\(\int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) [1256]

Optimal result

Integrand size = 29, antiderivative size = 525 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}-\frac {2 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^9 d}+\frac {\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac {a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))} \]

[Out]

1/8*a*(64*a^6-120*a^4*b^2+60*a^2*b^4-5*b^6)*x/b^9-2*a^2*(8*a^2-3*b^2)*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*d*x+
1/2*c))/(a^2-b^2)^(1/2))/b^9/d+1/105*(840*a^6-1435*a^4*b^2+588*a^2*b^4-15*b^6)*cos(d*x+c)/b^8/d-1/8*a*(32*a^4-
52*a^2*b^2+19*b^4)*cos(d*x+c)*sin(d*x+c)/b^7/d+1/105*(280*a^4-441*a^2*b^2+150*b^4)*cos(d*x+c)*sin(d*x+c)^2/b^6
/d-1/12*(24*a^4-37*a^2*b^2+12*b^4)*cos(d*x+c)*sin(d*x+c)^3/a/b^5/d+1/140*(224*a^4-340*a^2*b^2+105*b^4)*cos(d*x
+c)*sin(d*x+c)^4/a^2/b^4/d+1/4*cos(d*x+c)*sin(d*x+c)^4/a/d/(a+b*sin(d*x+c))-3/20*b*cos(d*x+c)*sin(d*x+c)^5/a^2
/d/(a+b*sin(d*x+c))-1/15*(20*a^4-30*a^2*b^2+9*b^4)*cos(d*x+c)*sin(d*x+c)^5/a^2/b^3/d/(a+b*sin(d*x+c))-4/21*a*c
os(d*x+c)*sin(d*x+c)^6/b^2/d/(a+b*sin(d*x+c))+1/7*cos(d*x+c)*sin(d*x+c)^7/b/d/(a+b*sin(d*x+c))

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2975, 3126, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^9 d}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^4 d}-\frac {a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \sin (c+d x) \cos (c+d x)}{8 b^7 d}+\frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{12 a b^5 d}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}+\frac {a x \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right )}{8 b^9}+\frac {\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))} \]

[In]

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

[Out]

(a*(64*a^6 - 120*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*x)/(8*b^9) - (2*a^2*(8*a^2 - 3*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b
+ a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^9*d) + ((840*a^6 - 1435*a^4*b^2 + 588*a^2*b^4 - 15*b^6)*Cos[c + d*
x])/(105*b^8*d) - (a*(32*a^4 - 52*a^2*b^2 + 19*b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*b^7*d) + ((280*a^4 - 441*a^2
*b^2 + 150*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^6*d) - ((24*a^4 - 37*a^2*b^2 + 12*b^4)*Cos[c + d*x]*Sin[c
+ d*x]^3)/(12*a*b^5*d) + ((224*a^4 - 340*a^2*b^2 + 105*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a^2*b^4*d) + (Co
s[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])) - (3*b*Cos[c + d*x]*Sin[c + d*x]^5)/(20*a^2*d*(a + b*S
in[c + d*x])) - ((20*a^4 - 30*a^2*b^2 + 9*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(15*a^2*b^3*d*(a + b*Sin[c + d*x])
) - (4*a*Cos[c + d*x]*Sin[c + d*x]^6)/(21*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x]^7)/(7*b*d*(
a + b*Sin[c + d*x]))

Rule 210

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2975

Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] +
(Dist[1/(a^2*b^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6)), Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*
x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m +
n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e +
f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(
n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x] - Simp[b*(m + n + 2)*Cos[e + f*x]*(d*S
in[e + f*x])^(n + 2)*((a + b*Sin[e + f*x])^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[a*(n + 5)*Cos[e + f
*x]*(d*Sin[e + f*x])^(n + 3)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3*f*(m + n + 5)*(m + n + 6))), x] + Simp[Cos
[e + f*x]*(d*Sin[e + f*x])^(n + 4)*((a + b*Sin[e + f*x])^(m + 1)/(b*d^4*f*(m + n + 6))), x]) /; FreeQ[{a, b, d
, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0]
&& NeQ[m + n + 6, 0] &&  !IGtQ[m, 0]

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3126

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 3128

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

Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\int \frac {\sin ^5(c+d x) \left (6 \left (160 a^4-245 a^2 b^2+84 b^4\right )-4 a b \left (10 a^2-21 b^2\right ) \sin (c+d x)-10 \left (112 a^4-180 a^2 b^2+63 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{840 a^2 b^2} \\ & = \frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin ^4(c+d x) \left (-280 \left (20 a^6-50 a^4 b^2+39 a^2 b^4-9 b^6\right )+10 a b \left (16 a^4-37 a^2 b^2+21 b^4\right ) \sin (c+d x)+30 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{840 a^2 b^3 \left (a^2-b^2\right )} \\ & = \frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin ^3(c+d x) \left (120 a \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right )-80 a^2 b \left (14 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x)-1400 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4200 a^2 b^4 \left (a^2-b^2\right )} \\ & = -\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin ^2(c+d x) \left (-4200 a^2 \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right )+120 a^3 b \left (56 a^4-121 a^2 b^2+65 b^4\right ) \sin (c+d x)+480 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{16800 a^2 b^5 \left (a^2-b^2\right )} \\ & = \frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin (c+d x) \left (960 a^3 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right )-120 a^2 b \left (280 a^6-637 a^4 b^2+417 a^2 b^4-60 b^6\right ) \sin (c+d x)-12600 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{50400 a^2 b^6 \left (a^2-b^2\right )} \\ & = -\frac {a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac {\int \frac {-12600 a^4 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right )+120 a^3 b \left (1120 a^6-2716 a^4 b^2+2001 a^2 b^4-405 b^6\right ) \sin (c+d x)+960 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{100800 a^2 b^7 \left (a^2-b^2\right )} \\ & = \frac {\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac {a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac {\int \frac {-12600 a^4 b \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right )-12600 a^3 \left (64 a^8-184 a^6 b^2+180 a^4 b^4-65 a^2 b^6+5 b^8\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{100800 a^2 b^8 \left (a^2-b^2\right )} \\ & = \frac {a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}+\frac {\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac {a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac {\left (a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^9} \\ & = \frac {a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}+\frac {\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac {a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}-\frac {\left (2 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^9 d} \\ & = \frac {a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}+\frac {\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac {a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\left (4 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^9 d} \\ & = \frac {a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}-\frac {2 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^9 d}+\frac {\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac {a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.85 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-26880 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+\frac {107520 a^8 c-201600 a^6 b^2 c+100800 a^4 b^4 c-8400 a^2 b^6 c+107520 a^8 d x-201600 a^6 b^2 d x+100800 a^4 b^4 d x-8400 a^2 b^6 d x+840 a b \left (128 a^6-224 a^4 b^2+98 a^2 b^4-5 b^6\right ) \cos (c+d x)+70 \left (64 a^5 b^3-96 a^3 b^5+27 a b^7\right ) \cos (3 (c+d x))-336 a^3 b^5 \cos (5 (c+d x))+350 a b^7 \cos (5 (c+d x))+40 a b^7 \cos (7 (c+d x))+107520 a^7 b c \sin (c+d x)-201600 a^5 b^3 c \sin (c+d x)+100800 a^3 b^5 c \sin (c+d x)-8400 a b^7 c \sin (c+d x)+107520 a^7 b d x \sin (c+d x)-201600 a^5 b^3 d x \sin (c+d x)+100800 a^3 b^5 d x \sin (c+d x)-8400 a b^7 d x \sin (c+d x)+26880 a^6 b^2 \sin (2 (c+d x))-45920 a^4 b^4 \sin (2 (c+d x))+18480 a^2 b^6 \sin (2 (c+d x))-210 b^8 \sin (2 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))+1428 a^2 b^6 \sin (4 (c+d x))-210 b^8 \sin (4 (c+d x))+112 a^2 b^6 \sin (6 (c+d x))-90 b^8 \sin (6 (c+d x))-15 b^8 \sin (8 (c+d x))}{a+b \sin (c+d x)}}{13440 b^9 d} \]

[In]

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

[Out]

(-26880*a^2*(8*a^2 - 3*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + (107520*a^8*c
- 201600*a^6*b^2*c + 100800*a^4*b^4*c - 8400*a^2*b^6*c + 107520*a^8*d*x - 201600*a^6*b^2*d*x + 100800*a^4*b^4
*d*x - 8400*a^2*b^6*d*x + 840*a*b*(128*a^6 - 224*a^4*b^2 + 98*a^2*b^4 - 5*b^6)*Cos[c + d*x] + 70*(64*a^5*b^3 -
96*a^3*b^5 + 27*a*b^7)*Cos[3*(c + d*x)] - 336*a^3*b^5*Cos[5*(c + d*x)] + 350*a*b^7*Cos[5*(c + d*x)] + 40*a*b^
7*Cos[7*(c + d*x)] + 107520*a^7*b*c*Sin[c + d*x] - 201600*a^5*b^3*c*Sin[c + d*x] + 100800*a^3*b^5*c*Sin[c + d*
x] - 8400*a*b^7*c*Sin[c + d*x] + 107520*a^7*b*d*x*Sin[c + d*x] - 201600*a^5*b^3*d*x*Sin[c + d*x] + 100800*a^3*
b^5*d*x*Sin[c + d*x] - 8400*a*b^7*d*x*Sin[c + d*x] + 26880*a^6*b^2*Sin[2*(c + d*x)] - 45920*a^4*b^4*Sin[2*(c +
d*x)] + 18480*a^2*b^6*Sin[2*(c + d*x)] - 210*b^8*Sin[2*(c + d*x)] - 1120*a^4*b^4*Sin[4*(c + d*x)] + 1428*a^2*
b^6*Sin[4*(c + d*x)] - 210*b^8*Sin[4*(c + d*x)] + 112*a^2*b^6*Sin[6*(c + d*x)] - 90*b^8*Sin[6*(c + d*x)] - 15*
b^8*Sin[8*(c + d*x)])/(a + b*Sin[c + d*x]))/(13440*b^9*d)

Maple [A] (verified)

Time = 4.22 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {\frac {\frac {4 \left (\left (\frac {3}{2} a^{5} b^{2}-\frac {9}{4} a^{3} b^{4}+\frac {11}{16} a \,b^{6}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {7}{2} a^{6} b -\frac {15}{2} a^{4} b^{3}+\frac {9}{2} a^{2} b^{5}-\frac {1}{2} b^{7}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{5} b^{2}-7 a^{3} b^{4}+\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{6} b -40 a^{4} b^{3}+18 a^{2} b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {15}{2} a^{5} b^{2}-\frac {29}{4} a^{3} b^{4}+\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{2} a^{6} b -\frac {545}{6} a^{4} b^{3}+\frac {73}{2} a^{2} b^{5}-\frac {5}{2} b^{7}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (70 a^{6} b -\frac {340}{3} a^{4} b^{3}+44 a^{2} b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {15}{2} a^{5} b^{2}+\frac {29}{4} a^{3} b^{4}-\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{2} a^{6} b -\frac {165}{2} a^{4} b^{3}+\frac {303}{10} a^{2} b^{5}-\frac {3}{2} b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6 a^{5} b^{2}+7 a^{3} b^{4}-\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{6} b -\frac {100}{3} a^{4} b^{3}+\frac {58}{5} a^{2} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{2} a^{5} b^{2}+\frac {9}{4} a^{3} b^{4}-\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {7 a^{6} b}{2}-\frac {35 a^{4} b^{3}}{6}+\frac {23 a^{2} b^{5}}{10}-\frac {b^{7}}{14}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {a \left (64 a^{6}-120 a^{4} b^{2}+60 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{9}}-\frac {4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2} \left (\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}-\frac {a b}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (8 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9}}}{d}\)	\(656\)
default	\(\frac {\frac {\frac {4 \left (\left (\frac {3}{2} a^{5} b^{2}-\frac {9}{4} a^{3} b^{4}+\frac {11}{16} a \,b^{6}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {7}{2} a^{6} b -\frac {15}{2} a^{4} b^{3}+\frac {9}{2} a^{2} b^{5}-\frac {1}{2} b^{7}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{5} b^{2}-7 a^{3} b^{4}+\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{6} b -40 a^{4} b^{3}+18 a^{2} b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {15}{2} a^{5} b^{2}-\frac {29}{4} a^{3} b^{4}+\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{2} a^{6} b -\frac {545}{6} a^{4} b^{3}+\frac {73}{2} a^{2} b^{5}-\frac {5}{2} b^{7}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (70 a^{6} b -\frac {340}{3} a^{4} b^{3}+44 a^{2} b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {15}{2} a^{5} b^{2}+\frac {29}{4} a^{3} b^{4}-\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{2} a^{6} b -\frac {165}{2} a^{4} b^{3}+\frac {303}{10} a^{2} b^{5}-\frac {3}{2} b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6 a^{5} b^{2}+7 a^{3} b^{4}-\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{6} b -\frac {100}{3} a^{4} b^{3}+\frac {58}{5} a^{2} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{2} a^{5} b^{2}+\frac {9}{4} a^{3} b^{4}-\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {7 a^{6} b}{2}-\frac {35 a^{4} b^{3}}{6}+\frac {23 a^{2} b^{5}}{10}-\frac {b^{7}}{14}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {a \left (64 a^{6}-120 a^{4} b^{2}+60 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{9}}-\frac {4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2} \left (\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}-\frac {a b}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (8 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9}}}{d}\)	\(656\)
risch	\(\frac {7 \,{\mathrm e}^{i \left (d x +c \right )} a^{6}}{2 d \,b^{8}}-\frac {45 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}}{8 d \,b^{6}}+\frac {33 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{16 d \,b^{4}}+\frac {7 \,{\mathrm e}^{-i \left (d x +c \right )} a^{6}}{2 d \,b^{8}}-\frac {45 \,{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{8 d \,b^{6}}+\frac {33 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{16 d \,b^{4}}+\frac {3 \cos \left (5 d x +5 c \right ) a^{2}}{80 d \,b^{4}}+\frac {a^{3} \sin \left (4 d x +4 c \right )}{8 b^{5} d}-\frac {3 a \sin \left (4 d x +4 c \right )}{32 b^{3} d}-\frac {5 \cos \left (3 d x +3 c \right ) a^{4}}{12 d \,b^{6}}+\frac {7 \cos \left (3 d x +3 c \right ) a^{2}}{16 d \,b^{4}}-\frac {5 a x}{8 b^{3}}-\frac {15 i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{64 b^{3} d}+\frac {3 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{7} d}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{b^{5} d}+\frac {15 i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{64 b^{3} d}+\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{b^{5} d}-\frac {3 i a^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{7} d}-\frac {8 i \sqrt {a^{2}-b^{2}}\, a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{9}}+\frac {11 i \sqrt {a^{2}-b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{7}}-\frac {3 i \sqrt {a^{2}-b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}+\frac {8 i \sqrt {a^{2}-b^{2}}\, a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{9}}-\frac {11 i \sqrt {a^{2}-b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{7}}+\frac {3 i \sqrt {a^{2}-b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}-\frac {\cos \left (5 d x +5 c \right )}{64 d \,b^{2}}+\frac {2 i a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{9} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )}-\frac {3 \cos \left (3 d x +3 c \right )}{64 d \,b^{2}}-\frac {a \sin \left (6 d x +6 c \right )}{96 b^{3} d}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{128 b^{2} d}+\frac {8 a^{7} x}{b^{9}}-\frac {15 a^{5} x}{b^{7}}+\frac {15 a^{3} x}{2 b^{5}}-\frac {\cos \left (7 d x +7 c \right )}{448 b^{2} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 d \,b^{2}}\)	\(875\)

[In]

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

[Out]

1/d*(4/b^9*(((3/2*a^5*b^2-9/4*a^3*b^4+11/16*a*b^6)*tan(1/2*d*x+1/2*c)^13+(7/2*a^6*b-15/2*a^4*b^3+9/2*a^2*b^5-1
/2*b^7)*tan(1/2*d*x+1/2*c)^12+(6*a^5*b^2-7*a^3*b^4+7/12*a*b^6)*tan(1/2*d*x+1/2*c)^11+(21*a^6*b-40*a^4*b^3+18*a
^2*b^5)*tan(1/2*d*x+1/2*c)^10+(15/2*a^5*b^2-29/4*a^3*b^4+85/48*a*b^6)*tan(1/2*d*x+1/2*c)^9+(105/2*a^6*b-545/6*
a^4*b^3+73/2*a^2*b^5-5/2*b^7)*tan(1/2*d*x+1/2*c)^8+(70*a^6*b-340/3*a^4*b^3+44*a^2*b^5)*tan(1/2*d*x+1/2*c)^6+(-
15/2*a^5*b^2+29/4*a^3*b^4-85/48*a*b^6)*tan(1/2*d*x+1/2*c)^5+(105/2*a^6*b-165/2*a^4*b^3+303/10*a^2*b^5-3/2*b^7)
*tan(1/2*d*x+1/2*c)^4+(-6*a^5*b^2+7*a^3*b^4-7/12*a*b^6)*tan(1/2*d*x+1/2*c)^3+(21*a^6*b-100/3*a^4*b^3+58/5*a^2*
b^5)*tan(1/2*d*x+1/2*c)^2+(-3/2*a^5*b^2+9/4*a^3*b^4-11/16*a*b^6)*tan(1/2*d*x+1/2*c)+7/2*a^6*b-35/6*a^4*b^3+23/
10*a^2*b^5-1/14*b^7)/(1+tan(1/2*d*x+1/2*c)^2)^7+1/16*a*(64*a^6-120*a^4*b^2+60*a^2*b^4-5*b^6)*arctan(tan(1/2*d*
x+1/2*c)))-4*(a^4-2*a^2*b^2+b^4)*a^2/b^9*((-1/2*tan(1/2*d*x+1/2*c)*b^2-1/2*a*b)/(tan(1/2*d*x+1/2*c)^2*a+2*b*ta
n(1/2*d*x+1/2*c)+a)+1/2*(8*a^2-3*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))
))

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.66 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [\frac {160 \, a b^{7} \cos \left (d x + c\right )^{7} - 14 \, {\left (24 \, a^{3} b^{5} - 5 \, a b^{7}\right )} \cos \left (d x + c\right )^{5} + 35 \, {\left (32 \, a^{5} b^{3} - 36 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \cos \left (d x + c\right )^{3} + 105 \, {\left (64 \, a^{8} - 120 \, a^{6} b^{2} + 60 \, a^{4} b^{4} - 5 \, a^{2} b^{6}\right )} d x + 420 \, {\left (8 \, a^{7} - 11 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + {\left (8 \, a^{6} b - 11 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )\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 ) + 105 \, {\left (64 \, a^{7} b - 120 \, a^{5} b^{3} + 60 \, a^{3} b^{5} - 5 \, a b^{7}\right )} \cos \left (d x + c\right ) - {\left (120 \, b^{8} \cos \left (d x + c\right )^{7} - 224 \, a^{2} b^{6} \cos \left (d x + c\right )^{5} + 70 \, {\left (8 \, a^{4} b^{4} - 7 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (64 \, a^{7} b - 120 \, a^{5} b^{3} + 60 \, a^{3} b^{5} - 5 \, a b^{7}\right )} d x - 105 \, {\left (32 \, a^{6} b^{2} - 52 \, a^{4} b^{4} + 19 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, {\left (b^{10} d \sin \left (d x + c\right ) + a b^{9} d\right )}}, \frac {160 \, a b^{7} \cos \left (d x + c\right )^{7} - 14 \, {\left (24 \, a^{3} b^{5} - 5 \, a b^{7}\right )} \cos \left (d x + c\right )^{5} + 35 \, {\left (32 \, a^{5} b^{3} - 36 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \cos \left (d x + c\right )^{3} + 105 \, {\left (64 \, a^{8} - 120 \, a^{6} b^{2} + 60 \, a^{4} b^{4} - 5 \, a^{2} b^{6}\right )} d x + 840 \, {\left (8 \, a^{7} - 11 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + {\left (8 \, a^{6} b - 11 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )\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 ) + 105 \, {\left (64 \, a^{7} b - 120 \, a^{5} b^{3} + 60 \, a^{3} b^{5} - 5 \, a b^{7}\right )} \cos \left (d x + c\right ) - {\left (120 \, b^{8} \cos \left (d x + c\right )^{7} - 224 \, a^{2} b^{6} \cos \left (d x + c\right )^{5} + 70 \, {\left (8 \, a^{4} b^{4} - 7 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (64 \, a^{7} b - 120 \, a^{5} b^{3} + 60 \, a^{3} b^{5} - 5 \, a b^{7}\right )} d x - 105 \, {\left (32 \, a^{6} b^{2} - 52 \, a^{4} b^{4} + 19 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, {\left (b^{10} d \sin \left (d x + c\right ) + a b^{9} d\right )}}\right ] \]

[In]

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

[Out]

[1/840*(160*a*b^7*cos(d*x + c)^7 - 14*(24*a^3*b^5 - 5*a*b^7)*cos(d*x + c)^5 + 35*(32*a^5*b^3 - 36*a^3*b^5 + 5*
a*b^7)*cos(d*x + c)^3 + 105*(64*a^8 - 120*a^6*b^2 + 60*a^4*b^4 - 5*a^2*b^6)*d*x + 420*(8*a^7 - 11*a^5*b^2 + 3*
a^3*b^4 + (8*a^6*b - 11*a^4*b^3 + 3*a^2*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)) + 105*(64*a^7*b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*cos(d*x +
c) - (120*b^8*cos(d*x + c)^7 - 224*a^2*b^6*cos(d*x + c)^5 + 70*(8*a^4*b^4 - 7*a^2*b^6)*cos(d*x + c)^3 - 105*(
64*a^7*b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*d*x - 105*(32*a^6*b^2 - 52*a^4*b^4 + 19*a^2*b^6)*cos(d*x + c))*
sin(d*x + c))/(b^10*d*sin(d*x + c) + a*b^9*d), 1/840*(160*a*b^7*cos(d*x + c)^7 - 14*(24*a^3*b^5 - 5*a*b^7)*cos
(d*x + c)^5 + 35*(32*a^5*b^3 - 36*a^3*b^5 + 5*a*b^7)*cos(d*x + c)^3 + 105*(64*a^8 - 120*a^6*b^2 + 60*a^4*b^4 -
5*a^2*b^6)*d*x + 840*(8*a^7 - 11*a^5*b^2 + 3*a^3*b^4 + (8*a^6*b - 11*a^4*b^3 + 3*a^2*b^5)*sin(d*x + c))*sqrt(
a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 105*(64*a^7*b - 120*a^5*b^3 + 60*a^3
*b^5 - 5*a*b^7)*cos(d*x + c) - (120*b^8*cos(d*x + c)^7 - 224*a^2*b^6*cos(d*x + c)^5 + 70*(8*a^4*b^4 - 7*a^2*b^
6)*cos(d*x + c)^3 - 105*(64*a^7*b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*d*x - 105*(32*a^6*b^2 - 52*a^4*b^4 + 1
9*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/(b^10*d*sin(d*x + c) + a*b^9*d)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 965, normalized size of antiderivative = 1.84 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/840*(105*(64*a^7 - 120*a^5*b^2 + 60*a^3*b^4 - 5*a*b^6)*(d*x + c)/b^9 - 1680*(8*a^8 - 19*a^6*b^2 + 14*a^4*b^4
- 3*a^2*b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))
/(sqrt(a^2 - b^2)*b^9) + 1680*(a^6*b*tan(1/2*d*x + 1/2*c) - 2*a^4*b^3*tan(1/2*d*x + 1/2*c) + a^2*b^5*tan(1/2*d
*x + 1/2*c) + a^7 - 2*a^5*b^2 + a^3*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*b^8) + 2*(
2520*a^5*b*tan(1/2*d*x + 1/2*c)^13 - 3780*a^3*b^3*tan(1/2*d*x + 1/2*c)^13 + 1155*a*b^5*tan(1/2*d*x + 1/2*c)^13
+ 5880*a^6*tan(1/2*d*x + 1/2*c)^12 - 12600*a^4*b^2*tan(1/2*d*x + 1/2*c)^12 + 7560*a^2*b^4*tan(1/2*d*x + 1/2*c
)^12 - 840*b^6*tan(1/2*d*x + 1/2*c)^12 + 10080*a^5*b*tan(1/2*d*x + 1/2*c)^11 - 11760*a^3*b^3*tan(1/2*d*x + 1/2
*c)^11 + 980*a*b^5*tan(1/2*d*x + 1/2*c)^11 + 35280*a^6*tan(1/2*d*x + 1/2*c)^10 - 67200*a^4*b^2*tan(1/2*d*x + 1
/2*c)^10 + 30240*a^2*b^4*tan(1/2*d*x + 1/2*c)^10 + 12600*a^5*b*tan(1/2*d*x + 1/2*c)^9 - 12180*a^3*b^3*tan(1/2*
d*x + 1/2*c)^9 + 2975*a*b^5*tan(1/2*d*x + 1/2*c)^9 + 88200*a^6*tan(1/2*d*x + 1/2*c)^8 - 152600*a^4*b^2*tan(1/2
*d*x + 1/2*c)^8 + 61320*a^2*b^4*tan(1/2*d*x + 1/2*c)^8 - 4200*b^6*tan(1/2*d*x + 1/2*c)^8 + 117600*a^6*tan(1/2*
d*x + 1/2*c)^6 - 190400*a^4*b^2*tan(1/2*d*x + 1/2*c)^6 + 73920*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 - 12600*a^5*b*ta
n(1/2*d*x + 1/2*c)^5 + 12180*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 - 2975*a*b^5*tan(1/2*d*x + 1/2*c)^5 + 88200*a^6*ta
n(1/2*d*x + 1/2*c)^4 - 138600*a^4*b^2*tan(1/2*d*x + 1/2*c)^4 + 50904*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 2520*b^6
*tan(1/2*d*x + 1/2*c)^4 - 10080*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 11760*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 980*a*b^
5*tan(1/2*d*x + 1/2*c)^3 + 35280*a^6*tan(1/2*d*x + 1/2*c)^2 - 56000*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 + 19488*a^2
*b^4*tan(1/2*d*x + 1/2*c)^2 - 2520*a^5*b*tan(1/2*d*x + 1/2*c) + 3780*a^3*b^3*tan(1/2*d*x + 1/2*c) - 1155*a*b^5
*tan(1/2*d*x + 1/2*c) + 5880*a^6 - 9800*a^4*b^2 + 3864*a^2*b^4 - 120*b^6)/((tan(1/2*d*x + 1/2*c)^2 + 1)^7*b^8)
)/d

Mupad [B] (verification not implemented)

Time = 17.95 (sec) , antiderivative size = 3724, normalized size of antiderivative = 7.09 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((tan(c/2 + (d*x)/2)^14*(7*a*b^6 + 32*a^7 + 4*a^3*b^4 - 44*a^5*b^2))/(2*b^8) - (2*(15*a*b^6 - 840*a^7 - 588*a^
3*b^4 + 1435*a^5*b^2))/(105*b^8) + (2*tan(c/2 + (d*x)/2)^12*(4*a*b^6 + 168*a^7 + 72*a^3*b^4 - 255*a^5*b^2))/(3
*b^8) - (2*tan(c/2 + (d*x)/2)^8*(15*a*b^6 - 840*a^7 - 588*a^3*b^4 + 1435*a^5*b^2))/(3*b^8) + (tan(c/2 + (d*x)/
2)^10*(25*a*b^6 + 2016*a^7 + 1212*a^3*b^4 - 3284*a^5*b^2))/(6*b^8) - (2*tan(c/2 + (d*x)/2)^4*(80*a*b^6 - 2520*
a^7 - 1992*a^3*b^4 + 4465*a^5*b^2))/(15*b^8) - (tan(c/2 + (d*x)/2)^6*(605*a*b^6 - 16800*a^7 - 12756*a^3*b^4 +
29500*a^5*b^2))/(30*b^8) - (tan(c/2 + (d*x)/2)^2*(1215*a*b^6 - 23520*a^7 - 18396*a^3*b^4 + 41300*a^5*b^2))/(21
0*b^8) + (tan(c/2 + (d*x)/2)*(10080*a^6 - 240*b^6 + 7413*a^2*b^4 - 17500*a^4*b^2))/(420*b^7) + (tan(c/2 + (d*x
)/2)^15*(32*a^6 + 19*a^2*b^4 - 52*a^4*b^2))/(4*b^7) + (tan(c/2 + (d*x)/2)^11*(3168*a^6 + 2345*a^2*b^4 - 5532*a
^4*b^2))/(12*b^7) + (tan(c/2 + (d*x)/2)^7*(7200*a^6 + 4979*a^2*b^4 - 12212*a^4*b^2))/(12*b^7) + (tan(c/2 + (d*
x)/2)^3*(9120*a^6 + 6103*a^2*b^4 - 15460*a^4*b^2))/(60*b^7) + (tan(c/2 + (d*x)/2)^13*(864*a^6 - 48*b^6 + 661*a
^2*b^4 - 1500*a^4*b^2))/(12*b^7) + (tan(c/2 + (d*x)/2)^9*(6240*a^6 - 240*b^6 + 4429*a^2*b^4 - 10748*a^4*b^2))/
(12*b^7) + (tan(c/2 + (d*x)/2)^5*(24480*a^6 - 720*b^6 + 16499*a^2*b^4 - 41220*a^4*b^2))/(60*b^7))/(d*(a + 2*b*
tan(c/2 + (d*x)/2) + 8*a*tan(c/2 + (d*x)/2)^2 + 28*a*tan(c/2 + (d*x)/2)^4 + 56*a*tan(c/2 + (d*x)/2)^6 + 70*a*t
an(c/2 + (d*x)/2)^8 + 56*a*tan(c/2 + (d*x)/2)^10 + 28*a*tan(c/2 + (d*x)/2)^12 + 8*a*tan(c/2 + (d*x)/2)^14 + a*
tan(c/2 + (d*x)/2)^16 + 14*b*tan(c/2 + (d*x)/2)^3 + 42*b*tan(c/2 + (d*x)/2)^5 + 70*b*tan(c/2 + (d*x)/2)^7 + 70
*b*tan(c/2 + (d*x)/2)^9 + 42*b*tan(c/2 + (d*x)/2)^11 + 14*b*tan(c/2 + (d*x)/2)^13 + 2*b*tan(c/2 + (d*x)/2)^15)
) + (a*atan(((a*(((25*a^4*b^20)/2 - 300*a^6*b^18 + 2400*a^8*b^16 - 7520*a^10*b^14 + 11040*a^12*b^12 - 7680*a^1
4*b^10 + 2048*a^16*b^8)/b^23 + (tan(c/2 + (d*x)/2)*(50*a^3*b^22 - 1801*a^5*b^20 + 15576*a^7*b^18 - 54720*a^9*b
^16 + 96320*a^11*b^14 - 90240*a^13*b^12 + 43008*a^15*b^10 - 8192*a^17*b^8))/(2*b^24) - (a*((20*a^2*b^24 - 164*
a^4*b^22 + 272*a^6*b^20 - 128*a^8*b^18)/b^23 + (tan(c/2 + (d*x)/2)*(384*a^3*b^24 - 1792*a^5*b^22 + 2432*a^7*b^
20 - 1024*a^9*b^18))/(2*b^24) - (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^28 - 128*a^3*b^26))/(2*b^24))*(6
4*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i)/(8*b^9))*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i)/(8*b^9)
)*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2))/(8*b^9) + (a*(((25*a^4*b^20)/2 - 300*a^6*b^18 + 2400*a^8*b^16 -
7520*a^10*b^14 + 11040*a^12*b^12 - 7680*a^14*b^10 + 2048*a^16*b^8)/b^23 + (tan(c/2 + (d*x)/2)*(50*a^3*b^22 -
1801*a^5*b^20 + 15576*a^7*b^18 - 54720*a^9*b^16 + 96320*a^11*b^14 - 90240*a^13*b^12 + 43008*a^15*b^10 - 8192*a
^17*b^8))/(2*b^24) + (a*((20*a^2*b^24 - 164*a^4*b^22 + 272*a^6*b^20 - 128*a^8*b^18)/b^23 + (tan(c/2 + (d*x)/2)
*(384*a^3*b^24 - 1792*a^5*b^22 + 2432*a^7*b^20 - 1024*a^9*b^18))/(2*b^24) + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/
2)*(192*a*b^28 - 128*a^3*b^26))/(2*b^24))*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i)/(8*b^9))*(64*a^6 - 5
*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i)/(8*b^9))*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2))/(8*b^9))/((16384*a^
22 + 285*a^6*b^16 - 5530*a^8*b^14 + 38085*a^10*b^12 - 133328*a^12*b^10 + 269608*a^14*b^8 - 329120*a^16*b^6 + 2
39872*a^18*b^4 - 96256*a^20*b^2)/b^23 + (tan(c/2 + (d*x)/2)*(65536*a^23 - 150*a^5*b^18 + 4300*a^7*b^16 - 46550
*a^9*b^14 + 247840*a^11*b^12 - 745600*a^13*b^10 + 1358720*a^15*b^8 - 1534336*a^17*b^6 + 1051648*a^19*b^4 - 401
408*a^21*b^2))/b^24 + (a*(((25*a^4*b^20)/2 - 300*a^6*b^18 + 2400*a^8*b^16 - 7520*a^10*b^14 + 11040*a^12*b^12 -
7680*a^14*b^10 + 2048*a^16*b^8)/b^23 + (tan(c/2 + (d*x)/2)*(50*a^3*b^22 - 1801*a^5*b^20 + 15576*a^7*b^18 - 54
720*a^9*b^16 + 96320*a^11*b^14 - 90240*a^13*b^12 + 43008*a^15*b^10 - 8192*a^17*b^8))/(2*b^24) - (a*((20*a^2*b^
24 - 164*a^4*b^22 + 272*a^6*b^20 - 128*a^8*b^18)/b^23 + (tan(c/2 + (d*x)/2)*(384*a^3*b^24 - 1792*a^5*b^22 + 24
32*a^7*b^20 - 1024*a^9*b^18))/(2*b^24) - (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^28 - 128*a^3*b^26))/(2*
b^24))*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i)/(8*b^9))*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i
)/(8*b^9))*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i)/(8*b^9) - (a*(((25*a^4*b^20)/2 - 300*a^6*b^18 + 240
0*a^8*b^16 - 7520*a^10*b^14 + 11040*a^12*b^12 - 7680*a^14*b^10 + 2048*a^16*b^8)/b^23 + (tan(c/2 + (d*x)/2)*(50
*a^3*b^22 - 1801*a^5*b^20 + 15576*a^7*b^18 - 54720*a^9*b^16 + 96320*a^11*b^14 - 90240*a^13*b^12 + 43008*a^15*b
^10 - 8192*a^17*b^8))/(2*b^24) + (a*((20*a^2*b^24 - 164*a^4*b^22 + 272*a^6*b^20 - 128*a^8*b^18)/b^23 + (tan(c/
2 + (d*x)/2)*(384*a^3*b^24 - 1792*a^5*b^22 + 2432*a^7*b^20 - 1024*a^9*b^18))/(2*b^24) + (a*(32*a^2*b^3 + (tan(
c/2 + (d*x)/2)*(192*a*b^28 - 128*a^3*b^26))/(2*b^24))*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i)/(8*b^9))
*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i)/(8*b^9))*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2)*1i)/(8*b
^9)))*(64*a^6 - 5*b^6 + 60*a^2*b^4 - 120*a^4*b^2))/(4*b^9*d) - (2*a^2*atanh((75*a^6*(b^6 - a^6 - 3*a^2*b^4 + 3
*a^4*b^2)^(1/2))/(75*a^6*b^3 - 422*a^8*b + (811*a^10)/b - (656*a^12)/b^3 + (192*a^14)/b^5 + 1622*a^9*tan(c/2 +
(d*x)/2) + 150*a^5*b^4*tan(c/2 + (d*x)/2) - 844*a^7*b^2*tan(c/2 + (d*x)/2) - (1312*a^11*tan(c/2 + (d*x)/2))/b
^2 + (384*a^13*tan(c/2 + (d*x)/2))/b^4) - (272*a^8*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(811*a^10*b + 75
*a^6*b^5 - 422*a^8*b^3 - (656*a^12)/b + (192*a^14)/b^3 - 1312*a^11*tan(c/2 + (d*x)/2) + 150*a^5*b^6*tan(c/2 +
(d*x)/2) - 844*a^7*b^4*tan(c/2 + (d*x)/2) + 1622*a^9*b^2*tan(c/2 + (d*x)/2) + (384*a^13*tan(c/2 + (d*x)/2))/b^
2) + (192*a^10*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(75*a^6*b^7 - 656*a^12*b - 422*a^8*b^5 + 811*a^10*b^
3 + (192*a^14)/b + 384*a^13*tan(c/2 + (d*x)/2) + 150*a^5*b^8*tan(c/2 + (d*x)/2) - 844*a^7*b^6*tan(c/2 + (d*x)/
2) + 1622*a^9*b^4*tan(c/2 + (d*x)/2) - 1312*a^11*b^2*tan(c/2 + (d*x)/2)) + (150*a^5*tan(c/2 + (d*x)/2)*(b^6 -
a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(75*a^6*b^2 - 422*a^8 + (811*a^10)/b^2 - (656*a^12)/b^4 + (192*a^14)/b^6 -
844*a^7*b*tan(c/2 + (d*x)/2) + 150*a^5*b^3*tan(c/2 + (d*x)/2) + (1622*a^9*tan(c/2 + (d*x)/2))/b - (1312*a^11*
tan(c/2 + (d*x)/2))/b^3 + (384*a^13*tan(c/2 + (d*x)/2))/b^5) - (619*a^7*tan(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*
b^4 + 3*a^4*b^2)^(1/2))/(811*a^10 + 75*a^6*b^4 - 422*a^8*b^2 - (656*a^12)/b^2 + (192*a^14)/b^4 + 1622*a^9*b*ta
n(c/2 + (d*x)/2) + 150*a^5*b^5*tan(c/2 + (d*x)/2) - 844*a^7*b^3*tan(c/2 + (d*x)/2) - (1312*a^11*tan(c/2 + (d*x
)/2))/b + (384*a^13*tan(c/2 + (d*x)/2))/b^3) + (656*a^9*tan(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)
^(1/2))/(75*a^6*b^6 - 656*a^12 - 422*a^8*b^4 + 811*a^10*b^2 + (192*a^14)/b^2 - 1312*a^11*b*tan(c/2 + (d*x)/2)
+ 150*a^5*b^7*tan(c/2 + (d*x)/2) - 844*a^7*b^5*tan(c/2 + (d*x)/2) + 1622*a^9*b^3*tan(c/2 + (d*x)/2) + (384*a^1
3*tan(c/2 + (d*x)/2))/b) - (192*a^11*tan(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(192*a^14 +
75*a^6*b^8 - 422*a^8*b^6 + 811*a^10*b^4 - 656*a^12*b^2 + 384*a^13*b*tan(c/2 + (d*x)/2) + 150*a^5*b^9*tan(c/2
+ (d*x)/2) - 844*a^7*b^7*tan(c/2 + (d*x)/2) + 1622*a^9*b^5*tan(c/2 + (d*x)/2) - 1312*a^11*b^3*tan(c/2 + (d*x)/
2)))*(8*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(b^9*d)