3.1320 \(\int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=467 \[ -\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{140 a b^4 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{192 b^5 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{240 a^2 b^3 d}+\frac {2 a^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^9 d}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{128 b^7 d}-\frac {x \left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right )}{128 b^9}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d} \]

[Out]

-1/128*(128*a^8-320*a^6*b^2+240*a^4*b^4-40*a^2*b^6-5*b^8)*x/b^9+2*a^3*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+
1/2*c))/(a^2-b^2)^(1/2))/b^9/d-1/105*a*(105*a^6-245*a^4*b^2+161*a^2*b^4-15*b^6)*cos(d*x+c)/b^8/d+1/128*(64*a^6
-144*a^4*b^2+88*a^2*b^4-5*b^6)*cos(d*x+c)*sin(d*x+c)/b^7/d-1/105*a*(35*a^4-77*a^2*b^2+45*b^4)*cos(d*x+c)*sin(d
*x+c)^2/b^6/d+1/192*(48*a^4-104*a^2*b^2+59*b^4)*cos(d*x+c)*sin(d*x+c)^3/b^5/d+1/4*cos(d*x+c)*sin(d*x+c)^4/a/d-
1/140*(28*a^4-60*a^2*b^2+35*b^4)*cos(d*x+c)*sin(d*x+c)^4/a/b^4/d-1/5*b*cos(d*x+c)*sin(d*x+c)^5/a^2/d+1/240*(40
*a^4-85*a^2*b^2+48*b^4)*cos(d*x+c)*sin(d*x+c)^5/a^2/b^3/d-1/7*a*cos(d*x+c)*sin(d*x+c)^6/b^2/d+1/8*cos(d*x+c)*s
in(d*x+c)^7/b/d

________________________________________________________________________________________

Rubi [A]  time = 1.81, antiderivative size = 467, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ -\frac {a \left (-245 a^4 b^2+161 a^2 b^4+105 a^6-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {2 a^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^9 d}+\frac {\left (-85 a^2 b^2+40 a^4+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{240 a^2 b^3 d}-\frac {\left (-60 a^2 b^2+28 a^4+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{140 a b^4 d}+\frac {\left (-104 a^2 b^2+48 a^4+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{192 b^5 d}-\frac {a \left (-77 a^2 b^2+35 a^4+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}+\frac {\left (-144 a^4 b^2+88 a^2 b^4+64 a^6-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{128 b^7 d}-\frac {x \left (-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6+128 a^8-5 b^8\right )}{128 b^9}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((128*a^8 - 320*a^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*x)/(128*b^9) + (2*a^3*(a^2 - b^2)^(5/2)*ArcTan[(b
+ a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^9*d) - (a*(105*a^6 - 245*a^4*b^2 + 161*a^2*b^4 - 15*b^6)*Cos[c + d
*x])/(105*b^8*d) + ((64*a^6 - 144*a^4*b^2 + 88*a^2*b^4 - 5*b^6)*Cos[c + d*x]*Sin[c + d*x])/(128*b^7*d) - (a*(3
5*a^4 - 77*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^6*d) + ((48*a^4 - 104*a^2*b^2 + 59*b^4)*Cos[c
+ d*x]*Sin[c + d*x]^3)/(192*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d) - ((28*a^4 - 60*a^2*b^2 + 35*b^4)*
Cos[c + d*x]*Sin[c + d*x]^4)/(140*a*b^4*d) - (b*Cos[c + d*x]*Sin[c + d*x]^5)/(5*a^2*d) + ((40*a^4 - 85*a^2*b^2
+ 48*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(240*a^2*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^6)/(7*b^2*d) + (Cos[c +
d*x]*Sin[c + d*x]^7)/(8*b*d)

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])

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 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 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 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 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 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])))

Rubi steps

\begin {align*} \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin ^5(c+d x) \left (40 \left (24 a^4-49 a^2 b^2+28 b^4\right )-4 a b \left (5 a^2-14 b^2\right ) \sin (c+d x)-28 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1120 a^2 b^2}\\ &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin ^4(c+d x) \left (-140 a \left (40 a^4-85 a^2 b^2+48 b^4\right )+20 a^2 b \left (8 a^2+7 b^2\right ) \sin (c+d x)+240 a \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}{6720 a^2 b^3}\\ &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a 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 b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin ^3(c+d x) \left (960 a^2 \left (28 a^4-60 a^2 b^2+35 b^4\right )-20 a^3 b \left (56 a^2-95 b^2\right ) \sin (c+d x)-700 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{33600 a^2 b^4}\\ &=\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a 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 b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin ^2(c+d x) \left (-2100 a^3 \left (48 a^4-104 a^2 b^2+59 b^4\right )+60 a^2 b \left (112 a^4-200 a^2 b^2+175 b^4\right ) \sin (c+d x)+3840 a^3 \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}{134400 a^2 b^5}\\ &=-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a 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 b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {\sin (c+d x) \left (7680 a^4 \left (35 a^4-77 a^2 b^2+45 b^4\right )-60 a^3 b \left (560 a^4-1064 a^2 b^2+435 b^4\right ) \sin (c+d x)-6300 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{403200 a^2 b^6}\\ &=\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a 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 b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {-6300 a^3 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right )+60 a^2 b \left (2240 a^6-4592 a^4 b^2+2280 a^2 b^4+525 b^6\right ) \sin (c+d x)+7680 a^3 \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}{806400 a^2 b^7}\\ &=-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a 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 b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\int \frac {-6300 a^3 b \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right )-6300 a^2 \left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{806400 a^2 b^8}\\ &=-\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a 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 b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\left (a^3 \left (a^2-b^2\right )^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^9}\\ &=-\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a 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 b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}+\frac {\left (2 a^3 \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^9 d}\\ &=-\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a 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 b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}-\frac {\left (4 a^3 \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^9 d}\\ &=-\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}+\frac {2 a^3 \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^9 d}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a 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 b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d}\\ \end {align*}

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Mathematica [A]  time = 3.32, size = 403, normalized size = 0.86 \[ \frac {-107520 a^8 c-107520 a^8 d x+26880 a^6 b^2 \sin (2 (c+d x))+268800 a^6 b^2 c+268800 a^6 b^2 d x-53760 a^4 b^4 \sin (2 (c+d x))-3360 a^4 b^4 \sin (4 (c+d x))-201600 a^4 b^4 c-201600 a^4 b^4 d x-1344 a^3 b^5 \cos (5 (c+d x))+25200 a^2 b^6 \sin (2 (c+d x))+5040 a^2 b^6 \sin (4 (c+d x))+560 a^2 b^6 \sin (6 (c+d x))+33600 a^2 b^6 c+33600 a^2 b^6 d x+560 \left (16 a^5 b^3-28 a^3 b^5+9 a b^7\right ) \cos (3 (c+d x))+215040 a^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )-1680 a b \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x)+1680 a b^7 \cos (5 (c+d x))+240 a b^7 \cos (7 (c+d x))+1680 b^8 \sin (2 (c+d x))-840 b^8 \sin (4 (c+d x))-560 b^8 \sin (6 (c+d x))-105 b^8 \sin (8 (c+d x))+4200 b^8 c+4200 b^8 d x}{107520 b^9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-107520*a^8*c + 268800*a^6*b^2*c - 201600*a^4*b^4*c + 33600*a^2*b^6*c + 4200*b^8*c - 107520*a^8*d*x + 268800*
a^6*b^2*d*x - 201600*a^4*b^4*d*x + 33600*a^2*b^6*d*x + 4200*b^8*d*x + 215040*a^3*(a^2 - b^2)^(5/2)*ArcTan[(b +
a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 1680*a*b*(64*a^6 - 144*a^4*b^2 + 88*a^2*b^4 - 5*b^6)*Cos[c + d*x] + 56
0*(16*a^5*b^3 - 28*a^3*b^5 + 9*a*b^7)*Cos[3*(c + d*x)] - 1344*a^3*b^5*Cos[5*(c + d*x)] + 1680*a*b^7*Cos[5*(c +
d*x)] + 240*a*b^7*Cos[7*(c + d*x)] + 26880*a^6*b^2*Sin[2*(c + d*x)] - 53760*a^4*b^4*Sin[2*(c + d*x)] + 25200*
a^2*b^6*Sin[2*(c + d*x)] + 1680*b^8*Sin[2*(c + d*x)] - 3360*a^4*b^4*Sin[4*(c + d*x)] + 5040*a^2*b^6*Sin[4*(c +
d*x)] - 840*b^8*Sin[4*(c + d*x)] + 560*a^2*b^6*Sin[6*(c + d*x)] - 560*b^8*Sin[6*(c + d*x)] - 105*b^8*Sin[8*(c
+ d*x)])/(107520*b^9*d)

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fricas [A]  time = 0.96, size = 706, normalized size = 1.51 \[ \left [\frac {1920 \, a b^{7} \cos \left (d x + c\right )^{7} - 2688 \, a^{3} b^{5} \cos \left (d x + c\right )^{5} + 4480 \, {\left (a^{5} b^{3} - a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (128 \, a^{8} - 320 \, a^{6} b^{2} + 240 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - 5 \, b^{8}\right )} d x + 6720 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} 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 ) - 13440 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right ) - 35 \, {\left (48 \, b^{8} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (48 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - 5 \, b^{8}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (64 \, a^{6} b^{2} - 112 \, a^{4} b^{4} + 40 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, b^{9} d}, \frac {1920 \, a b^{7} \cos \left (d x + c\right )^{7} - 2688 \, a^{3} b^{5} \cos \left (d x + c\right )^{5} + 4480 \, {\left (a^{5} b^{3} - a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (128 \, a^{8} - 320 \, a^{6} b^{2} + 240 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - 5 \, b^{8}\right )} d x - 13440 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} 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 ) - 13440 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right ) - 35 \, {\left (48 \, b^{8} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (48 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - 5 \, b^{8}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (64 \, a^{6} b^{2} - 112 \, a^{4} b^{4} + 40 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, b^{9} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/13440*(1920*a*b^7*cos(d*x + c)^7 - 2688*a^3*b^5*cos(d*x + c)^5 + 4480*(a^5*b^3 - a^3*b^5)*cos(d*x + c)^3 -
105*(128*a^8 - 320*a^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*d*x + 6720*(a^7 - 2*a^5*b^2 + a^3*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))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 13440*(a^7*b - 2*
a^5*b^3 + a^3*b^5)*cos(d*x + c) - 35*(48*b^8*cos(d*x + c)^7 - 8*(8*a^2*b^6 + b^8)*cos(d*x + c)^5 + 2*(48*a^4*b
^4 - 40*a^2*b^6 - 5*b^8)*cos(d*x + c)^3 - 3*(64*a^6*b^2 - 112*a^4*b^4 + 40*a^2*b^6 + 5*b^8)*cos(d*x + c))*sin(
d*x + c))/(b^9*d), 1/13440*(1920*a*b^7*cos(d*x + c)^7 - 2688*a^3*b^5*cos(d*x + c)^5 + 4480*(a^5*b^3 - a^3*b^5)
*cos(d*x + c)^3 - 105*(128*a^8 - 320*a^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*d*x - 13440*(a^7 - 2*a^5*b^2
+ a^3*b^4)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 13440*(a^7*b - 2*a^5
*b^3 + a^3*b^5)*cos(d*x + c) - 35*(48*b^8*cos(d*x + c)^7 - 8*(8*a^2*b^6 + b^8)*cos(d*x + c)^5 + 2*(48*a^4*b^4
- 40*a^2*b^6 - 5*b^8)*cos(d*x + c)^3 - 3*(64*a^6*b^2 - 112*a^4*b^4 + 40*a^2*b^6 + 5*b^8)*cos(d*x + c))*sin(d*x
+ c))/(b^9*d)]

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giac [B]  time = 0.21, size = 1244, normalized size = 2.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13440*(105*(128*a^8 - 320*a^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*(d*x + c)/b^9 - 26880*(a^9 - 3*a^7*b^
2 + 3*a^5*b^4 - a^3*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^9) + 2*(6720*a^6*b*tan(1/2*d*x + 1/2*c)^15 - 15120*a^4*b^3*tan(1/2*d*x + 1/2*c)
^15 + 9240*a^2*b^5*tan(1/2*d*x + 1/2*c)^15 - 525*b^7*tan(1/2*d*x + 1/2*c)^15 + 13440*a^7*tan(1/2*d*x + 1/2*c)^
14 - 40320*a^5*b^2*tan(1/2*d*x + 1/2*c)^14 + 40320*a^3*b^4*tan(1/2*d*x + 1/2*c)^14 - 13440*a*b^6*tan(1/2*d*x +
1/2*c)^14 + 33600*a^6*b*tan(1/2*d*x + 1/2*c)^13 - 62160*a^4*b^3*tan(1/2*d*x + 1/2*c)^13 + 17080*a^2*b^5*tan(1
/2*d*x + 1/2*c)^13 + 13895*b^7*tan(1/2*d*x + 1/2*c)^13 + 94080*a^7*tan(1/2*d*x + 1/2*c)^12 - 255360*a^5*b^2*ta
n(1/2*d*x + 1/2*c)^12 + 201600*a^3*b^4*tan(1/2*d*x + 1/2*c)^12 - 13440*a*b^6*tan(1/2*d*x + 1/2*c)^12 + 60480*a
^6*b*tan(1/2*d*x + 1/2*c)^11 - 95760*a^4*b^3*tan(1/2*d*x + 1/2*c)^11 + 31640*a^2*b^5*tan(1/2*d*x + 1/2*c)^11 -
31325*b^7*tan(1/2*d*x + 1/2*c)^11 + 282240*a^7*tan(1/2*d*x + 1/2*c)^10 - 703360*a^5*b^2*tan(1/2*d*x + 1/2*c)^
10 + 488320*a^3*b^4*tan(1/2*d*x + 1/2*c)^10 - 67200*a*b^6*tan(1/2*d*x + 1/2*c)^10 + 33600*a^6*b*tan(1/2*d*x +
1/2*c)^9 - 48720*a^4*b^3*tan(1/2*d*x + 1/2*c)^9 + 23800*a^2*b^5*tan(1/2*d*x + 1/2*c)^9 + 61775*b^7*tan(1/2*d*x
+ 1/2*c)^9 + 470400*a^7*tan(1/2*d*x + 1/2*c)^8 - 1097600*a^5*b^2*tan(1/2*d*x + 1/2*c)^8 + 721280*a^3*b^4*tan(
1/2*d*x + 1/2*c)^8 - 67200*a*b^6*tan(1/2*d*x + 1/2*c)^8 - 33600*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 48720*a^4*b^3*t
an(1/2*d*x + 1/2*c)^7 - 23800*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 61775*b^7*tan(1/2*d*x + 1/2*c)^7 + 470400*a^7*t
an(1/2*d*x + 1/2*c)^6 - 1052800*a^5*b^2*tan(1/2*d*x + 1/2*c)^6 + 665728*a^3*b^4*tan(1/2*d*x + 1/2*c)^6 - 40320
*a*b^6*tan(1/2*d*x + 1/2*c)^6 - 60480*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 95760*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 31
640*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 31325*b^7*tan(1/2*d*x + 1/2*c)^5 + 282240*a^7*tan(1/2*d*x + 1/2*c)^4 - 62
2720*a^5*b^2*tan(1/2*d*x + 1/2*c)^4 + 375424*a^3*b^4*tan(1/2*d*x + 1/2*c)^4 - 40320*a*b^6*tan(1/2*d*x + 1/2*c)
^4 - 33600*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 62160*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 17080*a^2*b^5*tan(1/2*d*x + 1
/2*c)^3 - 13895*b^7*tan(1/2*d*x + 1/2*c)^3 + 94080*a^7*tan(1/2*d*x + 1/2*c)^2 - 210560*a^5*b^2*tan(1/2*d*x + 1
/2*c)^2 + 124544*a^3*b^4*tan(1/2*d*x + 1/2*c)^2 - 1920*a*b^6*tan(1/2*d*x + 1/2*c)^2 - 6720*a^6*b*tan(1/2*d*x +
1/2*c) + 15120*a^4*b^3*tan(1/2*d*x + 1/2*c) - 9240*a^2*b^5*tan(1/2*d*x + 1/2*c) + 525*b^7*tan(1/2*d*x + 1/2*c
) + 13440*a^7 - 31360*a^5*b^2 + 20608*a^3*b^4 - 1920*a*b^6)/((tan(1/2*d*x + 1/2*c)^2 + 1)^8*b^8))/d

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maple [B]  time = 0.32, size = 2587, normalized size = 5.54 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/8/d/b^3*arctan(tan(1/2*d*x+1/2*c))*a^2+5/64/d/b*arctan(tan(1/2*d*x+1/2*c))+29/4/d/b^5/(1+tan(1/2*d*x+1/2*c)^
2)^8*tan(1/2*d*x+1/2*c)^9*a^4-14/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^12*a^7+38/d/b^6/(1+tan(1/
2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^12*a^5-30/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^12*a^3+11/8
/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)*a^2-14/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c
)^2*a^7-2/d*a^3/b^3/(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))-15/4/d/b^5*arctan
(tan(1/2*d*x+1/2*c))*a^4+5/d/b^7*arctan(tan(1/2*d*x+1/2*c))*a^6-2/d/b^9*arctan(tan(1/2*d*x+1/2*c))*a^8-1765/19
2/d/b/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^9+895/192/d/b/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c
)^11-397/192/d/b/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^13+5/64/d/b/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*
d*x+1/2*c)^15-46/15/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^8*a^3+2/7/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^8*a-2/d/b^8/(1+tan
(1/2*d*x+1/2*c)^2)^8*a^7+14/3/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^8*a^5-5/64/d/b/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2
*d*x+1/2*c)+397/192/d/b/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^3-895/192/d/b/(1+tan(1/2*d*x+1/2*c)^2)^8
*tan(1/2*d*x+1/2*c)^5+1765/192/d/b/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^7+113/24/d/b^3/(1+tan(1/2*d*x
+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^5*a^2-70/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^6*a^7+61/24/d/b^3
/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^3*a^2+94/3/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^
2*a^5-85/24/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^9*a^2-42/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(
1/2*d*x+1/2*c)^10*a^7+314/3/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^10*a^5-218/3/d/b^4/(1+tan(1/2*
d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^10*a^3+10/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^10*a+2/d/b^2/
(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^12*a-9/4/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)*a^4
-278/15/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^2*a^3+6/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d
*x+1/2*c)^14*a^5-6/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^14*a^3+2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)
^8*tan(1/2*d*x+1/2*c)^14*a+2/7/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^2*a+490/3/d/b^6/(1+tan(1/2*
d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^8*a^5-70/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^8*a^7-42/d/b^8
/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^4*a^7-1486/15/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*
c)^6*a^3+6/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^6*a+5/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*
d*x+1/2*c)^7*a^6-29/4/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^7*a^4+85/24/d/b^3/(1+tan(1/2*d*x+1/2
*c)^2)^8*tan(1/2*d*x+1/2*c)^7*a^2+2/d/b^9*a^9/(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))-322/3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^8*a^3+10/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^8*t
an(1/2*d*x+1/2*c)^8*a+1/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)*a^6+9/d/b^7/(1+tan(1/2*d*x+1/2*c)^
2)^8*tan(1/2*d*x+1/2*c)^5*a^6-6/d*a^7/b^7/(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^5/b^5/(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))-11/8/d/b^3/(1+tan(1/
2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^15*a^2+5/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^3*a^6-37/4/d
/b^5/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^3*a^4-5/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)
^9*a^6-57/4/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^5*a^4-2/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1
/2*d*x+1/2*c)^14*a^7+470/3/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^6*a^5-61/24/d/b^3/(1+tan(1/2*d*
x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^13*a^2-1/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^15*a^6+9/4/d/b^5
/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^15*a^4-5/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^13
*a^6+37/4/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^13*a^4+6/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/
2*d*x+1/2*c)^4*a+278/3/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^4*a^5-838/15/d/b^4/(1+tan(1/2*d*x+1
/2*c)^2)^8*tan(1/2*d*x+1/2*c)^4*a^3-9/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^11*a^6+57/4/d/b^5/(1
+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^11*a^4-113/24/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^
11*a^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(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 details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 15.08, size = 4505, normalized size = 9.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*(15*a*b^6 - 105*a^7 - 161*a^3*b^4 + 245*a^5*b^2))/(105*b^8) + (2*tan(c/2 + (d*x)/2)^14*(a*b^6 - a^7 - 3*a^
3*b^4 + 3*a^5*b^2))/b^8 + (2*tan(c/2 + (d*x)/2)^12*(a*b^6 - 7*a^7 - 15*a^3*b^4 + 19*a^5*b^2))/b^8 + (2*tan(c/2
+ (d*x)/2)^10*(15*a*b^6 - 63*a^7 - 109*a^3*b^4 + 157*a^5*b^2))/(3*b^8) + (2*tan(c/2 + (d*x)/2)^8*(15*a*b^6 -
105*a^7 - 161*a^3*b^4 + 245*a^5*b^2))/(3*b^8) + (2*tan(c/2 + (d*x)/2)^4*(45*a*b^6 - 315*a^7 - 419*a^3*b^4 + 69
5*a^5*b^2))/(15*b^8) + (2*tan(c/2 + (d*x)/2)^6*(45*a*b^6 - 525*a^7 - 743*a^3*b^4 + 1175*a^5*b^2))/(15*b^8) + (
2*tan(c/2 + (d*x)/2)^2*(15*a*b^6 - 735*a^7 - 973*a^3*b^4 + 1645*a^5*b^2))/(105*b^8) + (tan(c/2 + (d*x)/2)*(64*
a^6 - 5*b^6 + 88*a^2*b^4 - 144*a^4*b^2))/(64*b^7) - (tan(c/2 + (d*x)/2)^15*(64*a^6 - 5*b^6 + 88*a^2*b^4 - 144*
a^4*b^2))/(64*b^7) + (tan(c/2 + (d*x)/2)^3*(960*a^6 + 397*b^6 + 488*a^2*b^4 - 1776*a^4*b^2))/(192*b^7) - (tan(
c/2 + (d*x)/2)^13*(960*a^6 + 397*b^6 + 488*a^2*b^4 - 1776*a^4*b^2))/(192*b^7) + (tan(c/2 + (d*x)/2)^7*(960*a^6
+ 1765*b^6 + 680*a^2*b^4 - 1392*a^4*b^2))/(192*b^7) - (tan(c/2 + (d*x)/2)^9*(960*a^6 + 1765*b^6 + 680*a^2*b^4
- 1392*a^4*b^2))/(192*b^7) + (tan(c/2 + (d*x)/2)^5*(1728*a^6 - 895*b^6 + 904*a^2*b^4 - 2736*a^4*b^2))/(192*b^
7) - (tan(c/2 + (d*x)/2)^11*(1728*a^6 - 895*b^6 + 904*a^2*b^4 - 2736*a^4*b^2))/(192*b^7))/(d*(8*tan(c/2 + (d*x
)/2)^2 + 28*tan(c/2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*tan(c/2 + (d*x)/2)^1
0 + 28*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1)) - (atan((((((25*a^2*b^24)
/512 + (25*a^4*b^22)/32 - (25*a^6*b^20)/16 - (125*a^8*b^18)/4 + 160*a^10*b^16 - 320*a^12*b^14 + 320*a^14*b^12
- 160*a^16*b^10 + 32*a^18*b^8)/b^23 - ((((5*a*b^26)/4 + (35*a^3*b^24)/4 - 38*a^5*b^22 + 44*a^7*b^20 - 16*a^9*b
^18)/b^23 - ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(49152*a*b^28 - 32768*a^3*b^26))/(512*b^24))*(b^8*5i - a^8*128i
+ a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*320i))/(128*b^9) + (tan(c/2 + (d*x)/2)*(32768*a^4*b^24 - 98304*a^6*b^2
2 + 98304*a^8*b^20 - 32768*a^10*b^18))/(512*b^24))*(b^8*5i - a^8*128i + a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*3
20i))/(128*b^9) + (tan(c/2 + (d*x)/2)*(50*a*b^26 + 775*a^3*b^24 - 2000*a^5*b^22 - 47584*a^7*b^20 + 278144*a^9*
b^18 - 655360*a^11*b^16 + 819200*a^13*b^14 - 573440*a^15*b^12 + 212992*a^17*b^10 - 32768*a^19*b^8))/(512*b^24)
)*(b^8*5i - a^8*128i + a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*320i)*1i)/(128*b^9) + ((((25*a^2*b^24)/512 + (25*a
^4*b^22)/32 - (25*a^6*b^20)/16 - (125*a^8*b^18)/4 + 160*a^10*b^16 - 320*a^12*b^14 + 320*a^14*b^12 - 160*a^16*b
^10 + 32*a^18*b^8)/b^23 + ((((5*a*b^26)/4 + (35*a^3*b^24)/4 - 38*a^5*b^22 + 44*a^7*b^20 - 16*a^9*b^18)/b^23 +
((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(49152*a*b^28 - 32768*a^3*b^26))/(512*b^24))*(b^8*5i - a^8*128i + a^2*b^6*4
0i - a^4*b^4*240i + a^6*b^2*320i))/(128*b^9) + (tan(c/2 + (d*x)/2)*(32768*a^4*b^24 - 98304*a^6*b^22 + 98304*a^
8*b^20 - 32768*a^10*b^18))/(512*b^24))*(b^8*5i - a^8*128i + a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*320i))/(128*b
^9) + (tan(c/2 + (d*x)/2)*(50*a*b^26 + 775*a^3*b^24 - 2000*a^5*b^22 - 47584*a^7*b^20 + 278144*a^9*b^18 - 65536
0*a^11*b^16 + 819200*a^13*b^14 - 573440*a^15*b^12 + 212992*a^17*b^10 - 32768*a^19*b^8))/(512*b^24))*(b^8*5i -
a^8*128i + a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*320i)*1i)/(128*b^9))/((32*a^25 - (25*a^5*b^20)/256 + (315*a^7*
b^18)/256 + (3205*a^9*b^16)/256 - (39415*a^11*b^14)/256 + (10135*a^13*b^12)/16 - (11217*a^15*b^10)/8 + (3773*a
^17*b^8)/2 - (3195*a^19*b^6)/2 + 836*a^21*b^4 - 248*a^23*b^2)/b^23 + (tan(c/2 + (d*x)/2)*(32768*a^26 - 50*a^4*
b^22 - 650*a^6*b^20 + 3850*a^8*b^18 + 24850*a^10*b^16 - 254240*a^12*b^14 + 913600*a^14*b^12 - 1834240*a^16*b^1
0 + 2293760*a^18*b^8 - 1835008*a^20*b^6 + 917504*a^22*b^4 - 262144*a^24*b^2))/(256*b^24) + ((((25*a^2*b^24)/51
2 + (25*a^4*b^22)/32 - (25*a^6*b^20)/16 - (125*a^8*b^18)/4 + 160*a^10*b^16 - 320*a^12*b^14 + 320*a^14*b^12 - 1
60*a^16*b^10 + 32*a^18*b^8)/b^23 - ((((5*a*b^26)/4 + (35*a^3*b^24)/4 - 38*a^5*b^22 + 44*a^7*b^20 - 16*a^9*b^18
)/b^23 - ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(49152*a*b^28 - 32768*a^3*b^26))/(512*b^24))*(b^8*5i - a^8*128i +
a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*320i))/(128*b^9) + (tan(c/2 + (d*x)/2)*(32768*a^4*b^24 - 98304*a^6*b^22 +
98304*a^8*b^20 - 32768*a^10*b^18))/(512*b^24))*(b^8*5i - a^8*128i + a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*320i
))/(128*b^9) + (tan(c/2 + (d*x)/2)*(50*a*b^26 + 775*a^3*b^24 - 2000*a^5*b^22 - 47584*a^7*b^20 + 278144*a^9*b^1
8 - 655360*a^11*b^16 + 819200*a^13*b^14 - 573440*a^15*b^12 + 212992*a^17*b^10 - 32768*a^19*b^8))/(512*b^24))*(
b^8*5i - a^8*128i + a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*320i))/(128*b^9) - ((((25*a^2*b^24)/512 + (25*a^4*b^2
2)/32 - (25*a^6*b^20)/16 - (125*a^8*b^18)/4 + 160*a^10*b^16 - 320*a^12*b^14 + 320*a^14*b^12 - 160*a^16*b^10 +
32*a^18*b^8)/b^23 + ((((5*a*b^26)/4 + (35*a^3*b^24)/4 - 38*a^5*b^22 + 44*a^7*b^20 - 16*a^9*b^18)/b^23 + ((32*a
^2*b^3 + (tan(c/2 + (d*x)/2)*(49152*a*b^28 - 32768*a^3*b^26))/(512*b^24))*(b^8*5i - a^8*128i + a^2*b^6*40i - a
^4*b^4*240i + a^6*b^2*320i))/(128*b^9) + (tan(c/2 + (d*x)/2)*(32768*a^4*b^24 - 98304*a^6*b^22 + 98304*a^8*b^20
- 32768*a^10*b^18))/(512*b^24))*(b^8*5i - a^8*128i + a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*320i))/(128*b^9) +
(tan(c/2 + (d*x)/2)*(50*a*b^26 + 775*a^3*b^24 - 2000*a^5*b^22 - 47584*a^7*b^20 + 278144*a^9*b^18 - 655360*a^11
*b^16 + 819200*a^13*b^14 - 573440*a^15*b^12 + 212992*a^17*b^10 - 32768*a^19*b^8))/(512*b^24))*(b^8*5i - a^8*12
8i + a^2*b^6*40i - a^4*b^4*240i + a^6*b^2*320i))/(128*b^9)))*(b^8*5i - a^8*128i + a^2*b^6*40i - a^4*b^4*240i +
a^6*b^2*320i)*1i)/(64*b^9*d) - (a^3*atan(((a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(((25*a^2*b^24)/512 + (25*a^4*b^2
2)/32 - (25*a^6*b^20)/16 - (125*a^8*b^18)/4 + 160*a^10*b^16 - 320*a^12*b^14 + 320*a^14*b^12 - 160*a^16*b^10 +
32*a^18*b^8)/b^23 + (tan(c/2 + (d*x)/2)*(50*a*b^26 + 775*a^3*b^24 - 2000*a^5*b^22 - 47584*a^7*b^20 + 278144*a^
9*b^18 - 655360*a^11*b^16 + 819200*a^13*b^14 - 573440*a^15*b^12 + 212992*a^17*b^10 - 32768*a^19*b^8))/(512*b^2
4) + (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(((5*a*b^26)/4 + (35*a^3*b^24)/4 - 38*a^5*b^22 + 44*a^7*b^20 - 16*a^9*b
^18)/b^23 + (tan(c/2 + (d*x)/2)*(32768*a^4*b^24 - 98304*a^6*b^22 + 98304*a^8*b^20 - 32768*a^10*b^18))/(512*b^2
4) + (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(49152*a*b^28 - 32768*a^3*b^26))/(512
*b^24)))/b^9))/b^9)*1i)/b^9 + (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(((25*a^2*b^24)/512 + (25*a^4*b^22)/32 - (25*a
^6*b^20)/16 - (125*a^8*b^18)/4 + 160*a^10*b^16 - 320*a^12*b^14 + 320*a^14*b^12 - 160*a^16*b^10 + 32*a^18*b^8)/
b^23 + (tan(c/2 + (d*x)/2)*(50*a*b^26 + 775*a^3*b^24 - 2000*a^5*b^22 - 47584*a^7*b^20 + 278144*a^9*b^18 - 6553
60*a^11*b^16 + 819200*a^13*b^14 - 573440*a^15*b^12 + 212992*a^17*b^10 - 32768*a^19*b^8))/(512*b^24) - (a^3*(-(
a + b)^5*(a - b)^5)^(1/2)*(((5*a*b^26)/4 + (35*a^3*b^24)/4 - 38*a^5*b^22 + 44*a^7*b^20 - 16*a^9*b^18)/b^23 + (
tan(c/2 + (d*x)/2)*(32768*a^4*b^24 - 98304*a^6*b^22 + 98304*a^8*b^20 - 32768*a^10*b^18))/(512*b^24) - (a^3*(-(
a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(49152*a*b^28 - 32768*a^3*b^26))/(512*b^24)))/b^9)
)/b^9)*1i)/b^9)/((32*a^25 - (25*a^5*b^20)/256 + (315*a^7*b^18)/256 + (3205*a^9*b^16)/256 - (39415*a^11*b^14)/2
56 + (10135*a^13*b^12)/16 - (11217*a^15*b^10)/8 + (3773*a^17*b^8)/2 - (3195*a^19*b^6)/2 + 836*a^21*b^4 - 248*a
^23*b^2)/b^23 + (tan(c/2 + (d*x)/2)*(32768*a^26 - 50*a^4*b^22 - 650*a^6*b^20 + 3850*a^8*b^18 + 24850*a^10*b^16
- 254240*a^12*b^14 + 913600*a^14*b^12 - 1834240*a^16*b^10 + 2293760*a^18*b^8 - 1835008*a^20*b^6 + 917504*a^22
*b^4 - 262144*a^24*b^2))/(256*b^24) - (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(((25*a^2*b^24)/512 + (25*a^4*b^22)/32
- (25*a^6*b^20)/16 - (125*a^8*b^18)/4 + 160*a^10*b^16 - 320*a^12*b^14 + 320*a^14*b^12 - 160*a^16*b^10 + 32*a^
18*b^8)/b^23 + (tan(c/2 + (d*x)/2)*(50*a*b^26 + 775*a^3*b^24 - 2000*a^5*b^22 - 47584*a^7*b^20 + 278144*a^9*b^1
8 - 655360*a^11*b^16 + 819200*a^13*b^14 - 573440*a^15*b^12 + 212992*a^17*b^10 - 32768*a^19*b^8))/(512*b^24) +
(a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(((5*a*b^26)/4 + (35*a^3*b^24)/4 - 38*a^5*b^22 + 44*a^7*b^20 - 16*a^9*b^18)/
b^23 + (tan(c/2 + (d*x)/2)*(32768*a^4*b^24 - 98304*a^6*b^22 + 98304*a^8*b^20 - 32768*a^10*b^18))/(512*b^24) +
(a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(49152*a*b^28 - 32768*a^3*b^26))/(512*b^24
)))/b^9))/b^9))/b^9 + (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(((25*a^2*b^24)/512 + (25*a^4*b^22)/32 - (25*a^6*b^20)
/16 - (125*a^8*b^18)/4 + 160*a^10*b^16 - 320*a^12*b^14 + 320*a^14*b^12 - 160*a^16*b^10 + 32*a^18*b^8)/b^23 + (
tan(c/2 + (d*x)/2)*(50*a*b^26 + 775*a^3*b^24 - 2000*a^5*b^22 - 47584*a^7*b^20 + 278144*a^9*b^18 - 655360*a^11*
b^16 + 819200*a^13*b^14 - 573440*a^15*b^12 + 212992*a^17*b^10 - 32768*a^19*b^8))/(512*b^24) - (a^3*(-(a + b)^5
*(a - b)^5)^(1/2)*(((5*a*b^26)/4 + (35*a^3*b^24)/4 - 38*a^5*b^22 + 44*a^7*b^20 - 16*a^9*b^18)/b^23 + (tan(c/2
+ (d*x)/2)*(32768*a^4*b^24 - 98304*a^6*b^22 + 98304*a^8*b^20 - 32768*a^10*b^18))/(512*b^24) - (a^3*(-(a + b)^5
*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(49152*a*b^28 - 32768*a^3*b^26))/(512*b^24)))/b^9))/b^9))/
b^9))*(-(a + b)^5*(a - b)^5)^(1/2)*2i)/(b^9*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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