\(\int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx\) [1544]

Optimal result

Integrand size = 31, antiderivative size = 315 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}+\frac {\left (a^5 A b-3 a^3 A b^3+3 a A b^5-a^6 B+3 a^4 b^2 B-3 a^2 b^4 B+b^6 B\right ) \sin (c+d x)}{b^7 d}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^3 A b-3 a A b^3-a^4 B+3 a^2 b^2 B-3 b^4 B\right ) \sin ^3(c+d x)}{3 b^5 d}-\frac {\left (a^2-3 b^2\right ) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a A b-a^2 B+3 b^2 B\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac {B \sin ^7(c+d x)}{7 b d} \]

[Out]

-(a^2-b^2)^3*(A*b-B*a)*ln(a+b*sin(d*x+c))/b^8/d+(A*a^5*b-3*A*a^3*b^3+3*A*a*b^5-B*a^6+3*B*a^4*b^2-3*B*a^2*b^4+B
*b^6)*sin(d*x+c)/b^7/d-1/2*(a^4-3*a^2*b^2+3*b^4)*(A*b-B*a)*sin(d*x+c)^2/b^6/d+1/3*(A*a^3*b-3*A*a*b^3-B*a^4+3*B
*a^2*b^2-3*B*b^4)*sin(d*x+c)^3/b^5/d-1/4*(a^2-3*b^2)*(A*b-B*a)*sin(d*x+c)^4/b^4/d+1/5*(A*a*b-B*a^2+3*B*b^2)*si
n(d*x+c)^5/b^3/d-1/6*(A*b-B*a)*sin(d*x+c)^6/b^2/d-1/7*B*sin(d*x+c)^7/b/d

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2916, 786} \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}-\frac {\left (a^2-3 b^2\right ) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2 (-B)+a A b+3 b^2 B\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^4 (-B)+a^3 A b+3 a^2 b^2 B-3 a A b^3-3 b^4 B\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac {\left (a^6 (-B)+a^5 A b+3 a^4 b^2 B-3 a^3 A b^3-3 a^2 b^4 B+3 a A b^5+b^6 B\right ) \sin (c+d x)}{b^7 d}-\frac {(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac {B \sin ^7(c+d x)}{7 b d} \]

[In]

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

[Out]

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

Rule 786

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 2916

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*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (A+\frac {B x}{b}\right ) \left (b^2-x^2\right )^3}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^7 d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^5 A b-3 a^3 A b^3+3 a A b^5-a^6 B+3 a^4 b^2 B-3 a^2 b^4 B+b^6 B}{b}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) (A b-a B) x}{b}-\frac {\left (-a^3 A b+3 a A b^3+a^4 B-3 a^2 b^2 B+3 b^4 B\right ) x^2}{b}+\frac {\left (-a^2+3 b^2\right ) (A b-a B) x^3}{b}+\frac {\left (a A b-a^2 B+3 b^2 B\right ) x^4}{b}-\frac {(A b-a B) x^5}{b}-\frac {B x^6}{b}+\frac {\left (-a^2+b^2\right )^3 (A b-a B)}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d} \\ & = -\frac {\left (a^2-b^2\right )^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}+\frac {\left (a^5 A b-3 a^3 A b^3+3 a A b^5-a^6 B+3 a^4 b^2 B-3 a^2 b^4 B+b^6 B\right ) \sin (c+d x)}{b^7 d}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^3 A b-3 a A b^3-a^4 B+3 a^2 b^2 B-3 b^4 B\right ) \sin ^3(c+d x)}{3 b^5 d}-\frac {\left (a^2-3 b^2\right ) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a A b-a^2 B+3 b^2 B\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac {B \sin ^7(c+d x)}{7 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {\frac {(A b-a B) \left (15 b^4 \left (-a^2+b^2\right ) \cos ^4(c+d x)+10 b^6 \cos ^6(c+d x)-60 \left (a^2-b^2\right )^3 \log (a+b \sin (c+d x))+60 a b \left (a^4-3 a^2 b^2+3 b^4\right ) \sin (c+d x)-30 b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)+20 a b^3 \left (a^2-3 b^2\right ) \sin ^3(c+d x)+12 a b^5 \sin ^5(c+d x)\right )}{60 b}+\frac {b^6 B (1225 \sin (c+d x)+245 \sin (3 (c+d x))+49 \sin (5 (c+d x))+5 \sin (7 (c+d x)))}{2240}}{b^7 d} \]

[In]

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

[Out]

(((A*b - a*B)*(15*b^4*(-a^2 + b^2)*Cos[c + d*x]^4 + 10*b^6*Cos[c + d*x]^6 - 60*(a^2 - b^2)^3*Log[a + b*Sin[c +
d*x]] + 60*a*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Sin[c + d*x] - 30*b^2*(a^2 - b^2)^2*Sin[c + d*x]^2 + 20*a*b^3*(a^2 -
3*b^2)*Sin[c + d*x]^3 + 12*a*b^5*Sin[c + d*x]^5))/(60*b) + (b^6*B*(1225*Sin[c + d*x] + 245*Sin[3*(c + d*x)] +
49*Sin[5*(c + d*x)] + 5*Sin[7*(c + d*x)]))/2240)/(b^7*d)

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.19

method	result	size
parallelrisch	\(\frac {-6720 \left (a -b \right )^{3} \left (a +b \right )^{3} \left (A b -B a \right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )+6720 \left (a -b \right )^{3} \left (a +b \right )^{3} \left (A b -B a \right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 b \left (48 b \left (a^{4}-\frac {5}{2} a^{2} b^{2}+\frac {29}{16} b^{4}\right ) \left (A b -B a \right ) \cos \left (2 d x +2 c \right )+\left (-16 A \,a^{3} b^{3}+36 A a \,b^{5}+16 B \,a^{4} b^{2}-36 B \,a^{2} b^{4}+21 B \,b^{6}\right ) \sin \left (3 d x +3 c \right )-6 b^{3} \left (a^{2}-2 b^{2}\right ) \left (A b -B a \right ) \cos \left (4 d x +4 c \right )+\frac {12 b^{4} \left (A a b -B \,a^{2}+\frac {7}{4} B \,b^{2}\right ) \sin \left (5 d x +5 c \right )}{5}+\left (A \,b^{6}-B a \,b^{5}\right ) \cos \left (6 d x +6 c \right )+\frac {3 B \,b^{6} \sin \left (7 d x +7 c \right )}{7}+\left (192 A \,a^{5} b -528 A \,a^{3} b^{3}+456 A a \,b^{5}-192 B \,a^{6}+528 B \,a^{4} b^{2}-456 B \,a^{2} b^{4}+105 B \,b^{6}\right ) \sin \left (d x +c \right )-48 \left (a^{4}-\frac {21}{8} a^{2} b^{2}+\frac {25}{12} b^{4}\right ) b \left (A b -B a \right )\right )}{6720 d \,b^{8}}\)	\(374\)
derivativedivides	\(-\frac {-\frac {3 B \,a^{4} b^{2} \sin \left (d x +c \right )-3 B \,a^{2} b^{4} \sin \left (d x +c \right )+A \,a^{5} b \sin \left (d x +c \right )-3 A \,a^{3} b^{3} \sin \left (d x +c \right )+3 A a \,b^{5} \sin \left (d x +c \right )-B \,a^{6} \sin \left (d x +c \right )+B \,b^{6} \sin \left (d x +c \right )+B \,a^{2} b^{4} \left (\sin ^{3}\left (d x +c \right )\right )+\frac {3 B a \,b^{5} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+\frac {3 A \,a^{2} b^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+\frac {B \,a^{5} b \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\frac {3 B \,a^{3} b^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\frac {A \,a^{4} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\frac {3 B a \,b^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {A \,a^{3} b^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-A a \,b^{5} \left (\sin ^{3}\left (d x +c \right )\right )-\frac {B \,a^{4} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {A \,a^{2} b^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {B \,a^{3} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {B a \,b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {A a \,b^{5} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {B \,a^{2} b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {B \left (\sin ^{7}\left (d x +c \right )\right ) b^{6}}{7}-\frac {A \,b^{6} \left (\sin ^{6}\left (d x +c \right )\right )}{6}-B \,b^{6} \left (\sin ^{3}\left (d x +c \right )\right )+\frac {3 B \,b^{6} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {3 A \,b^{6} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+\frac {3 A \,b^{6} \left (\sin ^{4}\left (d x +c \right )\right )}{4}}{b^{7}}+\frac {\left (A \,a^{6} b -3 A \,a^{4} b^{3}+3 A \,a^{2} b^{5}-A \,b^{7}-B \,a^{7}+3 B \,a^{5} b^{2}-3 B \,a^{3} b^{4}+B a \,b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{8}}}{d}\)	\(508\)
default	\(-\frac {-\frac {3 B \,a^{4} b^{2} \sin \left (d x +c \right )-3 B \,a^{2} b^{4} \sin \left (d x +c \right )+A \,a^{5} b \sin \left (d x +c \right )-3 A \,a^{3} b^{3} \sin \left (d x +c \right )+3 A a \,b^{5} \sin \left (d x +c \right )-B \,a^{6} \sin \left (d x +c \right )+B \,b^{6} \sin \left (d x +c \right )+B \,a^{2} b^{4} \left (\sin ^{3}\left (d x +c \right )\right )+\frac {3 B a \,b^{5} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+\frac {3 A \,a^{2} b^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+\frac {B \,a^{5} b \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\frac {3 B \,a^{3} b^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\frac {A \,a^{4} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\frac {3 B a \,b^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {A \,a^{3} b^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-A a \,b^{5} \left (\sin ^{3}\left (d x +c \right )\right )-\frac {B \,a^{4} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {A \,a^{2} b^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {B \,a^{3} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {B a \,b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {A a \,b^{5} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {B \,a^{2} b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {B \left (\sin ^{7}\left (d x +c \right )\right ) b^{6}}{7}-\frac {A \,b^{6} \left (\sin ^{6}\left (d x +c \right )\right )}{6}-B \,b^{6} \left (\sin ^{3}\left (d x +c \right )\right )+\frac {3 B \,b^{6} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {3 A \,b^{6} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+\frac {3 A \,b^{6} \left (\sin ^{4}\left (d x +c \right )\right )}{4}}{b^{7}}+\frac {\left (A \,a^{6} b -3 A \,a^{4} b^{3}+3 A \,a^{2} b^{5}-A \,b^{7}-B \,a^{7}+3 B \,a^{5} b^{2}-3 B \,a^{3} b^{4}+B a \,b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{8}}}{d}\)	\(508\)
norman	\(\text {Expression too large to display}\)	\(1222\)
risch	\(\text {Expression too large to display}\)	\(1337\)

[In]

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

[Out]

1/6720*(-6720*(a-b)^3*(a+b)^3*(A*b-B*a)*ln(2*b*tan(1/2*d*x+1/2*c)+a*sec(1/2*d*x+1/2*c)^2)+6720*(a-b)^3*(a+b)^3
*(A*b-B*a)*ln(sec(1/2*d*x+1/2*c)^2)+35*b*(48*b*(a^4-5/2*a^2*b^2+29/16*b^4)*(A*b-B*a)*cos(2*d*x+2*c)+(-16*A*a^3
*b^3+36*A*a*b^5+16*B*a^4*b^2-36*B*a^2*b^4+21*B*b^6)*sin(3*d*x+3*c)-6*b^3*(a^2-2*b^2)*(A*b-B*a)*cos(4*d*x+4*c)+
12/5*b^4*(A*a*b-B*a^2+7/4*B*b^2)*sin(5*d*x+5*c)+(A*b^6-B*a*b^5)*cos(6*d*x+6*c)+3/7*B*b^6*sin(7*d*x+7*c)+(192*A
*a^5*b-528*A*a^3*b^3+456*A*a*b^5-192*B*a^6+528*B*a^4*b^2-456*B*a^2*b^4+105*B*b^6)*sin(d*x+c)-48*(a^4-21/8*a^2*
b^2+25/12*b^4)*b*(A*b-B*a)))/d/b^8

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {70 \, {\left (B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{6} - 105 \, {\left (B a^{3} b^{4} - A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \cos \left (d x + c\right )^{4} + 210 \, {\left (B a^{5} b^{2} - A a^{4} b^{3} - 2 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \, {\left (15 \, B b^{7} \cos \left (d x + c\right )^{6} - 105 \, B a^{6} b + 105 \, A a^{5} b^{2} + 280 \, B a^{4} b^{3} - 280 \, A a^{3} b^{4} - 231 \, B a^{2} b^{5} + 231 \, A a b^{6} + 48 \, B b^{7} - 3 \, {\left (7 \, B a^{2} b^{5} - 7 \, A a b^{6} - 6 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (35 \, B a^{4} b^{3} - 35 \, A a^{3} b^{4} - 63 \, B a^{2} b^{5} + 63 \, A a b^{6} + 24 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{420 \, b^{8} d} \]

[In]

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

[Out]

-1/420*(70*(B*a*b^6 - A*b^7)*cos(d*x + c)^6 - 105*(B*a^3*b^4 - A*a^2*b^5 - B*a*b^6 + A*b^7)*cos(d*x + c)^4 + 2
10*(B*a^5*b^2 - A*a^4*b^3 - 2*B*a^3*b^4 + 2*A*a^2*b^5 + B*a*b^6 - A*b^7)*cos(d*x + c)^2 - 420*(B*a^7 - A*a^6*b
- 3*B*a^5*b^2 + 3*A*a^4*b^3 + 3*B*a^3*b^4 - 3*A*a^2*b^5 - B*a*b^6 + A*b^7)*log(b*sin(d*x + c) + a) - 4*(15*B*
b^7*cos(d*x + c)^6 - 105*B*a^6*b + 105*A*a^5*b^2 + 280*B*a^4*b^3 - 280*A*a^3*b^4 - 231*B*a^2*b^5 + 231*A*a*b^6
+ 48*B*b^7 - 3*(7*B*a^2*b^5 - 7*A*a*b^6 - 6*B*b^7)*cos(d*x + c)^4 + (35*B*a^4*b^3 - 35*A*a^3*b^4 - 63*B*a^2*b
^5 + 63*A*a*b^6 + 24*B*b^7)*cos(d*x + c)^2)*sin(d*x + c))/(b^8*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\frac {60 \, B b^{6} \sin \left (d x + c\right )^{7} - 70 \, {\left (B a b^{5} - A b^{6}\right )} \sin \left (d x + c\right )^{6} + 84 \, {\left (B a^{2} b^{4} - A a b^{5} - 3 \, B b^{6}\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (B a^{3} b^{3} - A a^{2} b^{4} - 3 \, B a b^{5} + 3 \, A b^{6}\right )} \sin \left (d x + c\right )^{4} + 140 \, {\left (B a^{4} b^{2} - A a^{3} b^{3} - 3 \, B a^{2} b^{4} + 3 \, A a b^{5} + 3 \, B b^{6}\right )} \sin \left (d x + c\right )^{3} - 210 \, {\left (B a^{5} b - A a^{4} b^{2} - 3 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4} + 3 \, B a b^{5} - 3 \, A b^{6}\right )} \sin \left (d x + c\right )^{2} + 420 \, {\left (B a^{6} - A a^{5} b - 3 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3} + 3 \, B a^{2} b^{4} - 3 \, A a b^{5} - B b^{6}\right )} \sin \left (d x + c\right )}{b^{7}} - \frac {420 \, {\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{420 \, d} \]

[In]

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

[Out]

-1/420*((60*B*b^6*sin(d*x + c)^7 - 70*(B*a*b^5 - A*b^6)*sin(d*x + c)^6 + 84*(B*a^2*b^4 - A*a*b^5 - 3*B*b^6)*si
n(d*x + c)^5 - 105*(B*a^3*b^3 - A*a^2*b^4 - 3*B*a*b^5 + 3*A*b^6)*sin(d*x + c)^4 + 140*(B*a^4*b^2 - A*a^3*b^3 -
3*B*a^2*b^4 + 3*A*a*b^5 + 3*B*b^6)*sin(d*x + c)^3 - 210*(B*a^5*b - A*a^4*b^2 - 3*B*a^3*b^3 + 3*A*a^2*b^4 + 3*
B*a*b^5 - 3*A*b^6)*sin(d*x + c)^2 + 420*(B*a^6 - A*a^5*b - 3*B*a^4*b^2 + 3*A*a^3*b^3 + 3*B*a^2*b^4 - 3*A*a*b^5
- B*b^6)*sin(d*x + c))/b^7 - 420*(B*a^7 - A*a^6*b - 3*B*a^5*b^2 + 3*A*a^4*b^3 + 3*B*a^3*b^4 - 3*A*a^2*b^5 - B
*a*b^6 + A*b^7)*log(b*sin(d*x + c) + a)/b^8)/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.62 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\frac {60 \, B b^{6} \sin \left (d x + c\right )^{7} - 70 \, B a b^{5} \sin \left (d x + c\right )^{6} + 70 \, A b^{6} \sin \left (d x + c\right )^{6} + 84 \, B a^{2} b^{4} \sin \left (d x + c\right )^{5} - 84 \, A a b^{5} \sin \left (d x + c\right )^{5} - 252 \, B b^{6} \sin \left (d x + c\right )^{5} - 105 \, B a^{3} b^{3} \sin \left (d x + c\right )^{4} + 105 \, A a^{2} b^{4} \sin \left (d x + c\right )^{4} + 315 \, B a b^{5} \sin \left (d x + c\right )^{4} - 315 \, A b^{6} \sin \left (d x + c\right )^{4} + 140 \, B a^{4} b^{2} \sin \left (d x + c\right )^{3} - 140 \, A a^{3} b^{3} \sin \left (d x + c\right )^{3} - 420 \, B a^{2} b^{4} \sin \left (d x + c\right )^{3} + 420 \, A a b^{5} \sin \left (d x + c\right )^{3} + 420 \, B b^{6} \sin \left (d x + c\right )^{3} - 210 \, B a^{5} b \sin \left (d x + c\right )^{2} + 210 \, A a^{4} b^{2} \sin \left (d x + c\right )^{2} + 630 \, B a^{3} b^{3} \sin \left (d x + c\right )^{2} - 630 \, A a^{2} b^{4} \sin \left (d x + c\right )^{2} - 630 \, B a b^{5} \sin \left (d x + c\right )^{2} + 630 \, A b^{6} \sin \left (d x + c\right )^{2} + 420 \, B a^{6} \sin \left (d x + c\right ) - 420 \, A a^{5} b \sin \left (d x + c\right ) - 1260 \, B a^{4} b^{2} \sin \left (d x + c\right ) + 1260 \, A a^{3} b^{3} \sin \left (d x + c\right ) + 1260 \, B a^{2} b^{4} \sin \left (d x + c\right ) - 1260 \, A a b^{5} \sin \left (d x + c\right ) - 420 \, B b^{6} \sin \left (d x + c\right )}{b^{7}} - \frac {420 \, {\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}}}{420 \, d} \]

[In]

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

[Out]

-1/420*((60*B*b^6*sin(d*x + c)^7 - 70*B*a*b^5*sin(d*x + c)^6 + 70*A*b^6*sin(d*x + c)^6 + 84*B*a^2*b^4*sin(d*x
+ c)^5 - 84*A*a*b^5*sin(d*x + c)^5 - 252*B*b^6*sin(d*x + c)^5 - 105*B*a^3*b^3*sin(d*x + c)^4 + 105*A*a^2*b^4*s
in(d*x + c)^4 + 315*B*a*b^5*sin(d*x + c)^4 - 315*A*b^6*sin(d*x + c)^4 + 140*B*a^4*b^2*sin(d*x + c)^3 - 140*A*a
^3*b^3*sin(d*x + c)^3 - 420*B*a^2*b^4*sin(d*x + c)^3 + 420*A*a*b^5*sin(d*x + c)^3 + 420*B*b^6*sin(d*x + c)^3 -
210*B*a^5*b*sin(d*x + c)^2 + 210*A*a^4*b^2*sin(d*x + c)^2 + 630*B*a^3*b^3*sin(d*x + c)^2 - 630*A*a^2*b^4*sin(
d*x + c)^2 - 630*B*a*b^5*sin(d*x + c)^2 + 630*A*b^6*sin(d*x + c)^2 + 420*B*a^6*sin(d*x + c) - 420*A*a^5*b*sin(
d*x + c) - 1260*B*a^4*b^2*sin(d*x + c) + 1260*A*a^3*b^3*sin(d*x + c) + 1260*B*a^2*b^4*sin(d*x + c) - 1260*A*a*
b^5*sin(d*x + c) - 420*B*b^6*sin(d*x + c))/b^7 - 420*(B*a^7 - A*a^6*b - 3*B*a^5*b^2 + 3*A*a^4*b^3 + 3*B*a^3*b^
4 - 3*A*a^2*b^5 - B*a*b^6 + A*b^7)*log(abs(b*sin(d*x + c) + a))/b^8)/d

Mupad [B] (verification not implemented)

Time = 11.99 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.38 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {3\,A}{4\,b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}\right )}{4\,b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {B}{b}+\frac {a\,\left (\frac {3\,A}{b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}\right )}{b}\right )}{3\,b}\right )}{d}+\frac {\sin \left (c+d\,x\right )\,\left (\frac {B}{b}+\frac {a\,\left (\frac {3\,A}{b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {3\,A}{b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^6\,\left (\frac {A}{6\,b}-\frac {B\,a}{6\,b^2}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {3\,A}{2\,b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {3\,A}{b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )}{2\,b}\right )}{d}+\frac {{\sin \left (c+d\,x\right )}^5\,\left (\frac {3\,B}{5\,b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{5\,b}\right )}{d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (B\,a^7-A\,a^6\,b-3\,B\,a^5\,b^2+3\,A\,a^4\,b^3+3\,B\,a^3\,b^4-3\,A\,a^2\,b^5-B\,a\,b^6+A\,b^7\right )}{b^8\,d}-\frac {B\,{\sin \left (c+d\,x\right )}^7}{7\,b\,d} \]

[In]

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

[Out]

(sin(c + d*x)^4*((3*A)/(4*b) - (a*((3*B)/b + (a*(A/b - (B*a)/b^2))/b))/(4*b)))/d - (sin(c + d*x)^3*(B/b + (a*(
(3*A)/b - (a*((3*B)/b + (a*(A/b - (B*a)/b^2))/b))/b))/(3*b)))/d + (sin(c + d*x)*(B/b + (a*((3*A)/b - (a*((3*B)
/b + (a*((3*A)/b - (a*((3*B)/b + (a*(A/b - (B*a)/b^2))/b))/b))/b))/b))/b))/d - (sin(c + d*x)^6*(A/(6*b) - (B*a
)/(6*b^2)))/d - (sin(c + d*x)^2*((3*A)/(2*b) - (a*((3*B)/b + (a*((3*A)/b - (a*((3*B)/b + (a*(A/b - (B*a)/b^2))
/b))/b))/b))/(2*b)))/d + (sin(c + d*x)^5*((3*B)/(5*b) + (a*(A/b - (B*a)/b^2))/(5*b)))/d + (log(a + b*sin(c + d
*x))*(A*b^7 + B*a^7 - 3*A*a^2*b^5 + 3*A*a^4*b^3 + 3*B*a^3*b^4 - 3*B*a^5*b^2 - A*a^6*b - B*a*b^6))/(b^8*d) - (B
*sin(c + d*x)^7)/(7*b*d)