\(\int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1267]

Optimal result

Integrand size = 29, antiderivative size = 485 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}-\frac {\sqrt {a^2-b^2} \left (42 a^4-29 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^8 d}+\frac {\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac {\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))} \]

[Out]

1/8*a*(168*a^4-200*a^2*b^2+45*b^4)*x/b^8+1/30*(630*a^4-645*a^2*b^2+91*b^4)*cos(d*x+c)/b^7/d-1/8*(84*a^4-79*a^2
*b^2+8*b^4)*cos(d*x+c)*sin(d*x+c)/a/b^6/d+1/30*(210*a^4-187*a^2*b^2+15*b^4)*cos(d*x+c)*sin(d*x+c)^2/a^2/b^5/d+
1/3*cos(d*x+c)*sin(d*x+c)^3/a/d/(a+b*sin(d*x+c))^2-1/12*b*cos(d*x+c)*sin(d*x+c)^4/a^2/d/(a+b*sin(d*x+c))^2-1/6
0*(63*a^4-60*a^2*b^2+5*b^4)*cos(d*x+c)*sin(d*x+c)^4/a^2/b^3/d/(a+b*sin(d*x+c))^2-7/20*a*cos(d*x+c)*sin(d*x+c)^
5/b^2/d/(a+b*sin(d*x+c))^2+1/5*cos(d*x+c)*sin(d*x+c)^6/b/d/(a+b*sin(d*x+c))^2-1/12*(63*a^4-54*a^2*b^2+4*b^4)*c
os(d*x+c)*sin(d*x+c)^3/a^2/b^4/d/(a+b*sin(d*x+c))-(42*a^4-29*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a
^2-b^2)^(1/2))*(a^2-b^2)^(1/2)/b^8/d

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2975, 3126, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\sqrt {a^2-b^2} \left (42 a^4-29 a^2 b^2+2 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^8 d}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac {a x \left (168 a^4-200 a^2 b^2+45 b^4\right )}{8 b^8}+\frac {\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac {\left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{8 a b^6 d}+\frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{30 a^2 b^5 d}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2} \]

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]

[Out]

(a*(168*a^4 - 200*a^2*b^2 + 45*b^4)*x)/(8*b^8) - (Sqrt[a^2 - b^2]*(42*a^4 - 29*a^2*b^2 + 2*b^4)*ArcTan[(b + a*
Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^8*d) + ((630*a^4 - 645*a^2*b^2 + 91*b^4)*Cos[c + d*x])/(30*b^7*d) - ((8
4*a^4 - 79*a^2*b^2 + 8*b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*a*b^6*d) + ((210*a^4 - 187*a^2*b^2 + 15*b^4)*Cos[c +
d*x]*Sin[c + d*x]^2)/(30*a^2*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d*(a + b*Sin[c + d*x])^2) - (b*Cos[c
+ d*x]*Sin[c + d*x]^4)/(12*a^2*d*(a + b*Sin[c + d*x])^2) - ((63*a^4 - 60*a^2*b^2 + 5*b^4)*Cos[c + d*x]*Sin[c
+ d*x]^4)/(60*a^2*b^3*d*(a + b*Sin[c + d*x])^2) - (7*a*Cos[c + d*x]*Sin[c + d*x]^5)/(20*b^2*d*(a + b*Sin[c + d
*x])^2) + (Cos[c + d*x]*Sin[c + d*x]^6)/(5*b*d*(a + b*Sin[c + d*x])^2) - ((63*a^4 - 54*a^2*b^2 + 4*b^4)*Cos[c
+ d*x]*Sin[c + d*x]^3)/(12*a^2*b^4*d*(a + b*Sin[c + d*x]))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2975

Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] +
(Dist[1/(a^2*b^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6)), Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*
x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m +
n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e +
f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(
n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x] - Simp[b*(m + n + 2)*Cos[e + f*x]*(d*S
in[e + f*x])^(n + 2)*((a + b*Sin[e + f*x])^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[a*(n + 5)*Cos[e + f
*x]*(d*Sin[e + f*x])^(n + 3)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3*f*(m + n + 5)*(m + n + 6))), x] + Simp[Cos
[e + f*x]*(d*Sin[e + f*x])^(n + 4)*((a + b*Sin[e + f*x])^(m + 1)/(b*d^4*f*(m + n + 6))), x]) /; FreeQ[{a, b, d
, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0]
&& NeQ[m + n + 6, 0] &&  !IGtQ[m, 0]

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3126

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)
*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) +
(c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*
c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n +
1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\int \frac {\sin ^4(c+d x) \left (20 \left (21 a^4-20 a^2 b^2+2 b^4\right )-12 a b \left (3 a^2-5 b^2\right ) \sin (c+d x)-12 \left (42 a^4-44 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{240 a^2 b^2} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\int \frac {\sin ^3(c+d x) \left (-32 \left (63 a^6-123 a^4 b^2+65 a^2 b^4-5 b^6\right )+8 a b \left (21 a^4-41 a^2 b^2+20 b^4\right ) \sin (c+d x)+24 \left (105 a^6-209 a^4 b^2+114 a^2 b^4-10 b^6\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{480 a^2 b^3 \left (a^2-b^2\right )} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac {\int \frac {\sin ^2(c+d x) \left (120 \left (a^2-b^2\right )^2 \left (63 a^4-54 a^2 b^2+4 b^4\right )-24 a b \left (21 a^2-10 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-48 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{480 a^2 b^4 \left (a^2-b^2\right )^2} \\ & = \frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac {\int \frac {\sin (c+d x) \left (-96 a \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right )+24 a^2 b \left (105 a^2-62 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+360 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1440 a^2 b^5 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac {\int \frac {360 a^2 \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right )-24 a^3 b \left (420 a^2-311 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-96 a^2 \left (a^2-b^2\right )^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{2880 a^2 b^6 \left (a^2-b^2\right )^2} \\ & = \frac {\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac {\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac {\int \frac {360 a^2 b \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right )+360 a^3 \left (a^2-b^2\right )^2 \left (168 a^4-200 a^2 b^2+45 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{2880 a^2 b^7 \left (a^2-b^2\right )^2} \\ & = \frac {a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}+\frac {\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac {\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}-\frac {\left (\left (a^2-b^2\right ) \left (42 a^4-29 a^2 b^2+2 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 b^8} \\ & = \frac {a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}+\frac {\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac {\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}-\frac {\left (\left (a^2-b^2\right ) \left (42 a^4-29 a^2 b^2+2 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^8 d} \\ & = \frac {a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}+\frac {\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac {\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac {\left (2 \left (a^2-b^2\right ) \left (42 a^4-29 a^2 b^2+2 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^8 d} \\ & = \frac {a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}-\frac {\sqrt {a^2-b^2} \left (42 a^4-29 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^8 d}+\frac {\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac {\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.38 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-1920 \left (a^2-b^2\right )^{5/2} \left (42 a^4-29 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+\frac {\left (a^2-b^2\right )^2 \left (40320 a^7 c-27840 a^5 b^2 c-13200 a^3 b^4 c+5400 a b^6 c+40320 a^7 d x-27840 a^5 b^2 d x-13200 a^3 b^4 d x+5400 a b^6 d x+10 b \left (4032 a^6-3792 a^4 b^2+216 a^2 b^4+59 b^6\right ) \cos (c+d x)-120 a b^2 \left (168 a^4-200 a^2 b^2+45 b^4\right ) (c+d x) \cos (2 (c+d x))-3360 a^4 b^3 \cos (3 (c+d x))+3580 a^2 b^5 \cos (3 (c+d x))-526 b^7 \cos (3 (c+d x))+84 a^2 b^5 \cos (5 (c+d x))-58 b^7 \cos (5 (c+d x))-6 b^7 \cos (7 (c+d x))+80640 a^6 b c \sin (c+d x)-96000 a^4 b^3 c \sin (c+d x)+21600 a^2 b^5 c \sin (c+d x)+80640 a^6 b d x \sin (c+d x)-96000 a^4 b^3 d x \sin (c+d x)+21600 a^2 b^5 d x \sin (c+d x)+30240 a^5 b^2 \sin (2 (c+d x))-32640 a^3 b^4 \sin (2 (c+d x))+5675 a b^6 \sin (2 (c+d x))+420 a^3 b^4 \sin (4 (c+d x))-374 a b^6 \sin (4 (c+d x))-21 a b^6 \sin (6 (c+d x))\right )}{(a+b \sin (c+d x))^2}}{1920 (a-b)^2 b^8 (a+b)^2 d} \]

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]

[Out]

(-1920*(a^2 - b^2)^(5/2)*(42*a^4 - 29*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + ((a^
2 - b^2)^2*(40320*a^7*c - 27840*a^5*b^2*c - 13200*a^3*b^4*c + 5400*a*b^6*c + 40320*a^7*d*x - 27840*a^5*b^2*d*x
- 13200*a^3*b^4*d*x + 5400*a*b^6*d*x + 10*b*(4032*a^6 - 3792*a^4*b^2 + 216*a^2*b^4 + 59*b^6)*Cos[c + d*x] - 1
20*a*b^2*(168*a^4 - 200*a^2*b^2 + 45*b^4)*(c + d*x)*Cos[2*(c + d*x)] - 3360*a^4*b^3*Cos[3*(c + d*x)] + 3580*a^
2*b^5*Cos[3*(c + d*x)] - 526*b^7*Cos[3*(c + d*x)] + 84*a^2*b^5*Cos[5*(c + d*x)] - 58*b^7*Cos[5*(c + d*x)] - 6*
b^7*Cos[7*(c + d*x)] + 80640*a^6*b*c*Sin[c + d*x] - 96000*a^4*b^3*c*Sin[c + d*x] + 21600*a^2*b^5*c*Sin[c + d*x
] + 80640*a^6*b*d*x*Sin[c + d*x] - 96000*a^4*b^3*d*x*Sin[c + d*x] + 21600*a^2*b^5*d*x*Sin[c + d*x] + 30240*a^5
*b^2*Sin[2*(c + d*x)] - 32640*a^3*b^4*Sin[2*(c + d*x)] + 5675*a*b^6*Sin[2*(c + d*x)] + 420*a^3*b^4*Sin[4*(c +
d*x)] - 374*a*b^6*Sin[4*(c + d*x)] - 21*a*b^6*Sin[6*(c + d*x)]))/(a + b*Sin[c + d*x])^2)/(1920*(a - b)^2*b^8*(
a + b)^2*d)

Maple [A] (verified)

Time = 4.65 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {-\frac {a \,b^{2} \left (11 a^{4}-13 a^{2} b^{2}+2 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 b \left (4 a^{6}+3 a^{4} b^{2}-9 a^{2} b^{4}+2 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b^{2} a \left (37 a^{4}-47 a^{2} b^{2}+10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-6 a^{6} b +\frac {15 a^{4} b^{3}}{2}-\frac {3 a^{2} b^{5}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (42 a^{6}-71 a^{4} b^{2}+31 a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{8}}+\frac {\frac {2 \left (\left (5 a^{3} b^{2}-\frac {27}{8} a \,b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b -18 a^{2} b^{3}+3 b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{3} b^{2}-\frac {15}{4} a \,b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{4} b -60 a^{2} b^{3}+6 b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (90 a^{4} b -80 a^{2} b^{3}+\frac {28}{3} b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-10 a^{3} b^{2}+\frac {15}{4} a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{4} b -52 a^{2} b^{3}+\frac {14}{3} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-5 a^{3} b^{2}+\frac {27}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+15 a^{4} b -14 a^{2} b^{3}+\frac {23 b^{5}}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {a \left (168 a^{4}-200 a^{2} b^{2}+45 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{8}}}{d}\)	\(554\)
default	\(\frac {-\frac {2 \left (\frac {-\frac {a \,b^{2} \left (11 a^{4}-13 a^{2} b^{2}+2 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 b \left (4 a^{6}+3 a^{4} b^{2}-9 a^{2} b^{4}+2 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b^{2} a \left (37 a^{4}-47 a^{2} b^{2}+10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-6 a^{6} b +\frac {15 a^{4} b^{3}}{2}-\frac {3 a^{2} b^{5}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (42 a^{6}-71 a^{4} b^{2}+31 a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{8}}+\frac {\frac {2 \left (\left (5 a^{3} b^{2}-\frac {27}{8} a \,b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b -18 a^{2} b^{3}+3 b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{3} b^{2}-\frac {15}{4} a \,b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{4} b -60 a^{2} b^{3}+6 b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (90 a^{4} b -80 a^{2} b^{3}+\frac {28}{3} b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-10 a^{3} b^{2}+\frac {15}{4} a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{4} b -52 a^{2} b^{3}+\frac {14}{3} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-5 a^{3} b^{2}+\frac {27}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+15 a^{4} b -14 a^{2} b^{3}+\frac {23 b^{5}}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {a \left (168 a^{4}-200 a^{2} b^{2}+45 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{8}}}{d}\)	\(554\)
risch	\(\frac {15 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 b^{7} d}-\frac {27 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{4 b^{5} d}+\frac {15 \,{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 b^{7} d}-\frac {27 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{4 b^{5} d}-\frac {{\mathrm e}^{-3 i \left (d x +c \right )} a^{2}}{4 b^{5} d}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}-\frac {{\mathrm e}^{3 i \left (d x +c \right )} a^{2}}{4 b^{5} d}+\frac {\cos \left (5 d x +5 c \right )}{80 b^{3} d}+\frac {45 a x}{8 b^{4}}+\frac {3 a \sin \left (4 d x +4 c \right )}{32 b^{4} d}+\frac {11 \,{\mathrm e}^{i \left (d x +c \right )}}{16 b^{3} d}+\frac {7 \,{\mathrm e}^{3 i \left (d x +c \right )}}{96 b^{3} d}+\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 b^{3} d}+\frac {21 a^{5} x}{b^{8}}-\frac {25 a^{3} x}{b^{6}}+\frac {7 \,{\mathrm e}^{-3 i \left (d x +c \right )}}{96 b^{3} d}+\frac {5 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{6} d}-\frac {3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{4} d}-\frac {5 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{6} d}+\frac {3 i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{4} d}-\frac {21 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{4}}{d \,b^{8}}+\frac {29 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{2}}{2 d \,b^{6}}+\frac {21 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{4}}{d \,b^{8}}-\frac {29 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{2}}{2 d \,b^{6}}-\frac {i a \left (-14 i a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}+19 i a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-5 i a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+38 i a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-49 i a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+11 i a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+26 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-21 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+4 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-13 a^{4} b^{2}+17 a^{2} b^{4}-4 b^{6}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d \,b^{8}}\)	\(873\)

[In]

int(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2/b^8*((-1/2*a*b^2*(11*a^4-13*a^2*b^2+2*b^4)*tan(1/2*d*x+1/2*c)^3-3/2*b*(4*a^6+3*a^4*b^2-9*a^2*b^4+2*b^6
)*tan(1/2*d*x+1/2*c)^2-1/2*b^2*a*(37*a^4-47*a^2*b^2+10*b^4)*tan(1/2*d*x+1/2*c)-6*a^6*b+15/2*a^4*b^3-3/2*a^2*b^
5)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*(42*a^6-71*a^4*b^2+31*a^2*b^4-2*b^6)/(a^2-b^2)^(1/2
)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))+2/b^8*(((5*a^3*b^2-27/8*a*b^4)*tan(1/2*d*x+1/2*c)^
9+(15*a^4*b-18*a^2*b^3+3*b^5)*tan(1/2*d*x+1/2*c)^8+(10*a^3*b^2-15/4*a*b^4)*tan(1/2*d*x+1/2*c)^7+(60*a^4*b-60*a
^2*b^3+6*b^5)*tan(1/2*d*x+1/2*c)^6+(90*a^4*b-80*a^2*b^3+28/3*b^5)*tan(1/2*d*x+1/2*c)^4+(-10*a^3*b^2+15/4*a*b^4
)*tan(1/2*d*x+1/2*c)^3+(60*a^4*b-52*a^2*b^3+14/3*b^5)*tan(1/2*d*x+1/2*c)^2+(-5*a^3*b^2+27/8*a*b^4)*tan(1/2*d*x
+1/2*c)+15*a^4*b-14*a^2*b^3+23/15*b^5)/(1+tan(1/2*d*x+1/2*c)^2)^5+1/8*a*(168*a^4-200*a^2*b^2+45*b^4)*arctan(ta
n(1/2*d*x+1/2*c))))

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 995, normalized size of antiderivative = 2.05 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/120*(24*b^7*cos(d*x + c)^7 - 4*(21*a^2*b^5 - 4*b^7)*cos(d*x + c)^5 + 15*(168*a^5*b^2 - 200*a^3*b^4 + 45*a*b
^6)*d*x*cos(d*x + c)^2 + 10*(84*a^4*b^3 - 79*a^2*b^5 + 8*b^7)*cos(d*x + c)^3 - 15*(168*a^7 - 32*a^5*b^2 - 155*
a^3*b^4 + 45*a*b^6)*d*x - 30*(42*a^6 + 13*a^4*b^2 - 27*a^2*b^4 + 2*b^6 - (42*a^4*b^2 - 29*a^2*b^4 + 2*b^6)*cos
(d*x + c)^2 + 2*(42*a^5*b - 29*a^3*b^3 + 2*a*b^5)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(d*x +
c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 + 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^
2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 30*(84*a^6*b - 58*a^4*b^3 - 17*a^2*b^5 + 4*b^7)*cos(d*x
+ c) + (42*a*b^6*cos(d*x + c)^5 - 5*(42*a^3*b^4 - 29*a*b^6)*cos(d*x + c)^3 - 30*(168*a^6*b - 200*a^4*b^3 + 45*
a^2*b^5)*d*x - 15*(252*a^5*b^2 - 279*a^3*b^4 + 53*a*b^6)*cos(d*x + c))*sin(d*x + c))/(b^10*d*cos(d*x + c)^2 -
2*a*b^9*d*sin(d*x + c) - (a^2*b^8 + b^10)*d), 1/120*(24*b^7*cos(d*x + c)^7 - 4*(21*a^2*b^5 - 4*b^7)*cos(d*x +
c)^5 + 15*(168*a^5*b^2 - 200*a^3*b^4 + 45*a*b^6)*d*x*cos(d*x + c)^2 + 10*(84*a^4*b^3 - 79*a^2*b^5 + 8*b^7)*cos
(d*x + c)^3 - 15*(168*a^7 - 32*a^5*b^2 - 155*a^3*b^4 + 45*a*b^6)*d*x - 60*(42*a^6 + 13*a^4*b^2 - 27*a^2*b^4 +
2*b^6 - (42*a^4*b^2 - 29*a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 2*(42*a^5*b - 29*a^3*b^3 + 2*a*b^5)*sin(d*x + c))*s
qrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 30*(84*a^6*b - 58*a^4*b^3 - 17*a
^2*b^5 + 4*b^7)*cos(d*x + c) + (42*a*b^6*cos(d*x + c)^5 - 5*(42*a^3*b^4 - 29*a*b^6)*cos(d*x + c)^3 - 30*(168*a
^6*b - 200*a^4*b^3 + 45*a^2*b^5)*d*x - 15*(252*a^5*b^2 - 279*a^3*b^4 + 53*a*b^6)*cos(d*x + c))*sin(d*x + c))/(
b^10*d*cos(d*x + c)^2 - 2*a*b^9*d*sin(d*x + c) - (a^2*b^8 + b^10)*d)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**2/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.49 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/120*(15*(168*a^5 - 200*a^3*b^2 + 45*a*b^4)*(d*x + c)/b^8 - 120*(42*a^6 - 71*a^4*b^2 + 31*a^2*b^4 - 2*b^6)*(p
i*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2
)*b^8) + 120*(11*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 13*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 2*a*b^5*tan(1/2*d*x + 1/2*
c)^3 + 12*a^6*tan(1/2*d*x + 1/2*c)^2 + 9*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 - 27*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 +
6*b^6*tan(1/2*d*x + 1/2*c)^2 + 37*a^5*b*tan(1/2*d*x + 1/2*c) - 47*a^3*b^3*tan(1/2*d*x + 1/2*c) + 10*a*b^5*tan(
1/2*d*x + 1/2*c) + 12*a^6 - 15*a^4*b^2 + 3*a^2*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)
^2*b^7) + 2*(600*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 405*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 1800*a^4*tan(1/2*d*x + 1/2*
c)^8 - 2160*a^2*b^2*tan(1/2*d*x + 1/2*c)^8 + 360*b^4*tan(1/2*d*x + 1/2*c)^8 + 1200*a^3*b*tan(1/2*d*x + 1/2*c)^
7 - 450*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 7200*a^4*tan(1/2*d*x + 1/2*c)^6 - 7200*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 +
720*b^4*tan(1/2*d*x + 1/2*c)^6 + 10800*a^4*tan(1/2*d*x + 1/2*c)^4 - 9600*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 112
0*b^4*tan(1/2*d*x + 1/2*c)^4 - 1200*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 450*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 7200*a^4
*tan(1/2*d*x + 1/2*c)^2 - 6240*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 560*b^4*tan(1/2*d*x + 1/2*c)^2 - 600*a^3*b*tan
(1/2*d*x + 1/2*c) + 405*a*b^3*tan(1/2*d*x + 1/2*c) + 1800*a^4 - 1680*a^2*b^2 + 184*b^4)/((tan(1/2*d*x + 1/2*c)
^2 + 1)^5*b^7))/d

Mupad [B] (verification not implemented)

Time = 24.20 (sec) , antiderivative size = 3700, normalized size of antiderivative = 7.63 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int((cos(c + d*x)^6*sin(c + d*x)^2)/(a + b*sin(c + d*x))^3,x)

[Out]

((630*a^6 + 91*a^2*b^4 - 645*a^4*b^2)/(15*b^7) + (tan(c/2 + (d*x)/2)^13*(8*a*b^4 + 84*a^5 - 79*a^3*b^2))/(4*b^
6) + (tan(c/2 + (d*x)/2)^11*(17*a*b^4 + 252*a^5 - 237*a^3*b^2))/b^6 + (8*tan(c/2 + (d*x)/2)^7*(91*a*b^4 + 630*
a^5 - 645*a^3*b^2))/(3*b^6) + (9*tan(c/2 + (d*x)/2)^5*(112*a*b^4 + 700*a^5 - 733*a^3*b^2))/(4*b^6) + (tan(c/2
+ (d*x)/2)^9*(448*a*b^4 + 3780*a^5 - 3723*a^3*b^2))/(4*b^6) + (tan(c/2 + (d*x)/2)^3*(643*a*b^4 + 3780*a^5 - 39
75*a^3*b^2))/(5*b^6) + (tan(c/2 + (d*x)/2)^12*(42*a^6 + 6*b^6 - 48*a^2*b^4 + 13*a^4*b^2))/b^7 + (3*tan(c/2 + (
d*x)/2)^10*(84*a^6 + 18*b^6 - 103*a^2*b^4 + 26*a^4*b^2))/b^7 + (2*tan(c/2 + (d*x)/2)^6*(1260*a^6 + 202*b^6 - 1
187*a^2*b^4 + 54*a^4*b^2))/(3*b^7) + (tan(c/2 + (d*x)/2)^8*(1890*a^6 + 324*b^6 - 2149*a^2*b^4 + 417*a^4*b^2))/
(3*b^7) + (tan(c/2 + (d*x)/2)^2*(3780*a^6 + 274*b^6 - 1223*a^2*b^4 - 2190*a^4*b^2))/(15*b^7) + (tan(c/2 + (d*x
)/2)^4*(9450*a^6 + 1010*b^6 - 6354*a^2*b^4 - 2115*a^4*b^2))/(15*b^7) + (tan(c/2 + (d*x)/2)*(1336*a*b^4 + 8820*
a^5 - 9135*a^3*b^2))/(60*b^6))/(d*(tan(c/2 + (d*x)/2)^2*(7*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^12*(7*a^2 + 4*b^2
) + tan(c/2 + (d*x)/2)^4*(21*a^2 + 20*b^2) + tan(c/2 + (d*x)/2)^10*(21*a^2 + 20*b^2) + tan(c/2 + (d*x)/2)^6*(3
5*a^2 + 40*b^2) + tan(c/2 + (d*x)/2)^8*(35*a^2 + 40*b^2) + a^2*tan(c/2 + (d*x)/2)^14 + a^2 + 24*a*b*tan(c/2 +
(d*x)/2)^3 + 60*a*b*tan(c/2 + (d*x)/2)^5 + 80*a*b*tan(c/2 + (d*x)/2)^7 + 60*a*b*tan(c/2 + (d*x)/2)^9 + 24*a*b*
tan(c/2 + (d*x)/2)^11 + 4*a*b*tan(c/2 + (d*x)/2)^13 + 4*a*b*tan(c/2 + (d*x)/2))) + (a*atan(((a*(((2025*a^4*b^1
5)/2 - 9000*a^6*b^13 + 27560*a^8*b^11 - 33600*a^10*b^9 + 14112*a^12*b^7)/b^20 - (tan(c/2 + (d*x)/2)*(64*a*b^19
- 6034*a^3*b^17 + 57945*a^5*b^15 - 201360*a^7*b^13 + 311840*a^9*b^11 - 219072*a^11*b^9 + 56448*a^13*b^7))/(2*
b^21) + (a*(168*a^4 + 45*b^4 - 200*a^2*b^2)*((148*a^2*b^20 - 484*a^4*b^18 + 336*a^6*b^16)/b^20 - (tan(c/2 + (d
*x)/2)*(128*a*b^22 - 1984*a^3*b^20 + 4544*a^5*b^18 - 2688*a^7*b^16))/(2*b^21) + (a*(32*a^2*b^3 + (tan(c/2 + (d
*x)/2)*(192*a*b^25 - 128*a^3*b^23))/(2*b^21))*(168*a^4 + 45*b^4 - 200*a^2*b^2)*1i)/(8*b^8))*1i)/(8*b^8))*(168*
a^4 + 45*b^4 - 200*a^2*b^2))/(8*b^8) + (a*(((2025*a^4*b^15)/2 - 9000*a^6*b^13 + 27560*a^8*b^11 - 33600*a^10*b^
9 + 14112*a^12*b^7)/b^20 - (tan(c/2 + (d*x)/2)*(64*a*b^19 - 6034*a^3*b^17 + 57945*a^5*b^15 - 201360*a^7*b^13 +
311840*a^9*b^11 - 219072*a^11*b^9 + 56448*a^13*b^7))/(2*b^21) + (a*(168*a^4 + 45*b^4 - 200*a^2*b^2)*((tan(c/2
+ (d*x)/2)*(128*a*b^22 - 1984*a^3*b^20 + 4544*a^5*b^18 - 2688*a^7*b^16))/(2*b^21) - (148*a^2*b^20 - 484*a^4*b
^18 + 336*a^6*b^16)/b^20 + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^25 - 128*a^3*b^23))/(2*b^21))*(168*a^
4 + 45*b^4 - 200*a^2*b^2)*1i)/(8*b^8))*1i)/(8*b^8))*(168*a^4 + 45*b^4 - 200*a^2*b^2))/(8*b^8))/((296352*a^16 -
360*a^2*b^14 + 10735*a^4*b^12 - (227213*a^6*b^10)/2 + (1089913*a^8*b^8)/2 - 1331285*a^10*b^6 + 1725696*a^12*b
^4 - 1132488*a^14*b^2)/b^20 + (tan(c/2 + (d*x)/2)*(1185408*a^17 - 4050*a^3*b^14 + 98775*a^5*b^12 - 812015*a^7*
b^10 + 3206170*a^9*b^8 - 6809168*a^11*b^6 + 7961184*a^13*b^4 - 4826304*a^15*b^2))/b^21 + (a*(((2025*a^4*b^15)/
2 - 9000*a^6*b^13 + 27560*a^8*b^11 - 33600*a^10*b^9 + 14112*a^12*b^7)/b^20 - (tan(c/2 + (d*x)/2)*(64*a*b^19 -
6034*a^3*b^17 + 57945*a^5*b^15 - 201360*a^7*b^13 + 311840*a^9*b^11 - 219072*a^11*b^9 + 56448*a^13*b^7))/(2*b^2
1) + (a*(168*a^4 + 45*b^4 - 200*a^2*b^2)*((148*a^2*b^20 - 484*a^4*b^18 + 336*a^6*b^16)/b^20 - (tan(c/2 + (d*x)
/2)*(128*a*b^22 - 1984*a^3*b^20 + 4544*a^5*b^18 - 2688*a^7*b^16))/(2*b^21) + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)
/2)*(192*a*b^25 - 128*a^3*b^23))/(2*b^21))*(168*a^4 + 45*b^4 - 200*a^2*b^2)*1i)/(8*b^8))*1i)/(8*b^8))*(168*a^4
+ 45*b^4 - 200*a^2*b^2)*1i)/(8*b^8) - (a*(((2025*a^4*b^15)/2 - 9000*a^6*b^13 + 27560*a^8*b^11 - 33600*a^10*b^
9 + 14112*a^12*b^7)/b^20 - (tan(c/2 + (d*x)/2)*(64*a*b^19 - 6034*a^3*b^17 + 57945*a^5*b^15 - 201360*a^7*b^13 +
311840*a^9*b^11 - 219072*a^11*b^9 + 56448*a^13*b^7))/(2*b^21) + (a*(168*a^4 + 45*b^4 - 200*a^2*b^2)*((tan(c/2
+ (d*x)/2)*(128*a*b^22 - 1984*a^3*b^20 + 4544*a^5*b^18 - 2688*a^7*b^16))/(2*b^21) - (148*a^2*b^20 - 484*a^4*b
^18 + 336*a^6*b^16)/b^20 + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^25 - 128*a^3*b^23))/(2*b^21))*(168*a^
4 + 45*b^4 - 200*a^2*b^2)*1i)/(8*b^8))*1i)/(8*b^8))*(168*a^4 + 45*b^4 - 200*a^2*b^2)*1i)/(8*b^8)))*(168*a^4 +
45*b^4 - 200*a^2*b^2))/(4*b^8*d) + (atan((((-(a + b)*(a - b))^(1/2)*(21*a^4 + b^4 - (29*a^2*b^2)/2)*(((2025*a^
4*b^15)/2 - 9000*a^6*b^13 + 27560*a^8*b^11 - 33600*a^10*b^9 + 14112*a^12*b^7)/b^20 - (tan(c/2 + (d*x)/2)*(64*a
*b^19 - 6034*a^3*b^17 + 57945*a^5*b^15 - 201360*a^7*b^13 + 311840*a^9*b^11 - 219072*a^11*b^9 + 56448*a^13*b^7)
)/(2*b^21) + ((-(a + b)*(a - b))^(1/2)*(21*a^4 + b^4 - (29*a^2*b^2)/2)*((148*a^2*b^20 - 484*a^4*b^18 + 336*a^6
*b^16)/b^20 - (tan(c/2 + (d*x)/2)*(128*a*b^22 - 1984*a^3*b^20 + 4544*a^5*b^18 - 2688*a^7*b^16))/(2*b^21) + ((-
(a + b)*(a - b))^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^25 - 128*a^3*b^23))/(2*b^21))*(21*a^4 + b^4
- (29*a^2*b^2)/2))/b^8))/b^8)*1i)/b^8 + ((-(a + b)*(a - b))^(1/2)*(21*a^4 + b^4 - (29*a^2*b^2)/2)*(((2025*a^4*
b^15)/2 - 9000*a^6*b^13 + 27560*a^8*b^11 - 33600*a^10*b^9 + 14112*a^12*b^7)/b^20 - (tan(c/2 + (d*x)/2)*(64*a*b
^19 - 6034*a^3*b^17 + 57945*a^5*b^15 - 201360*a^7*b^13 + 311840*a^9*b^11 - 219072*a^11*b^9 + 56448*a^13*b^7))/
(2*b^21) + ((-(a + b)*(a - b))^(1/2)*(21*a^4 + b^4 - (29*a^2*b^2)/2)*((tan(c/2 + (d*x)/2)*(128*a*b^22 - 1984*a
^3*b^20 + 4544*a^5*b^18 - 2688*a^7*b^16))/(2*b^21) - (148*a^2*b^20 - 484*a^4*b^18 + 336*a^6*b^16)/b^20 + ((-(a
+ b)*(a - b))^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^25 - 128*a^3*b^23))/(2*b^21))*(21*a^4 + b^4 -
(29*a^2*b^2)/2))/b^8))/b^8)*1i)/b^8)/((296352*a^16 - 360*a^2*b^14 + 10735*a^4*b^12 - (227213*a^6*b^10)/2 + (10
89913*a^8*b^8)/2 - 1331285*a^10*b^6 + 1725696*a^12*b^4 - 1132488*a^14*b^2)/b^20 + (tan(c/2 + (d*x)/2)*(1185408
*a^17 - 4050*a^3*b^14 + 98775*a^5*b^12 - 812015*a^7*b^10 + 3206170*a^9*b^8 - 6809168*a^11*b^6 + 7961184*a^13*b
^4 - 4826304*a^15*b^2))/b^21 + ((-(a + b)*(a - b))^(1/2)*(21*a^4 + b^4 - (29*a^2*b^2)/2)*(((2025*a^4*b^15)/2 -
9000*a^6*b^13 + 27560*a^8*b^11 - 33600*a^10*b^9 + 14112*a^12*b^7)/b^20 - (tan(c/2 + (d*x)/2)*(64*a*b^19 - 603
4*a^3*b^17 + 57945*a^5*b^15 - 201360*a^7*b^13 + 311840*a^9*b^11 - 219072*a^11*b^9 + 56448*a^13*b^7))/(2*b^21)
+ ((-(a + b)*(a - b))^(1/2)*(21*a^4 + b^4 - (29*a^2*b^2)/2)*((148*a^2*b^20 - 484*a^4*b^18 + 336*a^6*b^16)/b^20
- (tan(c/2 + (d*x)/2)*(128*a*b^22 - 1984*a^3*b^20 + 4544*a^5*b^18 - 2688*a^7*b^16))/(2*b^21) + ((-(a + b)*(a
- b))^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^25 - 128*a^3*b^23))/(2*b^21))*(21*a^4 + b^4 - (29*a^2*b
^2)/2))/b^8))/b^8))/b^8 - ((-(a + b)*(a - b))^(1/2)*(21*a^4 + b^4 - (29*a^2*b^2)/2)*(((2025*a^4*b^15)/2 - 9000
*a^6*b^13 + 27560*a^8*b^11 - 33600*a^10*b^9 + 14112*a^12*b^7)/b^20 - (tan(c/2 + (d*x)/2)*(64*a*b^19 - 6034*a^3
*b^17 + 57945*a^5*b^15 - 201360*a^7*b^13 + 311840*a^9*b^11 - 219072*a^11*b^9 + 56448*a^13*b^7))/(2*b^21) + ((-
(a + b)*(a - b))^(1/2)*(21*a^4 + b^4 - (29*a^2*b^2)/2)*((tan(c/2 + (d*x)/2)*(128*a*b^22 - 1984*a^3*b^20 + 4544
*a^5*b^18 - 2688*a^7*b^16))/(2*b^21) - (148*a^2*b^20 - 484*a^4*b^18 + 336*a^6*b^16)/b^20 + ((-(a + b)*(a - b))
^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^25 - 128*a^3*b^23))/(2*b^21))*(21*a^4 + b^4 - (29*a^2*b^2)/2
))/b^8))/b^8))/b^8))*(-(a + b)*(a - b))^(1/2)*(21*a^4 + b^4 - (29*a^2*b^2)/2)*2i)/(b^8*d)