3.1552 \(\int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=324 \[ \frac {\left (a^2-b^2\right )^3 (A b-a B)}{b^8 d (a+b \sin (c+d x))}+\frac {\left (a^2-b^2\right )^2 \left (-7 a^2 B+6 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^8 d}+\frac {\left (-3 a^2 B+2 a A b+3 b^2 B\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac {\left (-4 a^3 B+3 a^2 A b+6 a b^2 B-3 A b^3\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac {\left (-5 a^4 B+4 a^3 A b+9 a^2 b^2 B-6 a A b^3-3 b^4 B\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac {\left (-6 a^5 B+5 a^4 A b+12 a^3 b^2 B-9 a^2 A b^3-6 a b^4 B+3 A b^5\right ) \sin (c+d x)}{b^7 d}-\frac {(A b-2 a B) \sin ^5(c+d x)}{5 b^3 d}-\frac {B \sin ^6(c+d x)}{6 b^2 d} \]

[Out]

(a^2-b^2)^2*(6*A*a*b-7*B*a^2+B*b^2)*ln(a+b*sin(d*x+c))/b^8/d-(5*A*a^4*b-9*A*a^2*b^3+3*A*b^5-6*B*a^5+12*B*a^3*b
^2-6*B*a*b^4)*sin(d*x+c)/b^7/d+1/2*(4*A*a^3*b-6*A*a*b^3-5*B*a^4+9*B*a^2*b^2-3*B*b^4)*sin(d*x+c)^2/b^6/d-1/3*(3
*A*a^2*b-3*A*b^3-4*B*a^3+6*B*a*b^2)*sin(d*x+c)^3/b^5/d+1/4*(2*A*a*b-3*B*a^2+3*B*b^2)*sin(d*x+c)^4/b^4/d-1/5*(A
*b-2*B*a)*sin(d*x+c)^5/b^3/d-1/6*B*sin(d*x+c)^6/b^2/d+(a^2-b^2)^3*(A*b-B*a)/b^8/d/(a+b*sin(d*x+c))

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Rubi [A]  time = 0.40, antiderivative size = 324, normalized size of antiderivative = 1.00, 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 = {2837, 772} \[ \frac {\left (-3 a^2 B+2 a A b+3 b^2 B\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac {\left (3 a^2 A b-4 a^3 B+6 a b^2 B-3 A b^3\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac {\left (4 a^3 A b+9 a^2 b^2 B-5 a^4 B-6 a A b^3-3 b^4 B\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac {\left (-9 a^2 A b^3+5 a^4 A b+12 a^3 b^2 B-6 a^5 B-6 a b^4 B+3 A b^5\right ) \sin (c+d x)}{b^7 d}+\frac {\left (a^2-b^2\right )^3 (A b-a B)}{b^8 d (a+b \sin (c+d x))}+\frac {\left (a^2-b^2\right )^2 \left (-7 a^2 B+6 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac {(A b-2 a B) \sin ^5(c+d x)}{5 b^3 d}-\frac {B \sin ^6(c+d x)}{6 b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]

[Out]

((a^2 - b^2)^2*(6*a*A*b - 7*a^2*B + b^2*B)*Log[a + b*Sin[c + d*x]])/(b^8*d) - ((5*a^4*A*b - 9*a^2*A*b^3 + 3*A*
b^5 - 6*a^5*B + 12*a^3*b^2*B - 6*a*b^4*B)*Sin[c + d*x])/(b^7*d) + ((4*a^3*A*b - 6*a*A*b^3 - 5*a^4*B + 9*a^2*b^
2*B - 3*b^4*B)*Sin[c + d*x]^2)/(2*b^6*d) - ((3*a^2*A*b - 3*A*b^3 - 4*a^3*B + 6*a*b^2*B)*Sin[c + d*x]^3)/(3*b^5
*d) + ((2*a*A*b - 3*a^2*B + 3*b^2*B)*Sin[c + d*x]^4)/(4*b^4*d) - ((A*b - 2*a*B)*Sin[c + d*x]^5)/(5*b^3*d) - (B
*Sin[c + d*x]^6)/(6*b^2*d) + ((a^2 - b^2)^3*(A*b - a*B))/(b^8*d*(a + b*Sin[c + d*x]))

Rule 772

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 2837

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*x)/b)^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*} \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (A+\frac {B x}{b}\right ) \left (b^2-x^2\right )^3}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {-5 a^4 A b+9 a^2 A b^3-3 A b^5+6 a^5 B-12 a^3 b^2 B+6 a b^4 B}{b}-\frac {\left (-4 a^3 A b+6 a A b^3+5 a^4 B-9 a^2 b^2 B+3 b^4 B\right ) x}{b}+\frac {\left (-3 a^2 A b+3 A b^3+4 a^3 B-6 a b^2 B\right ) x^2}{b}+\frac {\left (2 a A b-3 a^2 B+3 b^2 B\right ) x^3}{b}-\frac {(A b-2 a B) x^4}{b}-\frac {B x^5}{b}+\frac {\left (-a^2+b^2\right )^3 (A b-a B)}{b (a+x)^2}+\frac {\left (-a^2+b^2\right )^2 \left (6 a A b-7 a^2 B+b^2 B\right )}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac {\left (a^2-b^2\right )^2 \left (6 a A b-7 a^2 B+b^2 B\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac {\left (5 a^4 A b-9 a^2 A b^3+3 A b^5-6 a^5 B+12 a^3 b^2 B-6 a b^4 B\right ) \sin (c+d x)}{b^7 d}+\frac {\left (4 a^3 A b-6 a A b^3-5 a^4 B+9 a^2 b^2 B-3 b^4 B\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac {\left (3 a^2 A b-3 A b^3-4 a^3 B+6 a b^2 B\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac {\left (2 a A b-3 a^2 B+3 b^2 B\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac {(A b-2 a B) \sin ^5(c+d x)}{5 b^3 d}-\frac {B \sin ^6(c+d x)}{6 b^2 d}+\frac {\left (a^2-b^2\right )^3 (A b-a B)}{b^8 d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 1.63, size = 396, normalized size = 1.22 \[ \frac {\frac {6 (A b-a B) \left (-4 a^2 b^4 \sin ^4(c+d x)+4 \left (a^2-b^2\right )^2 \left (15 a^2 \log (a+b \sin (c+d x))+4 a^2-4 b^2\right )+b^4 \cos ^4(c+d x) \left (-a^2+3 a b \sin (c+d x)+4 b^2\right )+2 a b^3 \left (5 a^2-7 b^2\right ) \sin ^3(c+d x)-2 b^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)+4 a b \sin (c+d x) \left (-11 a^4+15 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))+18 a^2 b^2-4 b^4\right )+2 b^6 \cos ^6(c+d x)\right )}{a+b \sin (c+d x)}+B \left (-30 b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)-60 \left (a^2-b^2\right )^3 \log (a+b \sin (c+d x))+15 b^4 \left (b^2-a^2\right ) \cos ^4(c+d x)+20 a b^3 \left (a^2-3 b^2\right ) \sin ^3(c+d x)+60 a b \left (a^4-3 a^2 b^2+3 b^4\right ) \sin (c+d x)+12 a b^5 \sin ^5(c+d x)+10 b^6 \cos ^6(c+d x)\right )}{60 b^8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]

[Out]

(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) + (6*(A*b - a*B)*(2*b^6*Cos[c + d*x]^6 + 4*(a^2 - b^2)^2*(4*a^2 - 4*b^2
+ 15*a^2*Log[a + b*Sin[c + d*x]]) + 4*a*b*(-11*a^4 + 18*a^2*b^2 - 4*b^4 + 15*(a^2 - b^2)^2*Log[a + b*Sin[c +
d*x]])*Sin[c + d*x] - 2*b^2*(15*a^4 - 29*a^2*b^2 + 8*b^4)*Sin[c + d*x]^2 + 2*a*b^3*(5*a^2 - 7*b^2)*Sin[c + d*x
]^3 - 4*a^2*b^4*Sin[c + d*x]^4 + b^4*Cos[c + d*x]^4*(-a^2 + 4*b^2 + 3*a*b*Sin[c + d*x])))/(a + b*Sin[c + d*x])
)/(60*b^8*d)

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fricas [A]  time = 0.55, size = 508, normalized size = 1.57 \[ -\frac {480 \, B a^{7} - 480 \, A a^{6} b - 3720 \, B a^{5} b^{2} + 3360 \, A a^{4} b^{3} + 5705 \, B a^{3} b^{4} - 4710 \, A a^{2} b^{5} - 2402 \, B a b^{6} + 1536 \, A b^{7} + 16 \, {\left (7 \, B a b^{6} - 6 \, A b^{7}\right )} \cos \left (d x + c\right )^{6} - 8 \, {\left (35 \, B a^{3} b^{4} - 30 \, A a^{2} b^{5} - 33 \, B a b^{6} + 24 \, A b^{7}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (105 \, B a^{5} b^{2} - 90 \, A a^{4} b^{3} - 190 \, B a^{3} b^{4} + 150 \, A a^{2} b^{5} + 81 \, B a b^{6} - 48 \, A b^{7}\right )} \cos \left (d x + c\right )^{2} + 480 \, {\left (7 \, B a^{7} - 6 \, A a^{6} b - 15 \, B a^{5} b^{2} + 12 \, A a^{4} b^{3} + 9 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5} - B a b^{6} + {\left (7 \, B a^{6} b - 6 \, A a^{5} b^{2} - 15 \, B a^{4} b^{3} + 12 \, A a^{3} b^{4} + 9 \, B a^{2} b^{5} - 6 \, A a b^{6} - B b^{7}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left (80 \, B b^{7} \cos \left (d x + c\right )^{6} + 2880 \, B a^{6} b - 2400 \, A a^{5} b^{2} - 5720 \, B a^{4} b^{3} + 4320 \, A a^{3} b^{4} + 2967 \, B a^{2} b^{5} - 1626 \, A a b^{6} - 190 \, B b^{7} - 24 \, {\left (7 \, B a^{2} b^{5} - 6 \, A a b^{6} - 5 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (35 \, B a^{4} b^{3} - 30 \, A a^{3} b^{4} - 54 \, B a^{2} b^{5} + 42 \, A a b^{6} + 15 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, {\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/480*(480*B*a^7 - 480*A*a^6*b - 3720*B*a^5*b^2 + 3360*A*a^4*b^3 + 5705*B*a^3*b^4 - 4710*A*a^2*b^5 - 2402*B*a
*b^6 + 1536*A*b^7 + 16*(7*B*a*b^6 - 6*A*b^7)*cos(d*x + c)^6 - 8*(35*B*a^3*b^4 - 30*A*a^2*b^5 - 33*B*a*b^6 + 24
*A*b^7)*cos(d*x + c)^4 + 16*(105*B*a^5*b^2 - 90*A*a^4*b^3 - 190*B*a^3*b^4 + 150*A*a^2*b^5 + 81*B*a*b^6 - 48*A*
b^7)*cos(d*x + c)^2 + 480*(7*B*a^7 - 6*A*a^6*b - 15*B*a^5*b^2 + 12*A*a^4*b^3 + 9*B*a^3*b^4 - 6*A*a^2*b^5 - B*a
*b^6 + (7*B*a^6*b - 6*A*a^5*b^2 - 15*B*a^4*b^3 + 12*A*a^3*b^4 + 9*B*a^2*b^5 - 6*A*a*b^6 - B*b^7)*sin(d*x + c))
*log(b*sin(d*x + c) + a) - (80*B*b^7*cos(d*x + c)^6 + 2880*B*a^6*b - 2400*A*a^5*b^2 - 5720*B*a^4*b^3 + 4320*A*
a^3*b^4 + 2967*B*a^2*b^5 - 1626*A*a*b^6 - 190*B*b^7 - 24*(7*B*a^2*b^5 - 6*A*a*b^6 - 5*B*b^7)*cos(d*x + c)^4 +
16*(35*B*a^4*b^3 - 30*A*a^3*b^4 - 54*B*a^2*b^5 + 42*A*a*b^6 + 15*B*b^7)*cos(d*x + c)^2)*sin(d*x + c))/(b^9*d*s
in(d*x + c) + a*b^8*d)

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giac [A]  time = 0.27, size = 570, normalized size = 1.76 \[ -\frac {\frac {60 \, {\left (7 \, B a^{6} - 6 \, A a^{5} b - 15 \, B a^{4} b^{2} + 12 \, A a^{3} b^{3} + 9 \, B a^{2} b^{4} - 6 \, A a b^{5} - B b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}} - \frac {60 \, {\left (7 \, B a^{6} b \sin \left (d x + c\right ) - 6 \, A a^{5} b^{2} \sin \left (d x + c\right ) - 15 \, B a^{4} b^{3} \sin \left (d x + c\right ) + 12 \, A a^{3} b^{4} \sin \left (d x + c\right ) + 9 \, B a^{2} b^{5} \sin \left (d x + c\right ) - 6 \, A a b^{6} \sin \left (d x + c\right ) - B b^{7} \sin \left (d x + c\right ) + 6 \, B a^{7} - 5 \, A a^{6} b - 12 \, B a^{5} b^{2} + 9 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - A b^{7}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{8}} + \frac {10 \, B b^{10} \sin \left (d x + c\right )^{6} - 24 \, B a b^{9} \sin \left (d x + c\right )^{5} + 12 \, A b^{10} \sin \left (d x + c\right )^{5} + 45 \, B a^{2} b^{8} \sin \left (d x + c\right )^{4} - 30 \, A a b^{9} \sin \left (d x + c\right )^{4} - 45 \, B b^{10} \sin \left (d x + c\right )^{4} - 80 \, B a^{3} b^{7} \sin \left (d x + c\right )^{3} + 60 \, A a^{2} b^{8} \sin \left (d x + c\right )^{3} + 120 \, B a b^{9} \sin \left (d x + c\right )^{3} - 60 \, A b^{10} \sin \left (d x + c\right )^{3} + 150 \, B a^{4} b^{6} \sin \left (d x + c\right )^{2} - 120 \, A a^{3} b^{7} \sin \left (d x + c\right )^{2} - 270 \, B a^{2} b^{8} \sin \left (d x + c\right )^{2} + 180 \, A a b^{9} \sin \left (d x + c\right )^{2} + 90 \, B b^{10} \sin \left (d x + c\right )^{2} - 360 \, B a^{5} b^{5} \sin \left (d x + c\right ) + 300 \, A a^{4} b^{6} \sin \left (d x + c\right ) + 720 \, B a^{3} b^{7} \sin \left (d x + c\right ) - 540 \, A a^{2} b^{8} \sin \left (d x + c\right ) - 360 \, B a b^{9} \sin \left (d x + c\right ) + 180 \, A b^{10} \sin \left (d x + c\right )}{b^{12}}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/60*(60*(7*B*a^6 - 6*A*a^5*b - 15*B*a^4*b^2 + 12*A*a^3*b^3 + 9*B*a^2*b^4 - 6*A*a*b^5 - B*b^6)*log(abs(b*sin(
d*x + c) + a))/b^8 - 60*(7*B*a^6*b*sin(d*x + c) - 6*A*a^5*b^2*sin(d*x + c) - 15*B*a^4*b^3*sin(d*x + c) + 12*A*
a^3*b^4*sin(d*x + c) + 9*B*a^2*b^5*sin(d*x + c) - 6*A*a*b^6*sin(d*x + c) - B*b^7*sin(d*x + c) + 6*B*a^7 - 5*A*
a^6*b - 12*B*a^5*b^2 + 9*A*a^4*b^3 + 6*B*a^3*b^4 - 3*A*a^2*b^5 - A*b^7)/((b*sin(d*x + c) + a)*b^8) + (10*B*b^1
0*sin(d*x + c)^6 - 24*B*a*b^9*sin(d*x + c)^5 + 12*A*b^10*sin(d*x + c)^5 + 45*B*a^2*b^8*sin(d*x + c)^4 - 30*A*a
*b^9*sin(d*x + c)^4 - 45*B*b^10*sin(d*x + c)^4 - 80*B*a^3*b^7*sin(d*x + c)^3 + 60*A*a^2*b^8*sin(d*x + c)^3 + 1
20*B*a*b^9*sin(d*x + c)^3 - 60*A*b^10*sin(d*x + c)^3 + 150*B*a^4*b^6*sin(d*x + c)^2 - 120*A*a^3*b^7*sin(d*x +
c)^2 - 270*B*a^2*b^8*sin(d*x + c)^2 + 180*A*a*b^9*sin(d*x + c)^2 + 90*B*b^10*sin(d*x + c)^2 - 360*B*a^5*b^5*si
n(d*x + c) + 300*A*a^4*b^6*sin(d*x + c) + 720*B*a^3*b^7*sin(d*x + c) - 540*A*a^2*b^8*sin(d*x + c) - 360*B*a*b^
9*sin(d*x + c) + 180*A*b^10*sin(d*x + c))/b^12)/d

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maple [B]  time = 0.70, size = 721, normalized size = 2.23 \[ \frac {4 B \left (\sin ^{3}\left (d x +c \right )\right ) a^{3}}{3 d \,b^{5}}-\frac {3 B \left (\sin ^{2}\left (d x +c \right )\right )}{2 b^{2} d}-\frac {A \left (\sin ^{5}\left (d x +c \right )\right )}{5 d \,b^{2}}+\frac {A \left (\sin ^{3}\left (d x +c \right )\right )}{d \,b^{2}}-\frac {B \left (\sin ^{6}\left (d x +c \right )\right )}{6 b^{2} d}+\frac {3 B \left (\sin ^{4}\left (d x +c \right )\right )}{4 b^{2} d}-\frac {3 A \sin \left (d x +c \right )}{d \,b^{2}}-\frac {A}{d b \left (a +b \sin \left (d x +c \right )\right )}+\frac {6 B \,a^{5} \sin \left (d x +c \right )}{d \,b^{7}}-\frac {12 B \,a^{3} \sin \left (d x +c \right )}{d \,b^{5}}+\frac {9 A \,a^{2} \sin \left (d x +c \right )}{d \,b^{4}}+\frac {3 B \,a^{5}}{d \,b^{6} \left (a +b \sin \left (d x +c \right )\right )}-\frac {3 B \,a^{3}}{d \,b^{4} \left (a +b \sin \left (d x +c \right )\right )}+\frac {a B}{d \,b^{2} \left (a +b \sin \left (d x +c \right )\right )}-\frac {B \,a^{7}}{d \,b^{8} \left (a +b \sin \left (d x +c \right )\right )}-\frac {7 \ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{6}}{d \,b^{8}}+\frac {15 \ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{4}}{d \,b^{6}}-\frac {9 \ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{2}}{d \,b^{4}}+\frac {A \,a^{6}}{d \,b^{7} \left (a +b \sin \left (d x +c \right )\right )}-\frac {3 A \,a^{4}}{d \,b^{5} \left (a +b \sin \left (d x +c \right )\right )}-\frac {2 B \left (\sin ^{3}\left (d x +c \right )\right ) a}{d \,b^{3}}+\frac {2 A \left (\sin ^{2}\left (d x +c \right )\right ) a^{3}}{d \,b^{5}}-\frac {3 A \left (\sin ^{2}\left (d x +c \right )\right ) a}{d \,b^{3}}-\frac {5 B \left (\sin ^{2}\left (d x +c \right )\right ) a^{4}}{2 d \,b^{6}}+\frac {9 B \left (\sin ^{2}\left (d x +c \right )\right ) a^{2}}{2 d \,b^{4}}+\frac {2 B \left (\sin ^{5}\left (d x +c \right )\right ) a}{5 d \,b^{3}}+\frac {A \left (\sin ^{4}\left (d x +c \right )\right ) a}{2 d \,b^{3}}-\frac {3 B \left (\sin ^{4}\left (d x +c \right )\right ) a^{2}}{4 d \,b^{4}}-\frac {A \left (\sin ^{3}\left (d x +c \right )\right ) a^{2}}{d \,b^{4}}+\frac {3 A \,a^{2}}{d \,b^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {5 A \,a^{4} \sin \left (d x +c \right )}{d \,b^{6}}+\frac {6 B a \sin \left (d x +c \right )}{d \,b^{3}}+\frac {6 \ln \left (a +b \sin \left (d x +c \right )\right ) A \,a^{5}}{d \,b^{7}}-\frac {12 \ln \left (a +b \sin \left (d x +c \right )\right ) A \,a^{3}}{d \,b^{5}}+\frac {6 \ln \left (a +b \sin \left (d x +c \right )\right ) A a}{d \,b^{3}}+\frac {B \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x)

[Out]

2/5/d/b^3*B*sin(d*x+c)^5*a-1/5/d/b^2*A*sin(d*x+c)^5+1/d/b^2*A*sin(d*x+c)^3-3/d/b^2*A*sin(d*x+c)-1/d/b/(a+b*sin
(d*x+c))*A+1/2/d/b^3*A*sin(d*x+c)^4*a-3/4/d/b^4*B*sin(d*x+c)^4*a^2-1/d/b^4*A*sin(d*x+c)^3*a^2+6/d/b^7*B*a^5*si
n(d*x+c)-12/d/b^5*B*a^3*sin(d*x+c)+9/d/b^4*A*a^2*sin(d*x+c)+3/d/b^6/(a+b*sin(d*x+c))*B*a^5-3/d/b^4/(a+b*sin(d*
x+c))*B*a^3+1/d/b^2/(a+b*sin(d*x+c))*a*B-1/d/b^8/(a+b*sin(d*x+c))*B*a^7-7/d/b^8*ln(a+b*sin(d*x+c))*B*a^6+15/d/
b^6*ln(a+b*sin(d*x+c))*B*a^4-9/d/b^4*ln(a+b*sin(d*x+c))*B*a^2+1/d/b^7/(a+b*sin(d*x+c))*A*a^6-3/d/b^5/(a+b*sin(
d*x+c))*A*a^4-1/6*B*sin(d*x+c)^6/b^2/d+3/4*B*sin(d*x+c)^4/b^2/d-3/2*B*sin(d*x+c)^2/b^2/d+3/d/b^3/(a+b*sin(d*x+
c))*A*a^2+4/3/d/b^5*B*sin(d*x+c)^3*a^3-2/d/b^3*B*sin(d*x+c)^3*a+2/d/b^5*A*sin(d*x+c)^2*a^3-3/d/b^3*A*sin(d*x+c
)^2*a-5/2/d/b^6*B*sin(d*x+c)^2*a^4+9/2/d/b^4*B*sin(d*x+c)^2*a^2-5/d/b^6*A*a^4*sin(d*x+c)+6/d/b^3*B*a*sin(d*x+c
)+6/d/b^7*ln(a+b*sin(d*x+c))*A*a^5-12/d/b^5*ln(a+b*sin(d*x+c))*A*a^3+6/d/b^3*ln(a+b*sin(d*x+c))*A*a+B*ln(a+b*s
in(d*x+c))/b^2/d

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maxima [A]  time = 0.47, size = 377, normalized size = 1.16 \[ -\frac {\frac {60 \, {\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )}}{b^{9} \sin \left (d x + c\right ) + a b^{8}} + \frac {10 \, B b^{5} \sin \left (d x + c\right )^{6} - 12 \, {\left (2 \, B a b^{4} - A b^{5}\right )} \sin \left (d x + c\right )^{5} + 15 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4} - 3 \, B b^{5}\right )} \sin \left (d x + c\right )^{4} - 20 \, {\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - 6 \, B a b^{4} + 3 \, A b^{5}\right )} \sin \left (d x + c\right )^{3} + 30 \, {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2} - 9 \, B a^{2} b^{3} + 6 \, A a b^{4} + 3 \, B b^{5}\right )} \sin \left (d x + c\right )^{2} - 60 \, {\left (6 \, B a^{5} - 5 \, A a^{4} b - 12 \, B a^{3} b^{2} + 9 \, A a^{2} b^{3} + 6 \, B a b^{4} - 3 \, A b^{5}\right )} \sin \left (d x + c\right )}{b^{7}} + \frac {60 \, {\left (7 \, B a^{6} - 6 \, A a^{5} b - 15 \, B a^{4} b^{2} + 12 \, A a^{3} b^{3} + 9 \, B a^{2} b^{4} - 6 \, A a b^{5} - B b^{6}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/60*(60*(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)/(b^9*sin
(d*x + c) + a*b^8) + (10*B*b^5*sin(d*x + c)^6 - 12*(2*B*a*b^4 - A*b^5)*sin(d*x + c)^5 + 15*(3*B*a^2*b^3 - 2*A*
a*b^4 - 3*B*b^5)*sin(d*x + c)^4 - 20*(4*B*a^3*b^2 - 3*A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*sin(d*x + c)^3 + 30*(5*
B*a^4*b - 4*A*a^3*b^2 - 9*B*a^2*b^3 + 6*A*a*b^4 + 3*B*b^5)*sin(d*x + c)^2 - 60*(6*B*a^5 - 5*A*a^4*b - 12*B*a^3
*b^2 + 9*A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5)*sin(d*x + c))/b^7 + 60*(7*B*a^6 - 6*A*a^5*b - 15*B*a^4*b^2 + 12*A*a^
3*b^3 + 9*B*a^2*b^4 - 6*A*a*b^5 - B*b^6)*log(b*sin(d*x + c) + a)/b^8)/d

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mupad [B]  time = 12.15, size = 682, normalized size = 2.10 \[ \frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{3\,b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{3\,b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^5\,\left (\frac {A}{5\,b^2}-\frac {2\,B\,a}{5\,b^3}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {3\,B}{2\,b^2}+\frac {a\,\left (\frac {3\,A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}\right )}{b}+\frac {a^2\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{2\,b^2}\right )}{d}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {3\,B}{4\,b^2}+\frac {a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{2\,b}+\frac {B\,a^2}{4\,b^4}\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {3\,A}{b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {3\,A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}\right )}{b}+\frac {a^2\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {3\,A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}\right )}{b^2}\right )}{d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (-7\,B\,a^6+6\,A\,a^5\,b+15\,B\,a^4\,b^2-12\,A\,a^3\,b^3-9\,B\,a^2\,b^4+6\,A\,a\,b^5+B\,b^6\right )}{b^8\,d}-\frac {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}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^8+a\,b^7\right )}-\frac {B\,{\sin \left (c+d\,x\right )}^6}{6\,b^2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^7*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2,x)

[Out]

(sin(c + d*x)^3*(A/b^2 + (a^2*(A/b^2 - (2*B*a)/b^3))/(3*b^2) - (2*a*((3*B)/b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b
+ (B*a^2)/b^4))/(3*b)))/d - (sin(c + d*x)^5*(A/(5*b^2) - (2*B*a)/(5*b^3)))/d - (sin(c + d*x)^2*((3*B)/(2*b^2)
+ (a*((3*A)/b^2 + (a^2*(A/b^2 - (2*B*a)/b^3))/b^2 - (2*a*((3*B)/b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b + (B*a^2)
/b^4))/b))/b + (a^2*((3*B)/b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b + (B*a^2)/b^4))/(2*b^2)))/d + (sin(c + d*x)^4*(
(3*B)/(4*b^2) + (a*(A/b^2 - (2*B*a)/b^3))/(2*b) + (B*a^2)/(4*b^4)))/d - (sin(c + d*x)*((3*A)/b^2 - (2*a*((3*B)
/b^2 + (2*a*((3*A)/b^2 + (a^2*(A/b^2 - (2*B*a)/b^3))/b^2 - (2*a*((3*B)/b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b + (
B*a^2)/b^4))/b))/b + (a^2*((3*B)/b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b + (B*a^2)/b^4))/b^2))/b + (a^2*((3*A)/b^2
+ (a^2*(A/b^2 - (2*B*a)/b^3))/b^2 - (2*a*((3*B)/b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b + (B*a^2)/b^4))/b))/b^2))
/d + (log(a + b*sin(c + d*x))*(B*b^6 - 7*B*a^6 - 12*A*a^3*b^3 - 9*B*a^2*b^4 + 15*B*a^4*b^2 + 6*A*a*b^5 + 6*A*a
^5*b))/(b^8*d) - (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*d*(a*b^7 + b^8*sin(c + d*x))) - (B*sin(c + d*x)^6)/(6*b^2*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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