3.1003 \(\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx\)
Optimal. Leaf size=105 \[ -\frac{(A+5 B) (a-a \sin (c+d x))^6}{6 a^7 d}+\frac{4 (A+2 B) (a-a \sin (c+d x))^5}{5 a^6 d}-\frac{(A+B) (a-a \sin (c+d x))^4}{a^5 d}+\frac{B (a-a \sin (c+d x))^7}{7 a^8 d} \]
[Out]
-(((A + B)*(a - a*Sin[c + d*x])^4)/(a^5*d)) + (4*(A + 2*B)*(a - a*Sin[c + d*x])^5)/(5*a^6*d) - ((A + 5*B)*(a -
a*Sin[c + d*x])^6)/(6*a^7*d) + (B*(a - a*Sin[c + d*x])^7)/(7*a^8*d)
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Rubi [A] time = 0.14689, antiderivative size = 105, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used =
{2836, 77} \[ -\frac{(A+5 B) (a-a \sin (c+d x))^6}{6 a^7 d}+\frac{4 (A+2 B) (a-a \sin (c+d x))^5}{5 a^6 d}-\frac{(A+B) (a-a \sin (c+d x))^4}{a^5 d}+\frac{B (a-a \sin (c+d x))^7}{7 a^8 d} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]
[Out]
-(((A + B)*(a - a*Sin[c + d*x])^4)/(a^5*d)) + (4*(A + 2*B)*(a - a*Sin[c + d*x])^5)/(5*a^6*d) - ((A + 5*B)*(a -
a*Sin[c + d*x])^6)/(6*a^7*d) + (B*(a - a*Sin[c + d*x])^7)/(7*a^8*d)
Rule 2836
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))
Rubi steps
\begin{align*} \int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 (a+x)^2 \left (A+\frac{B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (A+B) (a-x)^3-4 a (A+2 B) (a-x)^4+(A+5 B) (a-x)^5-\frac{B (a-x)^6}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\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}\\ \end{align*}
Mathematica [A] time = 0.216757, size = 69, normalized size = 0.66 \[ -\frac{(\sin (c+d x)-1)^4 \left (5 (7 A+17 B) \sin ^2(c+d x)+(98 A+76 B) \sin (c+d x)+77 A+30 B \sin ^3(c+d x)+19 B\right )}{210 a d} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]
[Out]
-((-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))/(210*a*d)
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Maple [A] time = 0.108, size = 107, normalized size = 1. \begin{align*}{\frac{1}{da} \left ( -{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{ \left ( -A+B \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6}}+{\frac{ \left ( A+2\,B \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( 2\,A-2\,B \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{ \left ( -2\,A-B \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{ \left ( -A+B \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2}}+A\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x)
[Out]
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*A-2*B)*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))
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Maxima [A] time = 1.06914, size = 140, normalized size = 1.33 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
-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 + 105*(A - B)*sin(d*x + c)^2 - 210*A*sin(d*x + c))/(a*d)
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Fricas [A] time = 1.82632, size = 204, normalized size = 1.94 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
1/210*(35*(A - B)*cos(d*x + c)^6 + 2*(15*B*cos(d*x + c)^6 + 3*(7*A - B)*cos(d*x + c)^4 + 4*(7*A - B)*cos(d*x +
c)^2 + 56*A - 8*B)*sin(d*x + c))/(a*d)
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Sympy [A] time = 138.232, size = 3896, normalized size = 37.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x)
[Out]
Piecewise((-35*A*tan(c/2 + d*x/2)**14/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d
*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2
)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 420*A*tan(c/2 + d*x/2)**13/(210*a*d*tan(c/2 + d*x/2)**14 + 14
70*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2
+ d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) - 665*A*tan(c/2 + d*x/2)*
*12/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*t
an(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**
2 + 210*a*d) + 1400*A*tan(c/2 + d*x/2)**11/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 441
0*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 +
d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) - 1155*A*tan(c/2 + d*x/2)**10/(210*a*d*tan(c/2 + d*x/2)**1
4 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*ta
n(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 3164*A*tan(c/2 +
d*x/2)**9/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350
*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*
x/2)**2 + 210*a*d) - 2625*A*tan(c/2 + d*x/2)**8/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12
+ 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c
/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 4368*A*tan(c/2 + d*x/2)**7/(210*a*d*tan(c/2 + d*x/2
)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*
d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) - 2625*A*tan(c/
2 + d*x/2)**6/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 +
7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2
+ d*x/2)**2 + 210*a*d) + 3164*A*tan(c/2 + d*x/2)**5/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)*
*12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*t
an(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) - 1155*A*tan(c/2 + d*x/2)**4/(210*a*d*tan(c/2 + d
*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 735
0*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 1400*A*ta
n(c/2 + d*x/2)**3/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**1
0 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(
c/2 + d*x/2)**2 + 210*a*d) - 665*A*tan(c/2 + d*x/2)**2/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/
2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*
d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 420*A*tan(c/2 + d*x/2)/(210*a*d*tan(c/2 + d*
x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350
*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) - 35*A/(210*
a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 +
d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a
*d) - 10*B*tan(c/2 + d*x/2)**14/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c
/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 +
1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 350*B*tan(c/2 + d*x/2)**12/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d
*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/
2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) - 560*B*tan(c/2 + d*x/2)**11/(2
10*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2
+ d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 21
0*a*d) + 210*B*tan(c/2 + d*x/2)**10/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*t
an(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)*
*4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 448*B*tan(c/2 + d*x/2)**9/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*
a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d
*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 1050*B*tan(c/2 + d*x/2)**8
/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(
c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 +
210*a*d) - 1824*B*tan(c/2 + d*x/2)**7/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*
d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/
2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 1050*B*tan(c/2 + d*x/2)**6/(210*a*d*tan(c/2 + d*x/2)**14 + 1
470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2
+ d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 448*B*tan(c/2 + d*x/2)
**5/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*t
an(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**
2 + 210*a*d) + 210*B*tan(c/2 + d*x/2)**4/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*
a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*
x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) - 560*B*tan(c/2 + d*x/2)**3/(210*a*d*tan(c/2 + d*x/2)**14 +
1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/
2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)**2 + 210*a*d) + 350*B*tan(c/2 + d*x/2
)**2/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*
tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*tan(c/2 + d*x/2)*
*2 + 210*a*d) - 10*B/(210*a*d*tan(c/2 + d*x/2)**14 + 1470*a*d*tan(c/2 + d*x/2)**12 + 4410*a*d*tan(c/2 + d*x/2)
**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 7350*a*d*tan(c/2 + d*x/2)**6 + 4410*a*d*tan(c/2 + d*x/2)**4 + 1470*a*d*t
an(c/2 + d*x/2)**2 + 210*a*d), Ne(d, 0)), (x*(A + B*sin(c))*cos(c)**7/(a*sin(c) + a), True))
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Giac [A] time = 1.29216, size = 188, normalized size = 1.79 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
-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*sin(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)