\(\int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) [1257]

Optimal result

Integrand size = 29, antiderivative size = 471 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac {2 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^8 d}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))} \]

[Out]

-1/16*(112*a^6-200*a^4*b^2+90*a^2*b^4-5*b^6)*x/b^8+2*a*(7*a^2-2*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^8/d-1/15*a*(105*a^4-170*a^2*b^2+61*b^4)*cos(d*x+c)/b^7/d+1/16*(56*a^4-86*a^2*b^2+27*
b^4)*cos(d*x+c)*sin(d*x+c)/b^6/d-1/15*(35*a^4-52*a^2*b^2+15*b^4)*cos(d*x+c)*sin(d*x+c)^2/a/b^5/d+1/24*(42*a^4-
61*a^2*b^2+16*b^4)*cos(d*x+c)*sin(d*x+c)^3/a^2/b^4/d+1/3*cos(d*x+c)*sin(d*x+c)^3/a/d/(a+b*sin(d*x+c))-1/6*b*co
s(d*x+c)*sin(d*x+c)^4/a^2/d/(a+b*sin(d*x+c))-1/10*(14*a^4-20*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))-7/30*a*cos(d*x+c)*sin(d*x+c)^5/b^2/d/(a+b*sin(d*x+c))+1/6*cos(d*x+c)*sin(d*x+c)^6/b/d/(a+b*si
n(d*x+c))

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^8 d}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{6 a^2 d (a+b \sin (c+d x))}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^4 d}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{15 a b^5 d}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {x \left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right )}{16 b^8}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^6(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))} \]

[In]

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

[Out]

-1/16*((112*a^6 - 200*a^4*b^2 + 90*a^2*b^4 - 5*b^6)*x)/b^8 + (2*a*(7*a^2 - 2*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b
+ a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^8*d) - (a*(105*a^4 - 170*a^2*b^2 + 61*b^4)*Cos[c + d*x])/(15*b^7*d)
+ ((56*a^4 - 86*a^2*b^2 + 27*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^6*d) - ((35*a^4 - 52*a^2*b^2 + 15*b^4)*Cos
[c + d*x]*Sin[c + d*x]^2)/(15*a*b^5*d) + ((42*a^4 - 61*a^2*b^2 + 16*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*
b^4*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d*(a + b*Sin[c + d*x])) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(6*a^2*d
*(a + b*Sin[c + d*x])) - ((14*a^4 - 20*a^2*b^2 + 5*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(10*a^2*b^3*d*(a + b*Sin[
c + d*x])) - (7*a*Cos[c + d*x]*Sin[c + d*x]^5)/(30*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x]^6)
/(6*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 ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}+\frac {\int \frac {\sin ^4(c+d x) \left (60 \left (7 a^4-10 a^2 b^2+3 b^4\right )-12 a b \left (2 a^2-5 b^2\right ) \sin (c+d x)-12 \left (42 a^4-65 a^2 b^2+20 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{360 a^2 b^2} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin ^3(c+d x) \left (-144 \left (14 a^6-34 a^4 b^2+25 a^2 b^4-5 b^6\right )+12 a b \left (7 a^4-17 a^2 b^2+10 b^4\right ) \sin (c+d x)+60 \left (42 a^6-103 a^4 b^2+77 a^2 b^4-16 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{360 a^2 b^3 \left (a^2-b^2\right )} \\ & = \frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin ^2(c+d x) \left (180 a \left (42 a^6-103 a^4 b^2+77 a^2 b^4-16 b^6\right )-36 a^2 b \left (14 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x)-288 a \left (35 a^6-87 a^4 b^2+67 a^2 b^4-15 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1440 a^2 b^4 \left (a^2-b^2\right )} \\ & = -\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin (c+d x) \left (-576 a^2 \left (35 a^6-87 a^4 b^2+67 a^2 b^4-15 b^6\right )+36 a^3 b \left (70 a^4-153 a^2 b^2+83 b^4\right ) \sin (c+d x)+540 a^2 \left (56 a^6-142 a^4 b^2+113 a^2 b^4-27 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4320 a^2 b^5 \left (a^2-b^2\right )} \\ & = \frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {540 a^3 \left (56 a^6-142 a^4 b^2+113 a^2 b^4-27 b^6\right )-36 a^2 b \left (280 a^6-654 a^4 b^2+449 a^2 b^4-75 b^6\right ) \sin (c+d x)-576 a^3 \left (105 a^6-275 a^4 b^2+231 a^2 b^4-61 b^6\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{8640 a^2 b^6 \left (a^2-b^2\right )} \\ & = -\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\int \frac {540 a^3 b \left (56 a^6-142 a^4 b^2+113 a^2 b^4-27 b^6\right )+540 a^2 \left (112 a^8-312 a^6 b^2+290 a^4 b^4-95 a^2 b^6+5 b^8\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{8640 a^2 b^7 \left (a^2-b^2\right )} \\ & = -\frac {\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}+\frac {\left (a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^8} \\ & = -\frac {\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}+\frac {\left (2 a \left (7 a^2-2 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^8 d} \\ & = -\frac {\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac {\left (4 a \left (7 a^2-2 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^8 d} \\ & = -\frac {\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac {2 a \left (7 a^2-2 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^8 d}-\frac {a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac {\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac {\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac {\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac {\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.62 (sec) , antiderivative size = 462, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {3840 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-\frac {13440 a^7 c-24000 a^5 b^2 c+10800 a^3 b^4 c-600 a b^6 c+13440 a^7 d x-24000 a^5 b^2 d x+10800 a^3 b^4 d x-600 a b^6 d x+15 b \left (896 a^6-1488 a^4 b^2+576 a^2 b^4-15 b^6\right ) \cos (c+d x)+10 \left (56 a^4 b^3-79 a^2 b^5+18 b^7\right ) \cos (3 (c+d x))-42 a^2 b^5 \cos (5 (c+d x))+40 b^7 \cos (5 (c+d x))+5 b^7 \cos (7 (c+d x))+13440 a^6 b c \sin (c+d x)-24000 a^4 b^3 c \sin (c+d x)+10800 a^2 b^5 c \sin (c+d x)-600 b^7 c \sin (c+d x)+13440 a^6 b d x \sin (c+d x)-24000 a^4 b^3 d x \sin (c+d x)+10800 a^2 b^5 d x \sin (c+d x)-600 b^7 d x \sin (c+d x)+3360 a^5 b^2 \sin (2 (c+d x))-5440 a^3 b^4 \sin (2 (c+d x))+1910 a b^6 \sin (2 (c+d x))-140 a^3 b^4 \sin (4 (c+d x))+166 a b^6 \sin (4 (c+d x))+14 a b^6 \sin (6 (c+d x))}{a+b \sin (c+d x)}}{1920 b^8 d} \]

[In]

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

[Out]

(3840*a*(7*a^2 - 2*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - (13440*a^7*c - 24
000*a^5*b^2*c + 10800*a^3*b^4*c - 600*a*b^6*c + 13440*a^7*d*x - 24000*a^5*b^2*d*x + 10800*a^3*b^4*d*x - 600*a*
b^6*d*x + 15*b*(896*a^6 - 1488*a^4*b^2 + 576*a^2*b^4 - 15*b^6)*Cos[c + d*x] + 10*(56*a^4*b^3 - 79*a^2*b^5 + 18
*b^7)*Cos[3*(c + d*x)] - 42*a^2*b^5*Cos[5*(c + d*x)] + 40*b^7*Cos[5*(c + d*x)] + 5*b^7*Cos[7*(c + d*x)] + 1344
0*a^6*b*c*Sin[c + d*x] - 24000*a^4*b^3*c*Sin[c + d*x] + 10800*a^2*b^5*c*Sin[c + d*x] - 600*b^7*c*Sin[c + d*x]
+ 13440*a^6*b*d*x*Sin[c + d*x] - 24000*a^4*b^3*d*x*Sin[c + d*x] + 10800*a^2*b^5*d*x*Sin[c + d*x] - 600*b^7*d*x
*Sin[c + d*x] + 3360*a^5*b^2*Sin[2*(c + d*x)] - 5440*a^3*b^4*Sin[2*(c + d*x)] + 1910*a*b^6*Sin[2*(c + d*x)] -
140*a^3*b^4*Sin[4*(c + d*x)] + 166*a*b^6*Sin[4*(c + d*x)] + 14*a*b^6*Sin[6*(c + d*x)])/(a + b*Sin[c + d*x]))/(
1920*b^8*d)

Maple [A] (verified)

Time = 3.86 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (\frac {-b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{5} b +2 a^{3} b^{3}-a \,b^{5}}{\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 (7 a^{6}-16 a^{4} b^{2}+11 a^{2} b^{4}-2 b^{6}\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 )}{\sqrt {a^{2}-b^{2}}}\right )}{b^{8}}-\frac {2 \left (\frac {\left (\frac {5}{2} a^{4} b^{2}-\frac {27}{8} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{5} b -12 a^{3} b^{3}+6 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {15}{2} a^{4} b^{2}-\frac {57}{8} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{5} b -52 a^{3} b^{3}+18 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{4} b^{2}-\frac {15}{4} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{5} b -\frac {280}{3} a^{3} b^{3}+\frac {92}{3} a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-5 a^{4} b^{2}+\frac {15}{4} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{5} b -88 a^{3} b^{3}+28 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {15}{2} a^{4} b^{2}+\frac {57}{8} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{5} b -44 a^{3} b^{3}+\frac {62}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{4} b^{2}+\frac {27}{8} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+6 a^{5} b -\frac {28 a^{3} b^{3}}{3}+\frac {46 a \,b^{5}}{15}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (112 a^{6}-200 a^{4} b^{2}+90 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{8}}}{d}\)	\(611\)
default	\(\frac {\frac {2 a \left (\frac {-b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{5} b +2 a^{3} b^{3}-a \,b^{5}}{\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 (7 a^{6}-16 a^{4} b^{2}+11 a^{2} b^{4}-2 b^{6}\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 )}{\sqrt {a^{2}-b^{2}}}\right )}{b^{8}}-\frac {2 \left (\frac {\left (\frac {5}{2} a^{4} b^{2}-\frac {27}{8} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{5} b -12 a^{3} b^{3}+6 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {15}{2} a^{4} b^{2}-\frac {57}{8} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{5} b -52 a^{3} b^{3}+18 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{4} b^{2}-\frac {15}{4} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{5} b -\frac {280}{3} a^{3} b^{3}+\frac {92}{3} a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-5 a^{4} b^{2}+\frac {15}{4} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 a^{5} b -88 a^{3} b^{3}+28 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {15}{2} a^{4} b^{2}+\frac {57}{8} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{5} b -44 a^{3} b^{3}+\frac {62}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{4} b^{2}+\frac {27}{8} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+6 a^{5} b -\frac {28 a^{3} b^{3}}{3}+\frac {46 a \,b^{5}}{15}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (112 a^{6}-200 a^{4} b^{2}+90 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{8}}}{d}\)	\(611\)
risch	\(-\frac {a \cos \left (5 d x +5 c \right )}{40 b^{3} d}-\frac {11 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 b^{3} d}-\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )}}{128 b^{2} d}-\frac {3 \sin \left (4 d x +4 c \right ) a^{2}}{32 b^{4} d}+\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )}}{128 b^{2} d}-\frac {3 a^{5} {\mathrm e}^{i \left (d x +c \right )}}{b^{7} d}+\frac {a^{3} \cos \left (3 d x +3 c \right )}{3 b^{5} d}-\frac {7 a \cos \left (3 d x +3 c \right )}{24 b^{3} d}+\frac {9 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{5} d}-\frac {3 a^{5} {\mathrm e}^{-i \left (d x +c \right )}}{b^{7} d}+\frac {9 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{5} d}-\frac {11 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{3} d}+\frac {5 x}{16 b^{2}}+\frac {\sin \left (6 d x +6 c \right )}{192 b^{2} d}+\frac {5 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4}}{8 b^{6} d}-\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2}}{4 b^{4} d}-\frac {7 \sqrt {-a^{2}+b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{8}}+\frac {9 \sqrt {-a^{2}+b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{6}}-\frac {2 \sqrt {-a^{2}+b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}+\frac {7 \sqrt {-a^{2}+b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{8}}-\frac {9 \sqrt {-a^{2}+b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{6}}+\frac {2 \sqrt {-a^{2}+b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}-\frac {5 i {\mathrm e}^{2 i \left (d x +c \right )} a^{4}}{8 b^{6} d}+\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} a^{2}}{4 b^{4} d}+\frac {2 i a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (i b +a \,{\mathrm e}^{i \left (d x +c \right )}\right )}{b^{8} d \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )}-\frac {7 x \,a^{6}}{b^{8}}+\frac {25 x \,a^{4}}{2 b^{6}}-\frac {45 x \,a^{2}}{8 b^{4}}+\frac {3 \sin \left (4 d x +4 c \right )}{64 b^{2} d}\)	\(773\)

[In]

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

[Out]

1/d*(2*a/b^8*((-b^2*(a^4-2*a^2*b^2+b^4)*tan(1/2*d*x+1/2*c)-a^5*b+2*a^3*b^3-a*b^5)/(tan(1/2*d*x+1/2*c)^2*a+2*b*
tan(1/2*d*x+1/2*c)+a)+(7*a^6-16*a^4*b^2+11*a^2*b^4-2*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)))-2/b^8*(((5/2*a^4*b^2-27/8*a^2*b^4+11/16*b^6)*tan(1/2*d*x+1/2*c)^11+(6*a^5*b-12*a^3*b^3+6
*a*b^5)*tan(1/2*d*x+1/2*c)^10+(15/2*a^4*b^2-57/8*a^2*b^4-5/48*b^6)*tan(1/2*d*x+1/2*c)^9+(30*a^5*b-52*a^3*b^3+1
8*a*b^5)*tan(1/2*d*x+1/2*c)^8+(5*a^4*b^2-15/4*a^2*b^4+15/8*b^6)*tan(1/2*d*x+1/2*c)^7+(60*a^5*b-280/3*a^3*b^3+9
2/3*a*b^5)*tan(1/2*d*x+1/2*c)^6+(-5*a^4*b^2+15/4*a^2*b^4-15/8*b^6)*tan(1/2*d*x+1/2*c)^5+(60*a^5*b-88*a^3*b^3+2
8*a*b^5)*tan(1/2*d*x+1/2*c)^4+(-15/2*a^4*b^2+57/8*a^2*b^4+5/48*b^6)*tan(1/2*d*x+1/2*c)^3+(30*a^5*b-44*a^3*b^3+
62/5*a*b^5)*tan(1/2*d*x+1/2*c)^2+(-5/2*a^4*b^2+27/8*a^2*b^4-11/16*b^6)*tan(1/2*d*x+1/2*c)+6*a^5*b-28/3*a^3*b^3
+46/15*a*b^5)/(1+tan(1/2*d*x+1/2*c)^2)^6+1/16*(112*a^6-200*a^4*b^2+90*a^2*b^4-5*b^6)*arctan(tan(1/2*d*x+1/2*c)
)))

Fricas [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.73 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [-\frac {40 \, b^{7} \cos \left (d x + c\right )^{7} - 2 \, {\left (42 \, a^{2} b^{5} - 5 \, b^{7}\right )} \cos \left (d x + c\right )^{5} + 5 \, {\left (56 \, a^{4} b^{3} - 58 \, a^{2} b^{5} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (112 \, a^{7} - 200 \, a^{5} b^{2} + 90 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x - 120 \, {\left (7 \, a^{6} - 9 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (7 \, a^{5} b - 9 \, a^{3} b^{3} + 2 \, a 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 ) + 15 \, {\left (112 \, a^{6} b - 200 \, a^{4} b^{3} + 90 \, a^{2} b^{5} - 5 \, b^{7}\right )} \cos \left (d x + c\right ) + {\left (56 \, a b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (14 \, a^{3} b^{4} - 11 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (112 \, a^{6} b - 200 \, a^{4} b^{3} + 90 \, a^{2} b^{5} - 5 \, b^{7}\right )} d x + 15 \, {\left (56 \, a^{5} b^{2} - 86 \, a^{3} b^{4} + 27 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} d\right )}}, -\frac {40 \, b^{7} \cos \left (d x + c\right )^{7} - 2 \, {\left (42 \, a^{2} b^{5} - 5 \, b^{7}\right )} \cos \left (d x + c\right )^{5} + 5 \, {\left (56 \, a^{4} b^{3} - 58 \, a^{2} b^{5} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (112 \, a^{7} - 200 \, a^{5} b^{2} + 90 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x + 240 \, {\left (7 \, a^{6} - 9 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (7 \, a^{5} b - 9 \, a^{3} b^{3} + 2 \, a 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 ) + 15 \, {\left (112 \, a^{6} b - 200 \, a^{4} b^{3} + 90 \, a^{2} b^{5} - 5 \, b^{7}\right )} \cos \left (d x + c\right ) + {\left (56 \, a b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (14 \, a^{3} b^{4} - 11 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (112 \, a^{6} b - 200 \, a^{4} b^{3} + 90 \, a^{2} b^{5} - 5 \, b^{7}\right )} d x + 15 \, {\left (56 \, a^{5} b^{2} - 86 \, a^{3} b^{4} + 27 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} d\right )}}\right ] \]

[In]

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

[Out]

[-1/240*(40*b^7*cos(d*x + c)^7 - 2*(42*a^2*b^5 - 5*b^7)*cos(d*x + c)^5 + 5*(56*a^4*b^3 - 58*a^2*b^5 + 5*b^7)*c
os(d*x + c)^3 + 15*(112*a^7 - 200*a^5*b^2 + 90*a^3*b^4 - 5*a*b^6)*d*x - 120*(7*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + (
7*a^5*b - 9*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)) + 15*(112*a^6*b - 200*a^4*b^3 + 90*a^2*b^5 - 5*b^7)*cos(d*x + c) + (56*a*b^6
*cos(d*x + c)^5 - 10*(14*a^3*b^4 - 11*a*b^6)*cos(d*x + c)^3 + 15*(112*a^6*b - 200*a^4*b^3 + 90*a^2*b^5 - 5*b^7
)*d*x + 15*(56*a^5*b^2 - 86*a^3*b^4 + 27*a*b^6)*cos(d*x + c))*sin(d*x + c))/(b^9*d*sin(d*x + c) + a*b^8*d), -1
/240*(40*b^7*cos(d*x + c)^7 - 2*(42*a^2*b^5 - 5*b^7)*cos(d*x + c)^5 + 5*(56*a^4*b^3 - 58*a^2*b^5 + 5*b^7)*cos(
d*x + c)^3 + 15*(112*a^7 - 200*a^5*b^2 + 90*a^3*b^4 - 5*a*b^6)*d*x + 240*(7*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + (7*a
^5*b - 9*a^3*b^3 + 2*a*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))) + 15*(112*a^6*b - 200*a^4*b^3 + 90*a^2*b^5 - 5*b^7)*cos(d*x + c) + (56*a*b^6*cos(d*x + c)^5 - 10*(14*
a^3*b^4 - 11*a*b^6)*cos(d*x + c)^3 + 15*(112*a^6*b - 200*a^4*b^3 + 90*a^2*b^5 - 5*b^7)*d*x + 15*(56*a^5*b^2 -
86*a^3*b^4 + 27*a*b^6)*cos(d*x + c))*sin(d*x + c))/(b^9*d*sin(d*x + c) + a*b^8*d)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2/(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 = 0.34 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.77 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/240*(15*(112*a^6 - 200*a^4*b^2 + 90*a^2*b^4 - 5*b^6)*(d*x + c)/b^8 - 480*(7*a^7 - 16*a^5*b^2 + 11*a^3*b^4 -
2*a*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)))/(sq
rt(a^2 - b^2)*b^8) + 480*(a^5*b*tan(1/2*d*x + 1/2*c) - 2*a^3*b^3*tan(1/2*d*x + 1/2*c) + a*b^5*tan(1/2*d*x + 1/
2*c) + a^6 - 2*a^4*b^2 + 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)*b^7) + 2*(600*a^4
*b*tan(1/2*d*x + 1/2*c)^11 - 810*a^2*b^3*tan(1/2*d*x + 1/2*c)^11 + 165*b^5*tan(1/2*d*x + 1/2*c)^11 + 1440*a^5*
tan(1/2*d*x + 1/2*c)^10 - 2880*a^3*b^2*tan(1/2*d*x + 1/2*c)^10 + 1440*a*b^4*tan(1/2*d*x + 1/2*c)^10 + 1800*a^4
*b*tan(1/2*d*x + 1/2*c)^9 - 1710*a^2*b^3*tan(1/2*d*x + 1/2*c)^9 - 25*b^5*tan(1/2*d*x + 1/2*c)^9 + 7200*a^5*tan
(1/2*d*x + 1/2*c)^8 - 12480*a^3*b^2*tan(1/2*d*x + 1/2*c)^8 + 4320*a*b^4*tan(1/2*d*x + 1/2*c)^8 + 1200*a^4*b*ta
n(1/2*d*x + 1/2*c)^7 - 900*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 450*b^5*tan(1/2*d*x + 1/2*c)^7 + 14400*a^5*tan(1/2
*d*x + 1/2*c)^6 - 22400*a^3*b^2*tan(1/2*d*x + 1/2*c)^6 + 7360*a*b^4*tan(1/2*d*x + 1/2*c)^6 - 1200*a^4*b*tan(1/
2*d*x + 1/2*c)^5 + 900*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 450*b^5*tan(1/2*d*x + 1/2*c)^5 + 14400*a^5*tan(1/2*d*x
+ 1/2*c)^4 - 21120*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 + 6720*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 1800*a^4*b*tan(1/2*d*
x + 1/2*c)^3 + 1710*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 25*b^5*tan(1/2*d*x + 1/2*c)^3 + 7200*a^5*tan(1/2*d*x + 1/
2*c)^2 - 10560*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 2976*a*b^4*tan(1/2*d*x + 1/2*c)^2 - 600*a^4*b*tan(1/2*d*x + 1/
2*c) + 810*a^2*b^3*tan(1/2*d*x + 1/2*c) - 165*b^5*tan(1/2*d*x + 1/2*c) + 1440*a^5 - 2240*a^3*b^2 + 736*a*b^4)/
((tan(1/2*d*x + 1/2*c)^2 + 1)^6*b^7))/d

Mupad [B] (verification not implemented)

Time = 16.38 (sec) , antiderivative size = 4067, normalized size of antiderivative = 8.63 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- ((2*(105*a^6 + 61*a^2*b^4 - 170*a^4*b^2))/(15*b^7) + (tan(c/2 + (d*x)/2)^13*(27*a*b^4 + 56*a^5 - 86*a^3*b^2)
)/(8*b^6) + (8*tan(c/2 + (d*x)/2)^7*(61*a*b^4 + 105*a^5 - 170*a^3*b^2))/(3*b^6) + (tan(c/2 + (d*x)/2)^11*(223*
a*b^4 + 336*a^5 - 558*a^3*b^2))/(6*b^6) + (tan(c/2 + (d*x)/2)^3*(1813*a*b^4 + 3360*a^5 - 5370*a^3*b^2))/(30*b^
6) + (tan(c/2 + (d*x)/2)^9*(2533*a*b^4 + 4200*a^5 - 6954*a^3*b^2))/(24*b^6) + (tan(c/2 + (d*x)/2)^5*(3323*a*b^
4 + 5880*a^5 - 9366*a^3*b^2))/(24*b^6) + (tan(c/2 + (d*x)/2)^12*(56*a^6 + 11*b^6 + 2*a^2*b^4 - 72*a^4*b^2))/(4
*b^7) + (tan(c/2 + (d*x)/2)^10*(1008*a^6 - 5*b^6 + 378*a^2*b^4 - 1464*a^4*b^2))/(12*b^7) + (tan(c/2 + (d*x)/2)
^8*(1260*a^6 + 45*b^6 + 674*a^2*b^4 - 1984*a^4*b^2))/(6*b^7) + (tan(c/2 + (d*x)/2)^6*(1680*a^6 - 45*b^6 + 1034
*a^2*b^4 - 2776*a^4*b^2))/(6*b^7) + (tan(c/2 + (d*x)/2)^2*(5040*a^6 - 165*b^6 + 3386*a^2*b^4 - 8440*a^4*b^2))/
(60*b^7) + (tan(c/2 + (d*x)/2)^4*(12600*a^6 + 25*b^6 + 8358*a^2*b^4 - 21240*a^4*b^2))/(60*b^7) + (tan(c/2 + (d
*x)/2)*(1547*a*b^4 + 2520*a^5 - 4150*a^3*b^2))/(120*b^6))/(d*(a + 2*b*tan(c/2 + (d*x)/2) + 7*a*tan(c/2 + (d*x)
/2)^2 + 21*a*tan(c/2 + (d*x)/2)^4 + 35*a*tan(c/2 + (d*x)/2)^6 + 35*a*tan(c/2 + (d*x)/2)^8 + 21*a*tan(c/2 + (d*
x)/2)^10 + 7*a*tan(c/2 + (d*x)/2)^12 + a*tan(c/2 + (d*x)/2)^14 + 12*b*tan(c/2 + (d*x)/2)^3 + 30*b*tan(c/2 + (d
*x)/2)^5 + 40*b*tan(c/2 + (d*x)/2)^7 + 30*b*tan(c/2 + (d*x)/2)^9 + 12*b*tan(c/2 + (d*x)/2)^11 + 2*b*tan(c/2 +
(d*x)/2)^13)) - (atan((((((25*a^2*b^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b
^11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 - (((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 -
((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a
^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/
(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a
^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15
*b^7))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i)*1i)/(16*b^8) + ((((25*a^2*b^19)/8 - (225*a^4
*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 + (((10*a*
b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512
*a^3*b^23))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(1024*a
^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2
*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*
a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i -
a^4*b^2*200i)*1i)/(16*b^8))/((10976*a^19 + (135*a^3*b^16)/2 - (7205*a^5*b^14)/4 + 15115*a^7*b^12 - (244853*a^9
*b^10)/4 + 138577*a^11*b^8 - 184965*a^13*b^6 + 144788*a^15*b^4 - 61544*a^17*b^2)/b^20 + (tan(c/2 + (d*x)/2)*(1
75616*a^20 - 100*a^2*b^18 + 4150*a^4*b^16 - 61000*a^6*b^14 + 399830*a^8*b^12 - 1393080*a^10*b^10 + 2831960*a^1
2*b^8 - 3480576*a^14*b^6 + 2551808*a^16*b^4 - 1028608*a^18*b^2))/(4*b^21) + ((((25*a^2*b^19)/8 - (225*a^4*b^17
)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 - (((10*a*b^22
- 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 - ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*
b^23))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(1024*a^2*b^
22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i
))/(16*b^8) + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b
^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b
^2*200i))/(16*b^8) - ((((25*a^2*b^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^1
1 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 + (((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + ((
32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4
*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8
*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8) + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3
*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b
^7))/(8*b^21))*(a^6*112i - b^6*5i + a^2*b^4*90i - a^4*b^2*200i))/(16*b^8)))*(a^6*112i - b^6*5i + a^2*b^4*90i -
a^4*b^2*200i)*1i)/(8*b^8*d) - (a*atan(((a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(((25*a^2*b^19)/8 - (2
25*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 + (t
an(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b^13 - 254720*a^1
1*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21) + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((10*a*
b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 +
8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21) + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(32*a^2*b^3 + (tan
(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8)*1i)/b^8 + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a
- b)^3)^(1/2)*(((25*a^2*b^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 560
0*a^12*b^9 + 1568*a^14*b^7)/b^20 + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^
7*b^15 + 254720*a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21) - (a*(7*a^2 - 2*b^2)
*(-(a + b)^3*(a - b)^3)^(1/2)*((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + (tan(c/2 + (d*x
)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21) - (a*(7*a^2 - 2*b^2)*(-(a + b)^
3*(a - b)^3)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8)*1i)/b^
8)/((10976*a^19 + (135*a^3*b^16)/2 - (7205*a^5*b^14)/4 + 15115*a^7*b^12 - (244853*a^9*b^10)/4 + 138577*a^11*b^
8 - 184965*a^13*b^6 + 144788*a^15*b^4 - 61544*a^17*b^2)/b^20 + (tan(c/2 + (d*x)/2)*(175616*a^20 - 100*a^2*b^18
+ 4150*a^4*b^16 - 61000*a^6*b^14 + 399830*a^8*b^12 - 1393080*a^10*b^10 + 2831960*a^12*b^8 - 3480576*a^14*b^6
+ 2551808*a^16*b^4 - 1028608*a^18*b^2))/(4*b^21) - (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(((25*a^2*b
^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^11 - 5600*a^12*b^9 + 1568*a^14*b^7
)/b^20 + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 131700*a^7*b^15 + 254720*a^9*b^13 -
254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21) + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1
/2)*((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + (tan(c/2 + (d*x)/2)*(1024*a^2*b^22 - 5632
*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21) + (a*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(32*a^2
*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8))/b^8 + (a*(7*a^2 - 2*b^2)*(-(a +
b)^3*(a - b)^3)^(1/2)*(((25*a^2*b^19)/8 - (225*a^4*b^17)/2 + (2525*a^6*b^15)/2 - 4640*a^8*b^13 + 7520*a^10*b^
11 - 5600*a^12*b^9 + 1568*a^14*b^7)/b^20 + (tan(c/2 + (d*x)/2)*(50*a*b^21 - 2849*a^3*b^19 + 32364*a^5*b^17 - 1
31700*a^7*b^15 + 254720*a^9*b^13 - 254720*a^11*b^11 + 127232*a^13*b^9 - 25088*a^15*b^7))/(8*b^21) - (a*(7*a^2
- 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((10*a*b^22 - 126*a^3*b^20 + 228*a^5*b^18 - 112*a^7*b^16)/b^20 + (tan(c/
2 + (d*x)/2)*(1024*a^2*b^22 - 5632*a^4*b^20 + 8192*a^6*b^18 - 3584*a^8*b^16))/(8*b^21) - (a*(7*a^2 - 2*b^2)*(-
(a + b)^3*(a - b)^3)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^25 - 512*a^3*b^23))/(8*b^21)))/b^8))/b^8
))/b^8))*(7*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*2i)/(b^8*d)