Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^6(c+d x)}{6 a d}-\frac {\sin ^7(c+d x)}{7 a d}-\frac {\sin ^8(c+d x)}{4 a d}+\frac {2 \sin ^9(c+d x)}{9 a d}+\frac {\sin ^{10}(c+d x)}{10 a d}-\frac {\sin ^{11}(c+d x)}{11 a d} \] Output:
1/6*sin(d*x+c)^6/a/d-1/7*sin(d*x+c)^7/a/d-1/4*sin(d*x+c)^8/a/d+2/9*sin(d*x +c)^9/a/d+1/10*sin(d*x+c)^10/a/d-1/11*sin(d*x+c)^11/a/d
Time = 0.38 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^6(c+d x) \left (2310-1980 \sin (c+d x)-3465 \sin ^2(c+d x)+3080 \sin ^3(c+d x)+1386 \sin ^4(c+d x)-1260 \sin ^5(c+d x)\right )}{13860 a d} \] Input:
Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]
Output:
(Sin[c + d*x]^6*(2310 - 1980*Sin[c + d*x] - 3465*Sin[c + d*x]^2 + 3080*Sin [c + d*x]^3 + 1386*Sin[c + d*x]^4 - 1260*Sin[c + d*x]^5))/(13860*a*d)
Time = 0.35 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 27, 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.
\(\displaystyle \int \frac {\sin ^5(c+d x) \cos ^7(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x)^5 \cos (c+d x)^7}{a \sin (c+d x)+a}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \sin ^5(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int a^5 \sin ^5(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^{12} d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (-\sin ^{10}(c+d x) a^{10}+\sin ^9(c+d x) a^{10}+2 \sin ^8(c+d x) a^{10}-2 \sin ^7(c+d x) a^{10}-\sin ^6(c+d x) a^{10}+\sin ^5(c+d x) a^{10}\right )d(a \sin (c+d x))}{a^{12} d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{11} a^{11} \sin ^{11}(c+d x)+\frac {1}{10} a^{11} \sin ^{10}(c+d x)+\frac {2}{9} a^{11} \sin ^9(c+d x)-\frac {1}{4} a^{11} \sin ^8(c+d x)-\frac {1}{7} a^{11} \sin ^7(c+d x)+\frac {1}{6} a^{11} \sin ^6(c+d x)}{a^{12} d}\) |
Input:
Int[(Cos[c + d*x]^7*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]
Output:
((a^11*Sin[c + d*x]^6)/6 - (a^11*Sin[c + d*x]^7)/7 - (a^11*Sin[c + d*x]^8) /4 + (2*a^11*Sin[c + d*x]^9)/9 + (a^11*Sin[c + d*x]^10)/10 - (a^11*Sin[c + d*x]^11)/11)/(a^12*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 1.40 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(-\frac {\frac {\sin \left (d x +c \right )^{11}}{11}-\frac {\sin \left (d x +c \right )^{10}}{10}-\frac {2 \sin \left (d x +c \right )^{9}}{9}+\frac {\sin \left (d x +c \right )^{8}}{4}+\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{6}}{6}}{d a}\) | \(70\) |
default | \(-\frac {\frac {\sin \left (d x +c \right )^{11}}{11}-\frac {\sin \left (d x +c \right )^{10}}{10}-\frac {2 \sin \left (d x +c \right )^{9}}{9}+\frac {\sin \left (d x +c \right )^{8}}{4}+\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{6}}{6}}{d a}\) | \(70\) |
parallelrisch | \(\frac {\left (-10+\cos \left (3 d x +3 c \right )-6 \cos \left (2 d x +2 c \right )+15 \cos \left (d x +c \right )\right ) \left (315 \sin \left (5 d x +5 c \right )-4158 \cos \left (2 d x +2 c \right )+1830 \sin \left (d x +c \right )+1505 \sin \left (3 d x +3 c \right )-693 \cos \left (4 d x +4 c \right )-4389\right ) \left (\cos \left (3 d x +3 c \right )+6 \cos \left (2 d x +2 c \right )+15 \cos \left (d x +c \right )+10\right )}{887040 d a}\) | \(123\) |
risch | \(-\frac {5 \sin \left (d x +c \right )}{512 a d}+\frac {\sin \left (11 d x +11 c \right )}{11264 d a}-\frac {\cos \left (10 d x +10 c \right )}{5120 a d}-\frac {\sin \left (9 d x +9 c \right )}{9216 d a}-\frac {5 \sin \left (7 d x +7 c \right )}{7168 d a}+\frac {5 \cos \left (6 d x +6 c \right )}{3072 a d}+\frac {\sin \left (5 d x +5 c \right )}{1024 d a}+\frac {5 \sin \left (3 d x +3 c \right )}{1536 d a}-\frac {5 \cos \left (2 d x +2 c \right )}{512 a d}\) | \(152\) |
Input:
int(cos(d*x+c)^7*sin(d*x+c)^5/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/d/a*(1/11*sin(d*x+c)^11-1/10*sin(d*x+c)^10-2/9*sin(d*x+c)^9+1/4*sin(d*x +c)^8+1/7*sin(d*x+c)^7-1/6*sin(d*x+c)^6)
Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {1386 \, \cos \left (d x + c\right )^{10} - 3465 \, \cos \left (d x + c\right )^{8} + 2310 \, \cos \left (d x + c\right )^{6} - 20 \, {\left (63 \, \cos \left (d x + c\right )^{10} - 161 \, \cos \left (d x + c\right )^{8} + 113 \, \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right )}{13860 \, a d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/13860*(1386*cos(d*x + c)^10 - 3465*cos(d*x + c)^8 + 2310*cos(d*x + c)^6 - 20*(63*cos(d*x + c)^10 - 161*cos(d*x + c)^8 + 113*cos(d*x + c)^6 - 3*co s(d*x + c)^4 - 4*cos(d*x + c)^2 - 8)*sin(d*x + c))/(a*d)
Timed out. \[ \int \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**7*sin(d*x+c)**5/(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {1260 \, \sin \left (d x + c\right )^{11} - 1386 \, \sin \left (d x + c\right )^{10} - 3080 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} + 1980 \, \sin \left (d x + c\right )^{7} - 2310 \, \sin \left (d x + c\right )^{6}}{13860 \, a d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/13860*(1260*sin(d*x + c)^11 - 1386*sin(d*x + c)^10 - 3080*sin(d*x + c)^ 9 + 3465*sin(d*x + c)^8 + 1980*sin(d*x + c)^7 - 2310*sin(d*x + c)^6)/(a*d)
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {1260 \, \sin \left (d x + c\right )^{11} - 1386 \, \sin \left (d x + c\right )^{10} - 3080 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} + 1980 \, \sin \left (d x + c\right )^{7} - 2310 \, \sin \left (d x + c\right )^{6}}{13860 \, a d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/13860*(1260*sin(d*x + c)^11 - 1386*sin(d*x + c)^10 - 3080*sin(d*x + c)^ 9 + 3465*sin(d*x + c)^8 + 1980*sin(d*x + c)^7 - 2310*sin(d*x + c)^6)/(a*d)
Time = 31.53 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}-\frac {{\sin \left (c+d\,x\right )}^7}{7\,a}-\frac {{\sin \left (c+d\,x\right )}^8}{4\,a}+\frac {2\,{\sin \left (c+d\,x\right )}^9}{9\,a}+\frac {{\sin \left (c+d\,x\right )}^{10}}{10\,a}-\frac {{\sin \left (c+d\,x\right )}^{11}}{11\,a}}{d} \] Input:
int((cos(c + d*x)^7*sin(c + d*x)^5)/(a + a*sin(c + d*x)),x)
Output:
(sin(c + d*x)^6/(6*a) - sin(c + d*x)^7/(7*a) - sin(c + d*x)^8/(4*a) + (2*s in(c + d*x)^9)/(9*a) + sin(c + d*x)^10/(10*a) - sin(c + d*x)^11/(11*a))/d
Time = 1.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin \left (d x +c \right )^{6} \left (-1260 \sin \left (d x +c \right )^{5}+1386 \sin \left (d x +c \right )^{4}+3080 \sin \left (d x +c \right )^{3}-3465 \sin \left (d x +c \right )^{2}-1980 \sin \left (d x +c \right )+2310\right )}{13860 a d} \] Input:
int(cos(d*x+c)^7*sin(d*x+c)^5/(a+a*sin(d*x+c)),x)
Output:
(sin(c + d*x)**6*( - 1260*sin(c + d*x)**5 + 1386*sin(c + d*x)**4 + 3080*si n(c + d*x)**3 - 3465*sin(c + d*x)**2 - 1980*sin(c + d*x) + 2310))/(13860*a *d)