3.1321 \(\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx\)

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.45786, antiderivative size = 408, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2896, 3049, 3023, 2735, 2660, 618, 204} \[ \frac{\left (-245 a^4 b^2+161 a^2 b^4+105 a^6-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac{2 a^2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^8 d}+\frac{\left (-60 a^2 b^2+28 a^4+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^3 d}-\frac{\left (-13 a^2 b^2+6 a^4+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a b^4 d}+\frac{\left (-77 a^2 b^2+35 a^4+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{105 b^5 d}-\frac{a \left (-18 a^2 b^2+8 a^4+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^6 d}+\frac{a x \left (-40 a^4 b^2+30 a^2 b^4+16 a^6-5 b^6\right )}{16 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac{\sin ^6(c+d x) \cos (c+d x)}{7 b d} \]

Antiderivative was successfully verified.

[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 2896

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*
Sin[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[(Co
s[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 3049

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])))

Rule 3023

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 2735

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 2660

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 618

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 204

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

Rubi steps

\begin{align*} \int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\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 ) \operatorname{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 ) \operatorname{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} \tan ^{-1}\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]  time = 2.99382, size = 324, normalized size = 0.79 \[ -\frac{1680 a^5 b^2 \sin (2 (c+d x))-3360 a^3 b^4 \sin (2 (c+d x))-210 a^3 b^4 \sin (4 (c+d x))-84 a^2 b^5 \cos (5 (c+d x))+105 b \left (144 a^4 b^2-88 a^2 b^4-64 a^6+5 b^6\right ) \cos (c+d x)+35 \left (-28 a^2 b^5+16 a^4 b^3+9 b^7\right ) \cos (3 (c+d x))+13440 a^2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )+16800 a^5 b^2 c-12600 a^3 b^4 c+16800 a^5 b^2 d x-12600 a^3 b^4 d x-6720 a^7 c-6720 a^7 d x+1575 a b^6 \sin (2 (c+d x))+315 a b^6 \sin (4 (c+d x))+35 a b^6 \sin (6 (c+d x))+2100 a b^6 c+2100 a b^6 d x+105 b^7 \cos (5 (c+d x))+15 b^7 \cos (7 (c+d x))}{6720 b^8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-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*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 10
5*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)*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)] + 31
5*a*b^6*Sin[4*(c + d*x)] + 35*a*b^6*Sin[6*(c + d*x)])/(6720*b^8*d)

________________________________________________________________________________________

Maple [B]  time = 0.102, size = 1808, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-66/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4*a^4-4/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1
/2*c)^3*a^5-5/8/d/b^2*a*arctan(tan(1/2*d*x+1/2*c))+6/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^12*a^
2+12/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^10*a^6-272/3/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2
*d*x+1/2*c)^6*a^4+176/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^6*a^2-5/d/b^6/(1+tan(1/2*d*x+1/2*c
)^2)^7*tan(1/2*d*x+1/2*c)^5*a^5-5/d/b^6*arctan(tan(1/2*d*x+1/2*c))*a^5+2/d/b^2/(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))*a^2+15/4/d/b^4*arctan(tan(1/2*d*x+1/2*c))*a^3+2/d/b^8*arctan(tan(1/2
*d*x+1/2*c))*a^7+2/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^7*a^6-6/d/b/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4-
2/d/b/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^12-10/d/b/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^8-
14/3/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*a^4+46/15/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*a^2-2/7/d/b/(1+tan(1/2*d*x+1/
2*c)^2)^7-32/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^10*a^4+6/d*a^6/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))-6/d*a^4/b^4/(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/d*a^8/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))-85/
24/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^5*a+30/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2
*c)^4*a^6+24/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^10*a^2-218/3/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7
*tan(1/2*d*x+1/2*c)^8*a^4+146/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^8*a^2+40/d/b^7/(1+tan(1/2*
d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^6*a^6-7/6/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3*a+12/d/b^7/
(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2*a^6-80/3/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2
*a^4+30/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^8*a^6-6/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d
*x+1/2*c)^12*a^4+29/4/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^5*a^3+232/15/d/b^3/(1+tan(1/2*d*x+1/
2*c)^2)^7*tan(1/2*d*x+1/2*c)^2*a^2-1/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)*a^5+9/4/d/b^4/(1+tan(
1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)*a^3-11/8/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)*a+2/d/b^7/
(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^12*a^6+1/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13*
a^5-9/4/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13*a^3+11/8/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1
/2*d*x+1/2*c)^13*a+4/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^11*a^5-7/d/b^4/(1+tan(1/2*d*x+1/2*c)^
2)^7*tan(1/2*d*x+1/2*c)^11*a^3+7/6/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^11*a+5/d/b^6/(1+tan(1/2
*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9*a^5-29/4/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9*a^3+85/24
/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9*a+7/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)
^3*a^3+202/5/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4*a^2

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 1.87915, size = 1436, normalized size = 3.52 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.24234, size = 1165, normalized size = 2.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^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