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

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.81436, antiderivative size = 467, normalized size of antiderivative = 1., 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 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 ^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*}

Mathematica [A]  time = 3.32049, size = 403, normalized size = 0.86 \[ \frac{26880 a^6 b^2 \sin (2 (c+d x))-53760 a^4 b^4 \sin (2 (c+d x))-3360 a^4 b^4 \sin (4 (c+d x))+25200 a^2 b^6 \sin (2 (c+d x))+5040 a^2 b^6 \sin (4 (c+d x))+560 a^2 b^6 \sin (6 (c+d x))-1344 a^3 b^5 \cos (5 (c+d x))-1680 a b \left (-144 a^4 b^2+88 a^2 b^4+64 a^6-5 b^6\right ) \cos (c+d x)+560 \left (-28 a^3 b^5+16 a^5 b^3+9 a b^7\right ) \cos (3 (c+d x))+215040 a^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )+268800 a^6 b^2 c-201600 a^4 b^4 c+33600 a^2 b^6 c+268800 a^6 b^2 d x-201600 a^4 b^4 d x+33600 a^2 b^6 d x-107520 a^8 c-107520 a^8 d x+1680 a b^7 \cos (5 (c+d x))+240 a b^7 \cos (7 (c+d x))+1680 b^8 \sin (2 (c+d x))-840 b^8 \sin (4 (c+d x))-560 b^8 \sin (6 (c+d x))-105 b^8 \sin (8 (c+d x))+4200 b^8 c+4200 b^8 d x}{107520 b^9 d} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.099, size = 2587, normalized size = 5.5 \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)^3/(a+b*sin(d*x+c)),x)

[Out]

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+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^4-2/d/b^8/(1+tan(1/2*d*x+1/2*c)^2)^8*tan
(1/2*d*x+1/2*c)^14*a^7-322/3/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+5/64/d/b*arctan(tan(1/2
*d*x+1/2*c))-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+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^5-30/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+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+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+9/4/d/b^5/(1+t
an(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^15*a^4-6/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^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))+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*arctan(tan(1/2*d*x+1/2*c))*a^6-2/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^5-46/15/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^8*a^3+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)^15-5/64/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
-2/d/b^9*arctan(tan(1/2*d*x+1/2*c))*a^8+2/7/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^8*a+1765/192/d/b/(1+tan(1/2*d*x+1/2
*c)^2)^8*tan(1/2*d*x+1/2*c)^7-1765/192/d/b/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^9+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-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+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+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^6-113/24/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/
2*d*x+1/2*c)^11*a^2-5/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^4-15/4/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^4+5/8/d/b^3*arctan(tan(1/2*d*x+1/2*
c))*a^2+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+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+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^2-1486/15/d/b^4/(1+ta
n(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^6*a^3-2/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))-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+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+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^4-11/8/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+10/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^8
*a+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-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^4-85/24/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^7-61/24/d/b^3/(1+t
an(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-
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+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+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*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))-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))-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^7-9/4/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)*a^4+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+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^5-278/15/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^8
*tan(1/2*d*x+1/2*c)^2*a^3-5/d/b^7/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^9*a^6-37/4/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+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^6-57/4/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+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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.00193, size = 1659, normalized size = 3.55 \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)^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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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)**3/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.24775, size = 1679, normalized size = 3.6 \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)^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