\(\int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx\) [1000]

Optimal result

Integrand size = 31, antiderivative size = 105 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx=-\frac {(A+B) (a-a \sin (c+d x))^4}{a^5 d}+\frac {4 (A+2 B) (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(A+5 B) (a-a \sin (c+d x))^6}{6 a^7 d}+\frac {B (a-a \sin (c+d x))^7}{7 a^8 d} \] Output:

-(A+B)*(a-a*sin(d*x+c))^4/a^5/d+4/5*(A+2*B)*(a-a*sin(d*x+c))^5/a^6/d-1/6*( 
A+5*B)*(a-a*sin(d*x+c))^6/a^7/d+1/7*B*(a-a*sin(d*x+c))^7/a^8/d
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx=-\frac {(-1+\sin (c+d x))^4 \left (77 A+19 B+(98 A+76 B) \sin (c+d x)+5 (7 A+17 B) \sin ^2(c+d x)+30 B \sin ^3(c+d x)\right )}{210 a d} \] Input:

Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]
 

Output:

-1/210*((-1 + Sin[c + d*x])^4*(77*A + 19*B + (98*A + 76*B)*Sin[c + d*x] + 
5*(7*A + 17*B)*Sin[c + d*x]^2 + 30*B*Sin[c + d*x]^3))/(a*d)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 3315, 27, 86, 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.

\(\displaystyle \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^7 (A+B \sin (c+d x))}{a \sin (c+d x)+a}dx\)

\(\Big \downarrow \) 3315

\(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2 (a A+a B \sin (c+d x))}{a}d(a \sin (c+d x))}{a^7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2 (a A+a B \sin (c+d x))d(a \sin (c+d x))}{a^8 d}\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\int \left (-B (a-a \sin (c+d x))^6+a (A+5 B) (a-a \sin (c+d x))^5-4 a^2 (A+2 B) (a-a \sin (c+d x))^4+4 a^3 (A+B) (a-a \sin (c+d x))^3\right )d(a \sin (c+d x))}{a^8 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-a^3 (A+B) (a-a \sin (c+d x))^4+\frac {4}{5} a^2 (A+2 B) (a-a \sin (c+d x))^5-\frac {1}{6} a (A+5 B) (a-a \sin (c+d x))^6+\frac {1}{7} B (a-a \sin (c+d x))^7}{a^8 d}\)

Input:

Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]
 

Output:

(-(a^3*(A + B)*(a - a*Sin[c + d*x])^4) + (4*a^2*(A + 2*B)*(a - a*Sin[c + d 
*x])^5)/5 - (a*(A + 5*B)*(a - a*Sin[c + d*x])^6)/6 + (B*(a - a*Sin[c + d*x 
])^7)/7)/(a^8*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3315

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]
 

Maple [A] (verified)

Time = 1.44 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(-\frac {\frac {B \sin \left (d x +c \right )^{7}}{7}+\frac {\left (A -B \right ) \sin \left (d x +c \right )^{6}}{6}+\frac {\left (-A -2 B \right ) \sin \left (d x +c \right )^{5}}{5}+\frac {\left (2 B -2 A \right ) \sin \left (d x +c \right )^{4}}{4}+\frac {\left (2 A +B \right ) \sin \left (d x +c \right )^{3}}{3}+\frac {\left (A -B \right ) \sin \left (d x +c \right )^{2}}{2}-A \sin \left (d x +c \right )}{d a}\)	\(109\)
default	\(-\frac {\frac {B \sin \left (d x +c \right )^{7}}{7}+\frac {\left (A -B \right ) \sin \left (d x +c \right )^{6}}{6}+\frac {\left (-A -2 B \right ) \sin \left (d x +c \right )^{5}}{5}+\frac {\left (2 B -2 A \right ) \sin \left (d x +c \right )^{4}}{4}+\frac {\left (2 A +B \right ) \sin \left (d x +c \right )^{3}}{3}+\frac {\left (A -B \right ) \sin \left (d x +c \right )^{2}}{2}-A \sin \left (d x +c \right )}{d a}\)	\(109\)
parallelrisch	\(\frac {525 \left (A -B \right ) \cos \left (2 d x +2 c \right )+210 \left (A -B \right ) \cos \left (4 d x +4 c \right )+35 \left (A -B \right ) \cos \left (6 d x +6 c \right )+35 \left (20 A +B \right ) \sin \left (3 d x +3 c \right )+21 \left (4 A +3 B \right ) \sin \left (5 d x +5 c \right )+15 B \sin \left (7 d x +7 c \right )+525 \left (8 A -B \right ) \sin \left (d x +c \right )-770 A +770 B}{6720 d a}\)	\(125\)
risch	\(\frac {5 A \sin \left (d x +c \right )}{8 a d}-\frac {5 B \sin \left (d x +c \right )}{64 a d}+\frac {B \sin \left (7 d x +7 c \right )}{448 d a}+\frac {\cos \left (6 d x +6 c \right ) A}{192 a d}-\frac {\cos \left (6 d x +6 c \right ) B}{192 a d}+\frac {\sin \left (5 d x +5 c \right ) A}{80 d a}+\frac {3 \sin \left (5 d x +5 c \right ) B}{320 d a}+\frac {\cos \left (4 d x +4 c \right ) A}{32 a d}-\frac {\cos \left (4 d x +4 c \right ) B}{32 a d}+\frac {5 \sin \left (3 d x +3 c \right ) A}{48 d a}+\frac {\sin \left (3 d x +3 c \right ) B}{192 d a}+\frac {5 \cos \left (2 d x +2 c \right ) A}{64 a d}-\frac {5 \cos \left (2 d x +2 c \right ) B}{64 a d}\)	\(230\)
norman	\(\frac {\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{d a}+\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d a}+\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{d a}+\frac {2 \left (10 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d a}+\frac {2 \left (10 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{3 d a}+\frac {2 \left (7 A +2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d a}+\frac {2 \left (7 A +2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{3 d a}+\frac {2 \left (133 A +26 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d a}+\frac {2 \left (133 A +26 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{15 d a}+\frac {2 \left (98 A +61 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15 d a}+\frac {2 \left (98 A +61 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{15 d a}+\frac {2 \left (476 A +37 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d a}+\frac {2 \left (476 A +37 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{35 d a}+\frac {2 \left (1183 A +356 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{105 d a}+\frac {2 \left (1183 A +356 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{105 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\)	\(421\)

Input:

int(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE 
)
 

Output:

-1/d/a*(1/7*B*sin(d*x+c)^7+1/6*(A-B)*sin(d*x+c)^6+1/5*(-A-2*B)*sin(d*x+c)^ 
5+1/4*(2*B-2*A)*sin(d*x+c)^4+1/3*(2*A+B)*sin(d*x+c)^3+1/2*(A-B)*sin(d*x+c) 
^2-A*sin(d*x+c))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx=\frac {35 \, {\left (A - B\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (15 \, B \cos \left (d x + c\right )^{6} + 3 \, {\left (7 \, A - B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (7 \, A - B\right )} \cos \left (d x + c\right )^{2} + 56 \, A - 8 \, B\right )} \sin \left (d x + c\right )}{210 \, a d} \] Input:

integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x, algorithm="fri 
cas")
 

Output:

1/210*(35*(A - B)*cos(d*x + c)^6 + 2*(15*B*cos(d*x + c)^6 + 3*(7*A - B)*co 
s(d*x + c)^4 + 4*(7*A - B)*cos(d*x + c)^2 + 56*A - 8*B)*sin(d*x + c))/(a*d 
)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3363 vs. \(2 (94) = 188\).

Time = 35.70 (sec) , antiderivative size = 3363, normalized size of antiderivative = 32.03 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)**7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x)
 

Output:

Piecewise((210*A*tan(c/2 + d*x/2)**13/(105*a*d*tan(c/2 + d*x/2)**14 + 735* 
a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/ 
2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)** 
4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) - 210*A*tan(c/2 + d*x/2)**12/(1 
05*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan( 
c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2 
)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a* 
d) + 700*A*tan(c/2 + d*x/2)**11/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*ta 
n(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d* 
x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 73 
5*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) - 210*A*tan(c/2 + d*x/2)**10/(105*a*d 
*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + 
d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 
 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) + 1 
582*A*tan(c/2 + d*x/2)**9/(105*a*d*tan(c/2 + d*x/2)**14 + 735*a*d*tan(c/2 
+ d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)**10 + 3675*a*d*tan(c/2 + d*x/2)** 
8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 2205*a*d*tan(c/2 + d*x/2)**4 + 735*a*d* 
tan(c/2 + d*x/2)**2 + 105*a*d) - 700*A*tan(c/2 + d*x/2)**8/(105*a*d*tan(c/ 
2 + d*x/2)**14 + 735*a*d*tan(c/2 + d*x/2)**12 + 2205*a*d*tan(c/2 + d*x/2)* 
*10 + 3675*a*d*tan(c/2 + d*x/2)**8 + 3675*a*d*tan(c/2 + d*x/2)**6 + 220...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx=-\frac {30 \, B \sin \left (d x + c\right )^{7} + 35 \, {\left (A - B\right )} \sin \left (d x + c\right )^{6} - 42 \, {\left (A + 2 \, B\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (A - B\right )} \sin \left (d x + c\right )^{4} + 70 \, {\left (2 \, A + B\right )} \sin \left (d x + c\right )^{3} + 105 \, {\left (A - B\right )} \sin \left (d x + c\right )^{2} - 210 \, A \sin \left (d x + c\right )}{210 \, a d} \] Input:

integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x, algorithm="max 
ima")
 

Output:

-1/210*(30*B*sin(d*x + c)^7 + 35*(A - B)*sin(d*x + c)^6 - 42*(A + 2*B)*sin 
(d*x + c)^5 - 105*(A - B)*sin(d*x + c)^4 + 70*(2*A + B)*sin(d*x + c)^3 + 1 
05*(A - B)*sin(d*x + c)^2 - 210*A*sin(d*x + c))/(a*d)
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx=-\frac {30 \, B \sin \left (d x + c\right )^{7} + 35 \, A \sin \left (d x + c\right )^{6} - 35 \, B \sin \left (d x + c\right )^{6} - 42 \, A \sin \left (d x + c\right )^{5} - 84 \, B \sin \left (d x + c\right )^{5} - 105 \, A \sin \left (d x + c\right )^{4} + 105 \, B \sin \left (d x + c\right )^{4} + 140 \, A \sin \left (d x + c\right )^{3} + 70 \, B \sin \left (d x + c\right )^{3} + 105 \, A \sin \left (d x + c\right )^{2} - 105 \, B \sin \left (d x + c\right )^{2} - 210 \, A \sin \left (d x + c\right )}{210 \, a d} \] Input:

integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x, algorithm="gia 
c")
 

Output:

-1/210*(30*B*sin(d*x + c)^7 + 35*A*sin(d*x + c)^6 - 35*B*sin(d*x + c)^6 - 
42*A*sin(d*x + c)^5 - 84*B*sin(d*x + c)^5 - 105*A*sin(d*x + c)^4 + 105*B*s 
in(d*x + c)^4 + 140*A*sin(d*x + c)^3 + 70*B*sin(d*x + c)^3 + 105*A*sin(d*x 
 + c)^2 - 105*B*sin(d*x + c)^2 - 210*A*sin(d*x + c))/(a*d)
 

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx=-\frac {\frac {{\sin \left (c+d\,x\right )}^2\,\left (A-B\right )}{2\,a}+\frac {{\sin \left (c+d\,x\right )}^3\,\left (2\,A+B\right )}{3\,a}-\frac {{\sin \left (c+d\,x\right )}^5\,\left (A+2\,B\right )}{5\,a}+\frac {{\sin \left (c+d\,x\right )}^6\,\left (A-B\right )}{6\,a}+\frac {B\,{\sin \left (c+d\,x\right )}^7}{7\,a}-\frac {{\sin \left (c+d\,x\right )}^4\,\left (2\,A-2\,B\right )}{4\,a}-\frac {A\,\sin \left (c+d\,x\right )}{a}}{d} \] Input:

int((cos(c + d*x)^7*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x)),x)
 

Output:

-((sin(c + d*x)^2*(A - B))/(2*a) + (sin(c + d*x)^3*(2*A + B))/(3*a) - (sin 
(c + d*x)^5*(A + 2*B))/(5*a) + (sin(c + d*x)^6*(A - B))/(6*a) + (B*sin(c + 
 d*x)^7)/(7*a) - (sin(c + d*x)^4*(2*A - 2*B))/(4*a) - (A*sin(c + d*x))/a)/ 
d
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.29 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx=\frac {\sin \left (d x +c \right ) \left (-30 \sin \left (d x +c \right )^{6} b -35 \sin \left (d x +c \right )^{5} a +35 \sin \left (d x +c \right )^{5} b +42 \sin \left (d x +c \right )^{4} a +84 \sin \left (d x +c \right )^{4} b +105 \sin \left (d x +c \right )^{3} a -105 \sin \left (d x +c \right )^{3} b -140 \sin \left (d x +c \right )^{2} a -70 \sin \left (d x +c \right )^{2} b -105 \sin \left (d x +c \right ) a +105 \sin \left (d x +c \right ) b +210 a \right )}{210 a d} \] Input:

int(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x)
 

Output:

(sin(c + d*x)*( - 30*sin(c + d*x)**6*b - 35*sin(c + d*x)**5*a + 35*sin(c + 
 d*x)**5*b + 42*sin(c + d*x)**4*a + 84*sin(c + d*x)**4*b + 105*sin(c + d*x 
)**3*a - 105*sin(c + d*x)**3*b - 140*sin(c + d*x)**2*a - 70*sin(c + d*x)** 
2*b - 105*sin(c + d*x)*a + 105*sin(c + d*x)*b + 210*a))/(210*a*d)