 164\).
Time = 1.84 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.88 \[ \int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {16 a \sin ^{11}{\left (c + d x \right )}}{1155 d} + \frac {a \sin ^{10}{\left (c + d x \right )}}{40 d} + \frac {8 a \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{105 d} + \frac {a \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8 d} + \frac {6 a \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{5 d} + \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**7*sin(d*x+c)**3*(a+a*sin(d*x+c)),x)
Output:
Piecewise((16*a*sin(c + d*x)**11/(1155*d) + a*sin(c + d*x)**10/(40*d) + 8* a*sin(c + d*x)**9*cos(c + d*x)**2/(105*d) + a*sin(c + d*x)**8*cos(c + d*x) **2/(8*d) + 6*a*sin(c + d*x)**7*cos(c + d*x)**4/(35*d) + a*sin(c + d*x)**6 *cos(c + d*x)**4/(4*d) + a*sin(c + d*x)**5*cos(c + d*x)**6/(5*d) + a*sin(c + d*x)**4*cos(c + d*x)**6/(4*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**3*c os(c)**7, True))
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {840 \, a \sin \left (d x + c\right )^{11} + 924 \, a \sin \left (d x + c\right )^{10} - 3080 \, a \sin \left (d x + c\right )^{9} - 3465 \, a \sin \left (d x + c\right )^{8} + 3960 \, a \sin \left (d x + c\right )^{7} + 4620 \, a \sin \left (d x + c\right )^{6} - 1848 \, a \sin \left (d x + c\right )^{5} - 2310 \, a \sin \left (d x + c\right )^{4}}{9240 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/9240*(840*a*sin(d*x + c)^11 + 924*a*sin(d*x + c)^10 - 3080*a*sin(d*x + c)^9 - 3465*a*sin(d*x + c)^8 + 3960*a*sin(d*x + c)^7 + 4620*a*sin(d*x + c) ^6 - 1848*a*sin(d*x + c)^5 - 2310*a*sin(d*x + c)^4)/d
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {840 \, a \sin \left (d x + c\right )^{11} + 924 \, a \sin \left (d x + c\right )^{10} - 3080 \, a \sin \left (d x + c\right )^{9} - 3465 \, a \sin \left (d x + c\right )^{8} + 3960 \, a \sin \left (d x + c\right )^{7} + 4620 \, a \sin \left (d x + c\right )^{6} - 1848 \, a \sin \left (d x + c\right )^{5} - 2310 \, a \sin \left (d x + c\right )^{4}}{9240 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/9240*(840*a*sin(d*x + c)^11 + 924*a*sin(d*x + c)^10 - 3080*a*sin(d*x + c)^9 - 3465*a*sin(d*x + c)^8 + 3960*a*sin(d*x + c)^7 + 4620*a*sin(d*x + c) ^6 - 1848*a*sin(d*x + c)^5 - 2310*a*sin(d*x + c)^4)/d
Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^{11}}{11}-\frac {a\,{\sin \left (c+d\,x\right )}^{10}}{10}+\frac {a\,{\sin \left (c+d\,x\right )}^9}{3}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{2}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \] Input:
int(cos(c + d*x)^7*sin(c + d*x)^3*(a + a*sin(c + d*x)),x)
Output:
((a*sin(c + d*x)^4)/4 + (a*sin(c + d*x)^5)/5 - (a*sin(c + d*x)^6)/2 - (3*a *sin(c + d*x)^7)/7 + (3*a*sin(c + d*x)^8)/8 + (a*sin(c + d*x)^9)/3 - (a*si n(c + d*x)^10)/10 - (a*sin(c + d*x)^11)/11)/d
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {\sin \left (d x +c \right )^{4} a \left (-840 \sin \left (d x +c \right )^{7}-924 \sin \left (d x +c \right )^{6}+3080 \sin \left (d x +c \right )^{5}+3465 \sin \left (d x +c \right )^{4}-3960 \sin \left (d x +c \right )^{3}-4620 \sin \left (d x +c \right )^{2}+1848 \sin \left (d x +c \right )+2310\right )}{9240 d} \] Input:
int(cos(d*x+c)^7*sin(d*x+c)^3*(a+a*sin(d*x+c)),x)
Output:
(sin(c + d*x)**4*a*( - 840*sin(c + d*x)**7 - 924*sin(c + d*x)**6 + 3080*si n(c + d*x)**5 + 3465*sin(c + d*x)**4 - 3960*sin(c + d*x)**3 - 4620*sin(c + d*x)**2 + 1848*sin(c + d*x) + 2310))/(9240*d)