\(\int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [1321]

Optimal result

Integrand size = 29, antiderivative size = 408 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {2 a^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^8 d}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d} \]

[Out]

1/16*a*(16*a^6-40*a^4*b^2+30*a^2*b^4-5*b^6)*x/b^8-2*a^2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b
^2)^(1/2))/b^8/d+1/105*(105*a^6-245*a^4*b^2+161*a^2*b^4-15*b^6)*cos(d*x+c)/b^7/d-1/16*a*(8*a^4-18*a^2*b^2+11*b
^4)*cos(d*x+c)*sin(d*x+c)/b^6/d+1/105*(35*a^4-77*a^2*b^2+45*b^4)*cos(d*x+c)*sin(d*x+c)^2/b^5/d+1/3*cos(d*x+c)*
sin(d*x+c)^3/a/d-1/24*(6*a^4-13*a^2*b^2+8*b^4)*cos(d*x+c)*sin(d*x+c)^3/a/b^4/d-1/4*b*cos(d*x+c)*sin(d*x+c)^4/a
^2/d+1/140*(28*a^4-60*a^2*b^2+35*b^4)*cos(d*x+c)*sin(d*x+c)^4/a^2/b^3/d-1/6*a*cos(d*x+c)*sin(d*x+c)^5/b^2/d+1/
7*cos(d*x+c)*sin(d*x+c)^6/b/d

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2975, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 a^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^8 d}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a b^4 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{105 b^5 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^3 d}+\frac {a x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{16 b^8}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d} \]

[In]

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

[Out]

(a*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/(16*b^8) - (2*a^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*
x)/2])/Sqrt[a^2 - b^2]])/(b^8*d) + ((105*a^6 - 245*a^4*b^2 + 161*a^2*b^4 - 15*b^6)*Cos[c + d*x])/(105*b^7*d) -
(a*(8*a^4 - 18*a^2*b^2 + 11*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^6*d) + ((35*a^4 - 77*a^2*b^2 + 45*b^4)*Cos[
c + d*x]*Sin[c + d*x]^2)/(105*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d) - ((6*a^4 - 13*a^2*b^2 + 8*b^4)*C
os[c + d*x]*Sin[c + d*x]^3)/(24*a*b^4*d) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(4*a^2*d) + ((28*a^4 - 60*a^2*b^2 +
35*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a^2*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^5)/(6*b^2*d) + (Cos[c + d
*x]*Sin[c + d*x]^6)/(7*b*d)

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 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}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {\sin ^4(c+d x) \left (84 \left (5 a^4-10 a^2 b^2+6 b^4\right )-6 a b \left (2 a^2-7 b^2\right ) \sin (c+d x)-18 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{504 a^2 b^2} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {\sin ^3(c+d x) \left (-72 a \left (28 a^4-60 a^2 b^2+35 b^4\right )+12 a^2 b \left (7 a^2+10 b^2\right ) \sin (c+d x)+420 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2520 a^2 b^3} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {\sin ^2(c+d x) \left (1260 a^2 \left (6 a^4-13 a^2 b^2+8 b^4\right )-36 a^3 b \left (14 a^2-25 b^2\right ) \sin (c+d x)-288 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{10080 a^2 b^4} \\ & = \frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {\sin (c+d x) \left (-576 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right )+36 a^2 b \left (70 a^4-133 a^2 b^2+120 b^4\right ) \sin (c+d x)+3780 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{30240 a^2 b^5} \\ & = -\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {3780 a^4 \left (8 a^4-18 a^2 b^2+11 b^4\right )-36 a^3 b \left (280 a^4-574 a^2 b^2+285 b^4\right ) \sin (c+d x)-576 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{60480 a^2 b^6} \\ & = \frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {3780 a^4 b \left (8 a^4-18 a^2 b^2+11 b^4\right )+3780 a^3 \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{60480 a^2 b^7} \\ & = \frac {a \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}-\frac {\left (a^2 \left (a^2-b^2\right )^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^8} \\ & = \frac {a \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}-\frac {\left (2 a^2 \left (a^2-b^2\right )^3\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 (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\left (4 a^2 \left (a^2-b^2\right )^3\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 (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {2 a^2 \left (a^2-b^2\right )^{5/2} \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 (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.33 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {-6720 a^7 c+16800 a^5 b^2 c-12600 a^3 b^4 c+2100 a b^6 c-6720 a^7 d x+16800 a^5 b^2 d x-12600 a^3 b^4 d x+2100 a b^6 d x+13440 a^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+105 b \left (-64 a^6+144 a^4 b^2-88 a^2 b^4+5 b^6\right ) \cos (c+d x)+35 \left (16 a^4 b^3-28 a^2 b^5+9 b^7\right ) \cos (3 (c+d x))-84 a^2 b^5 \cos (5 (c+d x))+105 b^7 \cos (5 (c+d x))+15 b^7 \cos (7 (c+d x))+1680 a^5 b^2 \sin (2 (c+d x))-3360 a^3 b^4 \sin (2 (c+d x))+1575 a b^6 \sin (2 (c+d x))-210 a^3 b^4 \sin (4 (c+d x))+315 a b^6 \sin (4 (c+d x))+35 a b^6 \sin (6 (c+d x))}{6720 b^8 d} \]

[In]

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

[Out]

-1/6720*(-6720*a^7*c + 16800*a^5*b^2*c - 12600*a^3*b^4*c + 2100*a*b^6*c - 6720*a^7*d*x + 16800*a^5*b^2*d*x - 1
2600*a^3*b^4*d*x + 2100*a*b^6*d*x + 13440*a^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2
]] + 105*b*(-64*a^6 + 144*a^4*b^2 - 88*a^2*b^4 + 5*b^6)*Cos[c + d*x] + 35*(16*a^4*b^3 - 28*a^2*b^5 + 9*b^7)*Co
s[3*(c + d*x)] - 84*a^2*b^5*Cos[5*(c + d*x)] + 105*b^7*Cos[5*(c + d*x)] + 15*b^7*Cos[7*(c + d*x)] + 1680*a^5*b
^2*Sin[2*(c + d*x)] - 3360*a^3*b^4*Sin[2*(c + d*x)] + 1575*a*b^6*Sin[2*(c + d*x)] - 210*a^3*b^4*Sin[4*(c + d*x
)] + 315*a*b^6*Sin[4*(c + d*x)] + 35*a*b^6*Sin[6*(c + d*x)])/(b^8*d)

Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{5} b^{2}-\frac {9}{8} a^{3} b^{4}+\frac {11}{16} a \,b^{6}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{6} b -3 a^{4} b^{3}+3 a^{2} b^{5}-b^{7}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{5} b^{2}-\frac {7}{2} a^{3} b^{4}+\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{6} b -16 a^{4} b^{3}+12 a^{2} b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{5} b^{2}-\frac {29}{8} a^{3} b^{4}+\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{6} b -\frac {109}{3} a^{4} b^{3}+\frac {73}{3} a^{2} b^{5}-5 b^{7}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20 a^{6} b -\frac {136}{3} a^{4} b^{3}+\frac {88}{3} a^{2} b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{5} b^{2}+\frac {29}{8} a^{3} b^{4}-\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{6} b -33 a^{4} b^{3}+\frac {101}{5} a^{2} b^{5}-3 b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{5} b^{2}+\frac {7}{2} a^{3} b^{4}-\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{6} b -\frac {40}{3} a^{4} b^{3}+\frac {116}{15} a^{2} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{5} b^{2}+\frac {9}{8} a^{3} b^{4}-\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{6} b -\frac {7 a^{4} b^{3}}{3}+\frac {23 a^{2} b^{5}}{15}-\frac {b^{7}}{7}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {a \left (16 a^{6}-40 a^{4} b^{2}+30 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{b^{8}}-\frac {2 a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-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 )}{b^{8} \sqrt {a^{2}-b^{2}}}}{d}\)	\(601\)
default	\(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{5} b^{2}-\frac {9}{8} a^{3} b^{4}+\frac {11}{16} a \,b^{6}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{6} b -3 a^{4} b^{3}+3 a^{2} b^{5}-b^{7}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{5} b^{2}-\frac {7}{2} a^{3} b^{4}+\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{6} b -16 a^{4} b^{3}+12 a^{2} b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{5} b^{2}-\frac {29}{8} a^{3} b^{4}+\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{6} b -\frac {109}{3} a^{4} b^{3}+\frac {73}{3} a^{2} b^{5}-5 b^{7}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20 a^{6} b -\frac {136}{3} a^{4} b^{3}+\frac {88}{3} a^{2} b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{5} b^{2}+\frac {29}{8} a^{3} b^{4}-\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{6} b -33 a^{4} b^{3}+\frac {101}{5} a^{2} b^{5}-3 b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{5} b^{2}+\frac {7}{2} a^{3} b^{4}-\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{6} b -\frac {40}{3} a^{4} b^{3}+\frac {116}{15} a^{2} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{5} b^{2}+\frac {9}{8} a^{3} b^{4}-\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{6} b -\frac {7 a^{4} b^{3}}{3}+\frac {23 a^{2} b^{5}}{15}-\frac {b^{7}}{7}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {a \left (16 a^{6}-40 a^{4} b^{2}+30 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{b^{8}}-\frac {2 a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-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 )}{b^{8} \sqrt {a^{2}-b^{2}}}}{d}\)	\(601\)
risch	\(\frac {\cos \left (5 d x +5 c \right ) a^{2}}{80 b^{3} d}+\frac {a^{3} \sin \left (4 d x +4 c \right )}{32 b^{4} d}-\frac {\cos \left (7 d x +7 c \right )}{448 b d}-\frac {5 a x}{16 b^{2}}-\frac {\cos \left (3 d x +3 c \right ) a^{4}}{12 b^{5} d}-\frac {a^{5} \sin \left (2 d x +2 c \right )}{4 b^{6} d}-\frac {15 a \sin \left (2 d x +2 c \right )}{64 b^{2} d}-\frac {5 a^{5} x}{2 b^{6}}+\frac {15 a^{3} x}{8 b^{4}}-\frac {3 \cos \left (3 d x +3 c \right )}{64 b d}+\frac {7 \cos \left (3 d x +3 c \right ) a^{2}}{48 b^{3} d}-\frac {a \sin \left (6 d x +6 c \right )}{192 b^{2} d}-\frac {3 a \sin \left (4 d x +4 c \right )}{64 b^{2} d}+\frac {a^{3} \sin \left (2 d x +2 c \right )}{2 d \,b^{4}}-\frac {\cos \left (5 d x +5 c \right )}{64 b d}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{128 b d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 b d}-\frac {\sqrt {-a^{2}+b^{2}}\, a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{8}}+\frac {{\mathrm e}^{i \left (d x +c \right )} a^{6}}{2 b^{7} d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} a^{6}}{2 b^{7} d}-\frac {9 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}}{8 b^{5} d}+\frac {11 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{16 b^{3} d}-\frac {9 \,{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{8 b^{5} d}+\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{16 b^{3} d}+\frac {a^{7} x}{b^{8}}+\frac {\sqrt {-a^{2}+b^{2}}\, a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{8}}+\frac {\sqrt {-a^{2}+b^{2}}\, a^{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 {2 \sqrt {-a^{2}+b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{6}}+\frac {2 \sqrt {-a^{2}+b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{6}}-\frac {\sqrt {-a^{2}+b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}\)	\(736\)

[In]

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

[Out]

1/d*(2/b^8*(((1/2*a^5*b^2-9/8*a^3*b^4+11/16*a*b^6)*tan(1/2*d*x+1/2*c)^13+(a^6*b-3*a^4*b^3+3*a^2*b^5-b^7)*tan(1
/2*d*x+1/2*c)^12+(2*a^5*b^2-7/2*a^3*b^4+7/12*a*b^6)*tan(1/2*d*x+1/2*c)^11+(6*a^6*b-16*a^4*b^3+12*a^2*b^5)*tan(
1/2*d*x+1/2*c)^10+(5/2*a^5*b^2-29/8*a^3*b^4+85/48*a*b^6)*tan(1/2*d*x+1/2*c)^9+(15*a^6*b-109/3*a^4*b^3+73/3*a^2
*b^5-5*b^7)*tan(1/2*d*x+1/2*c)^8+(20*a^6*b-136/3*a^4*b^3+88/3*a^2*b^5)*tan(1/2*d*x+1/2*c)^6+(-5/2*a^5*b^2+29/8
*a^3*b^4-85/48*a*b^6)*tan(1/2*d*x+1/2*c)^5+(15*a^6*b-33*a^4*b^3+101/5*a^2*b^5-3*b^7)*tan(1/2*d*x+1/2*c)^4+(-2*
a^5*b^2+7/2*a^3*b^4-7/12*a*b^6)*tan(1/2*d*x+1/2*c)^3+(6*a^6*b-40/3*a^4*b^3+116/15*a^2*b^5)*tan(1/2*d*x+1/2*c)^
2+(-1/2*a^5*b^2+9/8*a^3*b^4-11/16*a*b^6)*tan(1/2*d*x+1/2*c)+a^6*b-7/3*a^4*b^3+23/15*a^2*b^5-1/7*b^7)/(1+tan(1/
2*d*x+1/2*c)^2)^7+1/16*a*(16*a^6-40*a^4*b^2+30*a^2*b^4-5*b^6)*arctan(tan(1/2*d*x+1/2*c)))-2*a^2*(a^6-3*a^4*b^2
+3*a^2*b^4-b^6)/b^8/(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)))

Fricas [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.52 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\left [-\frac {240 \, b^{7} \cos \left (d x + c\right )^{7} - 336 \, a^{2} b^{5} \cos \left (d x + c\right )^{5} + 560 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (16 \, a^{7} - 40 \, a^{5} b^{2} + 30 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x - 840 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 1680 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right ) + 35 \, {\left (8 \, a b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{5} b^{2} - 14 \, a^{3} b^{4} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, b^{8} d}, -\frac {240 \, b^{7} \cos \left (d x + c\right )^{7} - 336 \, a^{2} b^{5} \cos \left (d x + c\right )^{5} + 560 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (16 \, a^{7} - 40 \, a^{5} b^{2} + 30 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x - 1680 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 1680 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right ) + 35 \, {\left (8 \, a b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{5} b^{2} - 14 \, a^{3} b^{4} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, b^{8} d}\right ] \]

[In]

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

[Out]

[-1/1680*(240*b^7*cos(d*x + c)^7 - 336*a^2*b^5*cos(d*x + c)^5 + 560*(a^4*b^3 - a^2*b^5)*cos(d*x + c)^3 - 105*(
16*a^7 - 40*a^5*b^2 + 30*a^3*b^4 - 5*a*b^6)*d*x - 840*(a^6 - 2*a^4*b^2 + a^2*b^4)*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))*sqr
t(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 1680*(a^6*b - 2*a^4*b^3 + a^2*b^5)*cos
(d*x + c) + 35*(8*a*b^6*cos(d*x + c)^5 - 2*(6*a^3*b^4 - 5*a*b^6)*cos(d*x + c)^3 + 3*(8*a^5*b^2 - 14*a^3*b^4 +
5*a*b^6)*cos(d*x + c))*sin(d*x + c))/(b^8*d), -1/1680*(240*b^7*cos(d*x + c)^7 - 336*a^2*b^5*cos(d*x + c)^5 + 5
60*(a^4*b^3 - a^2*b^5)*cos(d*x + c)^3 - 105*(16*a^7 - 40*a^5*b^2 + 30*a^3*b^4 - 5*a*b^6)*d*x - 1680*(a^6 - 2*a
^4*b^2 + a^2*b^4)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 1680*(a^6*b -
2*a^4*b^3 + a^2*b^5)*cos(d*x + c) + 35*(8*a*b^6*cos(d*x + c)^5 - 2*(6*a^3*b^4 - 5*a*b^6)*cos(d*x + c)^3 + 3*(
8*a^5*b^2 - 14*a^3*b^4 + 5*a*b^6)*cos(d*x + c))*sin(d*x + c))/(b^8*d)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c)),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 863 vs. \(2 (383) = 766\).

Time = 0.35 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.12 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/1680*(105*(16*a^7 - 40*a^5*b^2 + 30*a^3*b^4 - 5*a*b^6)*(d*x + c)/b^8 - 3360*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a
^2*b^6)*(pi*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) + 2*(840*a^5*b*tan(1/2*d*x + 1/2*c)^13 - 1890*a^3*b^3*tan(1/2*d*x + 1/2*c)^13 + 1155*a*b^5*ta
n(1/2*d*x + 1/2*c)^13 + 1680*a^6*tan(1/2*d*x + 1/2*c)^12 - 5040*a^4*b^2*tan(1/2*d*x + 1/2*c)^12 + 5040*a^2*b^4
*tan(1/2*d*x + 1/2*c)^12 - 1680*b^6*tan(1/2*d*x + 1/2*c)^12 + 3360*a^5*b*tan(1/2*d*x + 1/2*c)^11 - 5880*a^3*b^
3*tan(1/2*d*x + 1/2*c)^11 + 980*a*b^5*tan(1/2*d*x + 1/2*c)^11 + 10080*a^6*tan(1/2*d*x + 1/2*c)^10 - 26880*a^4*
b^2*tan(1/2*d*x + 1/2*c)^10 + 20160*a^2*b^4*tan(1/2*d*x + 1/2*c)^10 + 4200*a^5*b*tan(1/2*d*x + 1/2*c)^9 - 6090
*a^3*b^3*tan(1/2*d*x + 1/2*c)^9 + 2975*a*b^5*tan(1/2*d*x + 1/2*c)^9 + 25200*a^6*tan(1/2*d*x + 1/2*c)^8 - 61040
*a^4*b^2*tan(1/2*d*x + 1/2*c)^8 + 40880*a^2*b^4*tan(1/2*d*x + 1/2*c)^8 - 8400*b^6*tan(1/2*d*x + 1/2*c)^8 + 336
00*a^6*tan(1/2*d*x + 1/2*c)^6 - 76160*a^4*b^2*tan(1/2*d*x + 1/2*c)^6 + 49280*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 -
4200*a^5*b*tan(1/2*d*x + 1/2*c)^5 + 6090*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 - 2975*a*b^5*tan(1/2*d*x + 1/2*c)^5 +
25200*a^6*tan(1/2*d*x + 1/2*c)^4 - 55440*a^4*b^2*tan(1/2*d*x + 1/2*c)^4 + 33936*a^2*b^4*tan(1/2*d*x + 1/2*c)^4
- 5040*b^6*tan(1/2*d*x + 1/2*c)^4 - 3360*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 5880*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 -
980*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 10080*a^6*tan(1/2*d*x + 1/2*c)^2 - 22400*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 +
12992*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 - 840*a^5*b*tan(1/2*d*x + 1/2*c) + 1890*a^3*b^3*tan(1/2*d*x + 1/2*c) - 11
55*a*b^5*tan(1/2*d*x + 1/2*c) + 1680*a^6 - 3920*a^4*b^2 + 2576*a^2*b^4 - 240*b^6)/((tan(1/2*d*x + 1/2*c)^2 + 1
)^7*b^7))/d

Mupad [B] (verification not implemented)

Time = 14.56 (sec) , antiderivative size = 3797, normalized size of antiderivative = 9.31 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*(105*a^6 - 15*b^6 + 161*a^2*b^4 - 245*a^4*b^2))/(105*b^7) + (tan(c/2 + (d*x)/2)^13*(11*a*b^4 + 8*a^5 - 18*
a^3*b^2))/(8*b^6) - (tan(c/2 + (d*x)/2)^3*(7*a*b^4 + 24*a^5 - 42*a^3*b^2))/(6*b^6) + (tan(c/2 + (d*x)/2)^11*(7
*a*b^4 + 24*a^5 - 42*a^3*b^2))/(6*b^6) - (tan(c/2 + (d*x)/2)^5*(85*a*b^4 + 120*a^5 - 174*a^3*b^2))/(24*b^6) +
(tan(c/2 + (d*x)/2)^9*(85*a*b^4 + 120*a^5 - 174*a^3*b^2))/(24*b^6) + (2*tan(c/2 + (d*x)/2)^12*(a^6 - b^6 + 3*a
^2*b^4 - 3*a^4*b^2))/b^7 + (4*tan(c/2 + (d*x)/2)^10*(3*a^6 + 6*a^2*b^4 - 8*a^4*b^2))/b^7 + (8*tan(c/2 + (d*x)/
2)^6*(15*a^6 + 22*a^2*b^4 - 34*a^4*b^2))/(3*b^7) + (4*tan(c/2 + (d*x)/2)^2*(45*a^6 + 58*a^2*b^4 - 100*a^4*b^2)
)/(15*b^7) + (2*tan(c/2 + (d*x)/2)^8*(45*a^6 - 15*b^6 + 73*a^2*b^4 - 109*a^4*b^2))/(3*b^7) + (2*tan(c/2 + (d*x
)/2)^4*(75*a^6 - 15*b^6 + 101*a^2*b^4 - 165*a^4*b^2))/(5*b^7) - (a*tan(c/2 + (d*x)/2)*(8*a^4 + 11*b^4 - 18*a^2
*b^2))/(8*b^6))/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan(c/2 +
(d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 + 1)) + (a^2*atan(((a^
2*(-(a + b)^5*(a - b)^5)^(1/2)*(((25*a^4*b^19)/8 - (75*a^6*b^17)/2 + (325*a^8*b^15)/2 - 320*a^10*b^13 + 320*a^
12*b^11 - 160*a^14*b^9 + 32*a^16*b^7)/b^20 + (tan(c/2 + (d*x)/2)*(50*a^3*b^21 - 881*a^5*b^19 + 4436*a^7*b^17 -
10260*a^9*b^15 + 12800*a^11*b^13 - 8960*a^13*b^11 + 3328*a^15*b^9 - 512*a^17*b^7))/(8*b^21) + (a^2*(-(a + b)^
5*(a - b)^5)^(1/2)*((10*a^2*b^22 - 38*a^4*b^20 + 44*a^6*b^18 - 16*a^8*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(512*a^
3*b^22 - 1536*a^5*b^20 + 1536*a^7*b^18 - 512*a^9*b^16))/(8*b^21) + (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b
^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8)*1i)/b^8 + (a^2*(-(a + b)^5*(a - b)
^5)^(1/2)*(((25*a^4*b^19)/8 - (75*a^6*b^17)/2 + (325*a^8*b^15)/2 - 320*a^10*b^13 + 320*a^12*b^11 - 160*a^14*b^
9 + 32*a^16*b^7)/b^20 + (tan(c/2 + (d*x)/2)*(50*a^3*b^21 - 881*a^5*b^19 + 4436*a^7*b^17 - 10260*a^9*b^15 + 128
00*a^11*b^13 - 8960*a^13*b^11 + 3328*a^15*b^9 - 512*a^17*b^7))/(8*b^21) - (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*((
10*a^2*b^22 - 38*a^4*b^20 + 44*a^6*b^18 - 16*a^8*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(512*a^3*b^22 - 1536*a^5*b^2
0 + 1536*a^7*b^18 - 512*a^9*b^16))/(8*b^21) - (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)
/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8)*1i)/b^8)/((32*a^22 + (55*a^6*b^16)/4 - (585*a^8*b^14)/4
+ (2445*a^10*b^12)/4 - (5511*a^12*b^10)/4 + 1874*a^14*b^8 - 1595*a^16*b^6 + 836*a^18*b^4 - 248*a^20*b^2)/b^20
+ (tan(c/2 + (d*x)/2)*(512*a^23 - 50*a^5*b^18 + 750*a^7*b^16 - 4550*a^9*b^14 + 14770*a^11*b^12 - 28880*a^13*b
^10 + 35880*a^15*b^8 - 28672*a^17*b^6 + 14336*a^19*b^4 - 4096*a^21*b^2))/(4*b^21) - (a^2*(-(a + b)^5*(a - b)^5
)^(1/2)*(((25*a^4*b^19)/8 - (75*a^6*b^17)/2 + (325*a^8*b^15)/2 - 320*a^10*b^13 + 320*a^12*b^11 - 160*a^14*b^9
+ 32*a^16*b^7)/b^20 + (tan(c/2 + (d*x)/2)*(50*a^3*b^21 - 881*a^5*b^19 + 4436*a^7*b^17 - 10260*a^9*b^15 + 12800
*a^11*b^13 - 8960*a^13*b^11 + 3328*a^15*b^9 - 512*a^17*b^7))/(8*b^21) + (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*((10
*a^2*b^22 - 38*a^4*b^20 + 44*a^6*b^18 - 16*a^8*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(512*a^3*b^22 - 1536*a^5*b^20
+ 1536*a^7*b^18 - 512*a^9*b^16))/(8*b^21) + (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2
)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8))/b^8 + (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*(((25*a^4*b^19)/
8 - (75*a^6*b^17)/2 + (325*a^8*b^15)/2 - 320*a^10*b^13 + 320*a^12*b^11 - 160*a^14*b^9 + 32*a^16*b^7)/b^20 + (t
an(c/2 + (d*x)/2)*(50*a^3*b^21 - 881*a^5*b^19 + 4436*a^7*b^17 - 10260*a^9*b^15 + 12800*a^11*b^13 - 8960*a^13*b
^11 + 3328*a^15*b^9 - 512*a^17*b^7))/(8*b^21) - (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*((10*a^2*b^22 - 38*a^4*b^20
+ 44*a^6*b^18 - 16*a^8*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(512*a^3*b^22 - 1536*a^5*b^20 + 1536*a^7*b^18 - 512*a^
9*b^16))/(8*b^21) - (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*
b^23))/(8*b^21)))/b^8))/b^8))/b^8))*(-(a + b)^5*(a - b)^5)^(1/2)*2i)/(b^8*d) + (a*atan(((a*(((25*a^4*b^19)/8 -
(75*a^6*b^17)/2 + (325*a^8*b^15)/2 - 320*a^10*b^13 + 320*a^12*b^11 - 160*a^14*b^9 + 32*a^16*b^7)/b^20 + (tan(
c/2 + (d*x)/2)*(50*a^3*b^21 - 881*a^5*b^19 + 4436*a^7*b^17 - 10260*a^9*b^15 + 12800*a^11*b^13 - 8960*a^13*b^11
+ 3328*a^15*b^9 - 512*a^17*b^7))/(8*b^21) - (a*((10*a^2*b^22 - 38*a^4*b^20 + 44*a^6*b^18 - 16*a^8*b^16)/b^20
+ (tan(c/2 + (d*x)/2)*(512*a^3*b^22 - 1536*a^5*b^20 + 1536*a^7*b^18 - 512*a^9*b^16))/(8*b^21) - (a*(32*a^2*b^3
+ (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40*a^4*b^2)*1i)/(
16*b^8))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40*a^4*b^2)*1i)/(16*b^8))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40*a^4*b^2))/
(16*b^8) + (a*(((25*a^4*b^19)/8 - (75*a^6*b^17)/2 + (325*a^8*b^15)/2 - 320*a^10*b^13 + 320*a^12*b^11 - 160*a^1
4*b^9 + 32*a^16*b^7)/b^20 + (tan(c/2 + (d*x)/2)*(50*a^3*b^21 - 881*a^5*b^19 + 4436*a^7*b^17 - 10260*a^9*b^15 +
12800*a^11*b^13 - 8960*a^13*b^11 + 3328*a^15*b^9 - 512*a^17*b^7))/(8*b^21) + (a*((10*a^2*b^22 - 38*a^4*b^20 +
44*a^6*b^18 - 16*a^8*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(512*a^3*b^22 - 1536*a^5*b^20 + 1536*a^7*b^18 - 512*a^9
*b^16))/(8*b^21) + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21))*(16*a^6 - 5*b^6
+ 30*a^2*b^4 - 40*a^4*b^2)*1i)/(16*b^8))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40*a^4*b^2)*1i)/(16*b^8))*(16*a^6 - 5
*b^6 + 30*a^2*b^4 - 40*a^4*b^2))/(16*b^8))/((32*a^22 + (55*a^6*b^16)/4 - (585*a^8*b^14)/4 + (2445*a^10*b^12)/4
- (5511*a^12*b^10)/4 + 1874*a^14*b^8 - 1595*a^16*b^6 + 836*a^18*b^4 - 248*a^20*b^2)/b^20 + (tan(c/2 + (d*x)/2
)*(512*a^23 - 50*a^5*b^18 + 750*a^7*b^16 - 4550*a^9*b^14 + 14770*a^11*b^12 - 28880*a^13*b^10 + 35880*a^15*b^8
- 28672*a^17*b^6 + 14336*a^19*b^4 - 4096*a^21*b^2))/(4*b^21) + (a*(((25*a^4*b^19)/8 - (75*a^6*b^17)/2 + (325*a
^8*b^15)/2 - 320*a^10*b^13 + 320*a^12*b^11 - 160*a^14*b^9 + 32*a^16*b^7)/b^20 + (tan(c/2 + (d*x)/2)*(50*a^3*b^
21 - 881*a^5*b^19 + 4436*a^7*b^17 - 10260*a^9*b^15 + 12800*a^11*b^13 - 8960*a^13*b^11 + 3328*a^15*b^9 - 512*a^
17*b^7))/(8*b^21) - (a*((10*a^2*b^22 - 38*a^4*b^20 + 44*a^6*b^18 - 16*a^8*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(51
2*a^3*b^22 - 1536*a^5*b^20 + 1536*a^7*b^18 - 512*a^9*b^16))/(8*b^21) - (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(7
68*a*b^25 - 512*a^3*b^23))/(8*b^21))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40*a^4*b^2)*1i)/(16*b^8))*(16*a^6 - 5*b^6
+ 30*a^2*b^4 - 40*a^4*b^2)*1i)/(16*b^8))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40*a^4*b^2)*1i)/(16*b^8) - (a*(((25*a^
4*b^19)/8 - (75*a^6*b^17)/2 + (325*a^8*b^15)/2 - 320*a^10*b^13 + 320*a^12*b^11 - 160*a^14*b^9 + 32*a^16*b^7)/b
^20 + (tan(c/2 + (d*x)/2)*(50*a^3*b^21 - 881*a^5*b^19 + 4436*a^7*b^17 - 10260*a^9*b^15 + 12800*a^11*b^13 - 896
0*a^13*b^11 + 3328*a^15*b^9 - 512*a^17*b^7))/(8*b^21) + (a*((10*a^2*b^22 - 38*a^4*b^20 + 44*a^6*b^18 - 16*a^8*
b^16)/b^20 + (tan(c/2 + (d*x)/2)*(512*a^3*b^22 - 1536*a^5*b^20 + 1536*a^7*b^18 - 512*a^9*b^16))/(8*b^21) + (a*
(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40*a^4
*b^2)*1i)/(16*b^8))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40*a^4*b^2)*1i)/(16*b^8))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40
*a^4*b^2)*1i)/(16*b^8)))*(16*a^6 - 5*b^6 + 30*a^2*b^4 - 40*a^4*b^2))/(8*b^8*d)