3.1267 \(\int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.72, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2896, 3047, 3049, 3023, 2735, 2660, 618, 204} \[ \frac {\left (-645 a^2 b^2+630 a^4+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac {\sqrt {a^2-b^2} \left (-29 a^2 b^2+42 a^4+2 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^8 d}-\frac {\left (-60 a^2 b^2+63 a^4+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {\left (-54 a^2 b^2+63 a^4+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac {\left (-187 a^2 b^2+210 a^4+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{30 a^2 b^5 d}-\frac {\left (-79 a^2 b^2+84 a^4+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{8 a b^6 d}+\frac {a x \left (-200 a^2 b^2+168 a^4+45 b^4\right )}{8 b^8}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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 3047

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

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

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Mathematica [A]  time = 12.87, size = 517, normalized size = 1.07 \[ \frac {\frac {\left (a^2-b^2\right )^2 \left (40320 a^7 c+40320 a^7 d x+80640 a^6 b c \sin (c+d x)+80640 a^6 b d x \sin (c+d x)+30240 a^5 b^2 \sin (2 (c+d x))-27840 a^5 b^2 c-27840 a^5 b^2 d x-96000 a^4 b^3 c \sin (c+d x)-96000 a^4 b^3 d x \sin (c+d x)-3360 a^4 b^3 \cos (3 (c+d x))-32640 a^3 b^4 \sin (2 (c+d x))+420 a^3 b^4 \sin (4 (c+d x))-13200 a^3 b^4 c-13200 a^3 b^4 d x+21600 a^2 b^5 c \sin (c+d x)+21600 a^2 b^5 d x \sin (c+d x)+3580 a^2 b^5 \cos (3 (c+d x))+84 a^2 b^5 \cos (5 (c+d x))-120 a b^2 \left (168 a^4-200 a^2 b^2+45 b^4\right ) (c+d x) \cos (2 (c+d x))+10 b \left (4032 a^6-3792 a^4 b^2+216 a^2 b^4+59 b^6\right ) \cos (c+d x)+5675 a b^6 \sin (2 (c+d x))-374 a b^6 \sin (4 (c+d x))-21 a b^6 \sin (6 (c+d x))+5400 a b^6 c+5400 a b^6 d x-526 b^7 \cos (3 (c+d x))-58 b^7 \cos (5 (c+d x))-6 b^7 \cos (7 (c+d x))\right )}{(a+b \sin (c+d x))^2}-1920 \left (a^2-b^2\right )^{5/2} \left (42 a^4-29 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{1920 b^8 d (a-b)^2 (a+b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.25, size = 995, normalized size = 2.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.28, size = 724, normalized size = 1.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.64, size = 1676, normalized size = 3.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-36/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^8*a^2+46/15/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5+20/d/b^6/
(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^7*a^3-15/2/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^7
*a+120/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^6*a^4-120/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*
d*x+1/2*c)^6*a^2+180/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4*a^4-160/d/b^5/(1+tan(1/2*d*x+1/2*c)
^2)^5*tan(1/2*d*x+1/2*c)^4*a^2-20/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^3*a^3+15/2/d/b^4/(1+tan(
1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^3*a-13/d/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/
2*d*x+1/2*c)^3*a^3+2/d/b^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3*a+10/d/b^6
/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9*a^3-27/4/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^
9*a+42/d/b^8*arctan(tan(1/2*d*x+1/2*c))*a^5-15/d/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*a^4+3
0/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^5*a^4+28/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^2+56/3/d/b^3/(
1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4+6/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^8+12/d/b^
3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^6-28/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*a^2+3/d/b^3/(tan(1/2*d*x
+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*a^2-42/d/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))*a^6+71/d/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))*a^4-31/d
/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))*a^2+11/d/b^6/(tan(1/2*d*x+1/2*c)
^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3*a^5+30/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*
c)^8*a^4+12/d/b^7/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2*a^6+9/d/b^5/(tan(1/
2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2*a^4-27/d/b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1
/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2*a^2+37/d/b^6/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan
(1/2*d*x+1/2*c)*a^5-47/d/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)*a^3+10/d/b
^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)*a+120/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)
^5*tan(1/2*d*x+1/2*c)^2*a^4-104/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^2*a^2-10/d/b^6/(1+tan(1/2*
d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)*a^3+27/4/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)*a+45/4/d/b^4*a
rctan(tan(1/2*d*x+1/2*c))*a+6/d/b/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2+12/
d/b^7/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*a^6+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))-50/d/b^6*arctan(tan(1/2*d*x+1/2*c))*a^3

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

[Out]

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

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mupad [B]  time = 23.85, size = 3700, normalized size = 7.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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