\(\int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1136]

Optimal result

Integrand size = 29, antiderivative size = 284 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {a \left (9-\frac {20 a^2}{b^2}\right ) x}{2 b^4}+\frac {\left (20 a^4-19 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^6 \sqrt {a^2-b^2} d}-\frac {\left (60 a^2-17 b^2\right ) \cos (c+d x)}{6 b^5 d}+\frac {\left (5 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{a b^4 d}-\frac {\left (20 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{6 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))} \]

[Out]

1/2*a*(9-20*a^2/b^2)*x/b^4-1/6*(60*a^2-17*b^2)*cos(d*x+c)/b^5/d+(5*a^2-b^2)*cos(d*x+c)*sin(d*x+c)/a/b^4/d-1/6*
(20*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)^2/a^2/b^3/d-1/2*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^3/a/b^2/d/(a+b*sin(d*x+c)
)^2+1/2*(6*a^2-b^2)*cos(d*x+c)*sin(d*x+c)^3/a^2/b^2/d/(a+b*sin(d*x+c))+(20*a^4-19*a^2*b^2+2*b^4)*arctan((b+a*t
an(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^6/d/(a^2-b^2)^(1/2)

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2970, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac {\left (60 a^2-17 b^2\right ) \cos (c+d x)}{6 b^5 d}+\frac {\left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{a b^4 d}+\frac {a x \left (9-\frac {20 a^2}{b^2}\right )}{2 b^4}-\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{6 a^2 b^3 d}+\frac {\left (20 a^4-19 a^2 b^2+2 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^6 d \sqrt {a^2-b^2}} \]

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]

[Out]

(a*(9 - (20*a^2)/b^2)*x)/(2*b^4) + ((20*a^4 - 19*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b
^2]])/(b^6*Sqrt[a^2 - b^2]*d) - ((60*a^2 - 17*b^2)*Cos[c + d*x])/(6*b^5*d) + ((5*a^2 - b^2)*Cos[c + d*x]*Sin[c
+ d*x])/(a*b^4*d) - ((20*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(6*a^2*b^3*d) - ((a^2 - b^2)*Cos[c + d*x]*
Sin[c + d*x]^3)/(2*a*b^2*d*(a + b*Sin[c + d*x])^2) + ((6*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*a^2*b^2*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 2970

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f
*(m + 1))), x] + (-Dist[1/(a^2*b^2*(m + 1)*(m + 2)), Int[(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^n*Simp[
a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m +
n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] + Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b
*Sin[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2))), x]) /; FreeQ[{a, b, d, e, f,
n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && (LtQ[m, -2] || EqQ[m + n +
4, 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 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 {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\int \frac {\sin ^2(c+d x) \left (15 a^2-2 b^2-a b \sin (c+d x)-\left (20 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^2 b^2} \\ & = -\frac {\left (20 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{6 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\int \frac {\sin (c+d x) \left (-2 a \left (20 a^2-3 b^2\right )+5 a^2 b \sin (c+d x)+12 a \left (5 a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^2 b^3} \\ & = \frac {\left (5 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{a b^4 d}-\frac {\left (20 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{6 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\int \frac {12 a^2 \left (5 a^2-b^2\right )-20 a^3 b \sin (c+d x)-2 a^2 \left (60 a^2-17 b^2\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{12 a^2 b^4} \\ & = -\frac {\left (60 a^2-17 b^2\right ) \cos (c+d x)}{6 b^5 d}+\frac {\left (5 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{a b^4 d}-\frac {\left (20 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{6 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\int \frac {12 a^2 b \left (5 a^2-b^2\right )+6 a^3 \left (20 a^2-9 b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{12 a^2 b^5} \\ & = -\frac {a \left (20 a^2-9 b^2\right ) x}{2 b^6}-\frac {\left (60 a^2-17 b^2\right ) \cos (c+d x)}{6 b^5 d}+\frac {\left (5 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{a b^4 d}-\frac {\left (20 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{6 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}+\frac {\left (6 a^4 \left (20 a^2-9 b^2\right )-12 a^2 b^2 \left (5 a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{12 a^2 b^6} \\ & = -\frac {a \left (20 a^2-9 b^2\right ) x}{2 b^6}-\frac {\left (60 a^2-17 b^2\right ) \cos (c+d x)}{6 b^5 d}+\frac {\left (5 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{a b^4 d}-\frac {\left (20 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{6 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}+\frac {\left (20 a^4-19 a^2 b^2+2 b^4\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^6 d} \\ & = -\frac {a \left (20 a^2-9 b^2\right ) x}{2 b^6}-\frac {\left (60 a^2-17 b^2\right ) \cos (c+d x)}{6 b^5 d}+\frac {\left (5 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{a b^4 d}-\frac {\left (20 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{6 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (2 \left (20 a^4-19 a^2 b^2+2 b^4\right )\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^6 d} \\ & = -\frac {a \left (20 a^2-9 b^2\right ) x}{2 b^6}+\frac {\left (20 a^4-19 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^6 \sqrt {a^2-b^2} d}-\frac {\left (60 a^2-17 b^2\right ) \cos (c+d x)}{6 b^5 d}+\frac {\left (5 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{a b^4 d}-\frac {\left (20 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{6 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1030\) vs. \(2(284)=568\).

Time = 5.15 (sec) , antiderivative size = 1030, normalized size of antiderivative = 3.63 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-\frac {12 \left (-48 a (c+d x)+\frac {6 \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-16 b \cos (c+d x)+\frac {b \left (8 a^4-8 a^2 b^2+b^4\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}+\frac {a b \left (-40 a^4+72 a^2 b^2-29 b^4\right ) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}\right )}{b^4}+12 \left (\frac {2 \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {b \cos (c+d x) \left (4 a^2-b^2+3 a b \sin (c+d x)\right )}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))^2}\right )+\frac {6 \left (-\frac {6 b^2 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {\cos (c+d x) \left (-b \left (2 a^2+b^2\right )+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}\right )}{(a-b)^2 (a+b)^2}-\frac {-\frac {12 \left (640 a^8-1792 a^6 b^2+1680 a^4 b^4-560 a^2 b^6+35 b^8\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {3840 a^9 (c+d x)-6912 a^7 b^2 (c+d x)+1728 a^5 b^4 (c+d x)+1920 a^3 b^6 (c+d x)-576 a b^8 (c+d x)+3840 a^8 b \cos (c+d x)-7872 a^6 b^3 \cos (c+d x)+4256 a^4 b^5 \cos (c+d x)-172 a^2 b^7 \cos (c+d x)-70 b^9 \cos (c+d x)-1920 a^7 b^2 (c+d x) \cos (2 (c+d x))+4416 a^5 b^4 (c+d x) \cos (2 (c+d x))-3072 a^3 b^6 (c+d x) \cos (2 (c+d x))+576 a b^8 (c+d x) \cos (2 (c+d x))-320 a^6 b^3 \cos (3 (c+d x))+696 a^4 b^5 \cos (3 (c+d x))-432 a^2 b^7 \cos (3 (c+d x))+56 b^9 \cos (3 (c+d x))+8 a^4 b^5 \cos (5 (c+d x))-16 a^2 b^7 \cos (5 (c+d x))+8 b^9 \cos (5 (c+d x))+7680 a^8 b (c+d x) \sin (c+d x)-17664 a^6 b^3 (c+d x) \sin (c+d x)+12288 a^4 b^5 (c+d x) \sin (c+d x)-2304 a^2 b^7 (c+d x) \sin (c+d x)+2880 a^7 b^2 \sin (2 (c+d x))-6304 a^5 b^4 \sin (2 (c+d x))+4022 a^3 b^6 \sin (2 (c+d x))-607 a b^8 \sin (2 (c+d x))+40 a^5 b^4 \sin (4 (c+d x))-80 a^3 b^6 \sin (4 (c+d x))+40 a b^8 \sin (4 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}}{b^6}}{384 d} \]

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]

[Out]

((-12*(-48*a*(c + d*x) + (6*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^
2 - b^2]])/(a^2 - b^2)^(5/2) - 16*b*Cos[c + d*x] + (b*(8*a^4 - 8*a^2*b^2 + b^4)*Cos[c + d*x])/((a - b)*(a + b)
*(a + b*Sin[c + d*x])^2) + (a*b*(-40*a^4 + 72*a^2*b^2 - 29*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[
c + d*x]))))/b^4 + 12*((2*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) +
(b*Cos[c + d*x]*(4*a^2 - b^2 + 3*a*b*Sin[c + d*x]))/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^2)) + (6*((-6*b^
2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(-(b*(2*a^2 + b^2)) + a*(2
*a^2 - 5*b^2)*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2))/((a - b)^2*(a + b)^2) - ((-12*(640*a^8 - 1792*a^6*b^2 +
1680*a^4*b^4 - 560*a^2*b^6 + 35*b^8)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (38
40*a^9*(c + d*x) - 6912*a^7*b^2*(c + d*x) + 1728*a^5*b^4*(c + d*x) + 1920*a^3*b^6*(c + d*x) - 576*a*b^8*(c + d
*x) + 3840*a^8*b*Cos[c + d*x] - 7872*a^6*b^3*Cos[c + d*x] + 4256*a^4*b^5*Cos[c + d*x] - 172*a^2*b^7*Cos[c + d*
x] - 70*b^9*Cos[c + d*x] - 1920*a^7*b^2*(c + d*x)*Cos[2*(c + d*x)] + 4416*a^5*b^4*(c + d*x)*Cos[2*(c + d*x)] -
3072*a^3*b^6*(c + d*x)*Cos[2*(c + d*x)] + 576*a*b^8*(c + d*x)*Cos[2*(c + d*x)] - 320*a^6*b^3*Cos[3*(c + d*x)]
+ 696*a^4*b^5*Cos[3*(c + d*x)] - 432*a^2*b^7*Cos[3*(c + d*x)] + 56*b^9*Cos[3*(c + d*x)] + 8*a^4*b^5*Cos[5*(c
+ d*x)] - 16*a^2*b^7*Cos[5*(c + d*x)] + 8*b^9*Cos[5*(c + d*x)] + 7680*a^8*b*(c + d*x)*Sin[c + d*x] - 17664*a^6
*b^3*(c + d*x)*Sin[c + d*x] + 12288*a^4*b^5*(c + d*x)*Sin[c + d*x] - 2304*a^2*b^7*(c + d*x)*Sin[c + d*x] + 288
0*a^7*b^2*Sin[2*(c + d*x)] - 6304*a^5*b^4*Sin[2*(c + d*x)] + 4022*a^3*b^6*Sin[2*(c + d*x)] - 607*a*b^8*Sin[2*(
c + d*x)] + 40*a^5*b^4*Sin[4*(c + d*x)] - 80*a^3*b^6*Sin[4*(c + d*x)] + 40*a*b^8*Sin[4*(c + d*x)])/((a^2 - b^2
)^2*(a + b*Sin[c + d*x])^2))/b^6)/(384*d)

Maple [A] (verified)

Time = 2.34 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {\frac {\frac {4 \left (-\frac {a \,b^{2} \left (7 a^{2}-2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {b \left (8 a^{4}+13 a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {5 b^{2} a \left (5 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-2 a^{4} b +\frac {3 a^{2} b^{3}}{4}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (20 a^{4}-19 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{b^{6}}-\frac {4 \left (\frac {\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{4}+\left (3 a^{2} b -b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{2} b -b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{4}+3 a^{2} b -\frac {2 b^{3}}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {a \left (20 a^{2}-9 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}\right )}{b^{6}}}{d}\)	\(344\)
default	\(\frac {\frac {\frac {4 \left (-\frac {a \,b^{2} \left (7 a^{2}-2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {b \left (8 a^{4}+13 a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {5 b^{2} a \left (5 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-2 a^{4} b +\frac {3 a^{2} b^{3}}{4}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (20 a^{4}-19 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{b^{6}}-\frac {4 \left (\frac {\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{4}+\left (3 a^{2} b -b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{2} b -b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{4}+3 a^{2} b -\frac {2 b^{3}}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {a \left (20 a^{2}-9 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}\right )}{b^{6}}}{d}\)	\(344\)
risch	\(-\frac {10 x \,a^{3}}{b^{6}}+\frac {9 a x}{2 b^{4}}+\frac {{\mathrm e}^{3 i \left (d x +c \right )}}{24 b^{3} d}+\frac {i a \left (-10 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+5 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+26 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-11 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+18 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{2} b^{2}+4 b^{4}\right )}{\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 )^{2} d \,b^{6}}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{b^{5} d}+\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{8 b^{3} d}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{b^{5} d}+\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{3} d}-\frac {3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{4} d}+\frac {{\mathrm e}^{-3 i \left (d x +c \right )}}{24 b^{3} d}+\frac {3 i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{4} d}+\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{4}}{\sqrt {-a^{2}+b^{2}}\, d \,b^{6}}-\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{2}}{2 \sqrt {-a^{2}+b^{2}}\, d \,b^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}-\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{4}}{\sqrt {-a^{2}+b^{2}}\, d \,b^{6}}+\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{2}}{2 \sqrt {-a^{2}+b^{2}}\, d \,b^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}\)	\(753\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(4/b^6*((-1/4*a*b^2*(7*a^2-2*b^2)*tan(1/2*d*x+1/2*c)^3-1/4*b*(8*a^4+13*a^2*b^2-6*b^4)*tan(1/2*d*x+1/2*c)^2
-5/4*b^2*a*(5*a^2-2*b^2)*tan(1/2*d*x+1/2*c)-2*a^4*b+3/4*a^2*b^3)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c
)+a)^2+1/4*(20*a^4-19*a^2*b^2+2*b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))
-4/b^6*((3/4*tan(1/2*d*x+1/2*c)^5*a*b^2+(3*a^2*b-b^3)*tan(1/2*d*x+1/2*c)^4+(6*a^2*b-b^3)*tan(1/2*d*x+1/2*c)^2-
3/4*tan(1/2*d*x+1/2*c)*a*b^2+3*a^2*b-2/3*b^3)/(1+tan(1/2*d*x+1/2*c)^2)^3+1/4*a*(20*a^2-9*b^2)*arctan(tan(1/2*d
*x+1/2*c))))

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 976, normalized size of antiderivative = 3.44 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/12*(4*(a^2*b^5 - b^7)*cos(d*x + c)^5 - 6*(20*a^5*b^2 - 29*a^3*b^4 + 9*a*b^6)*d*x*cos(d*x + c)^2 - 8*(5*a^4*
b^3 - 6*a^2*b^5 + b^7)*cos(d*x + c)^3 + 6*(20*a^7 - 9*a^5*b^2 - 20*a^3*b^4 + 9*a*b^6)*d*x + 3*(20*a^6 + a^4*b^
2 - 17*a^2*b^4 + 2*b^6 - (20*a^4*b^2 - 19*a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 2*(20*a^5*b - 19*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)) + 6*(20*a^6*b - 19*a^4*b^3 - 3*a^2*b^5 + 2*b^7)*cos(d*x + c) + 2*(5*(a^3*b^4 - a*b^6)*cos(d*x + c)^3 + 6*(
20*a^6*b - 29*a^4*b^3 + 9*a^2*b^5)*d*x + 3*(30*a^5*b^2 - 41*a^3*b^4 + 11*a*b^6)*cos(d*x + c))*sin(d*x + c))/((
a^2*b^8 - b^10)*d*cos(d*x + c)^2 - 2*(a^3*b^7 - a*b^9)*d*sin(d*x + c) - (a^4*b^6 - b^10)*d), 1/6*(2*(a^2*b^5 -
b^7)*cos(d*x + c)^5 - 3*(20*a^5*b^2 - 29*a^3*b^4 + 9*a*b^6)*d*x*cos(d*x + c)^2 - 4*(5*a^4*b^3 - 6*a^2*b^5 + b
^7)*cos(d*x + c)^3 + 3*(20*a^7 - 9*a^5*b^2 - 20*a^3*b^4 + 9*a*b^6)*d*x + 3*(20*a^6 + a^4*b^2 - 17*a^2*b^4 + 2*
b^6 - (20*a^4*b^2 - 19*a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 2*(20*a^5*b - 19*a^3*b^3 + 2*a*b^5)*sin(d*x + c))*sqr
t(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 3*(20*a^6*b - 19*a^4*b^3 - 3*a^2*b
^5 + 2*b^7)*cos(d*x + c) + (5*(a^3*b^4 - a*b^6)*cos(d*x + c)^3 + 6*(20*a^6*b - 29*a^4*b^3 + 9*a^2*b^5)*d*x + 3
*(30*a^5*b^2 - 41*a^3*b^4 + 11*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^2*b^8 - b^10)*d*cos(d*x + c)^2 - 2*(a^3*
b^7 - a*b^9)*d*sin(d*x + c) - (a^4*b^6 - b^10)*d)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)**2/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.38 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {3 \, {\left (20 \, a^{3} - 9 \, a b^{2}\right )} {\left (d x + c\right )}}{b^{6}} - \frac {6 \, {\left (20 \, a^{4} - 19 \, a^{2} b^{2} + 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{6}} + \frac {6 \, {\left (7 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 13 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 25 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 10 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a^{4} - 3 \, a^{2} b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} b^{5}} + \frac {2 \, {\left (9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 72 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} - 8 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{5}}}{6 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/6*(3*(20*a^3 - 9*a*b^2)*(d*x + c)/b^6 - 6*(20*a^4 - 19*a^2*b^2 + 2*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*s
gn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^6) + 6*(7*a^3*b*tan(1/2*d*x +
1/2*c)^3 - 2*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 8*a^4*tan(1/2*d*x + 1/2*c)^2 + 13*a^2*b^2*tan(1/2*d*x + 1/2*c)^2
- 6*b^4*tan(1/2*d*x + 1/2*c)^2 + 25*a^3*b*tan(1/2*d*x + 1/2*c) - 10*a*b^3*tan(1/2*d*x + 1/2*c) + 8*a^4 - 3*a^2
*b^2)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*b^5) + 2*(9*a*b*tan(1/2*d*x + 1/2*c)^5 + 36
*a^2*tan(1/2*d*x + 1/2*c)^4 - 12*b^2*tan(1/2*d*x + 1/2*c)^4 + 72*a^2*tan(1/2*d*x + 1/2*c)^2 - 12*b^2*tan(1/2*d
*x + 1/2*c)^2 - 9*a*b*tan(1/2*d*x + 1/2*c) + 36*a^2 - 8*b^2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*b^5))/d

Mupad [B] (verification not implemented)

Time = 16.86 (sec) , antiderivative size = 2034, normalized size of antiderivative = 7.16 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int((cos(c + d*x)^4*sin(c + d*x)^2)/(a + b*sin(c + d*x))^3,x)

[Out]

- ((60*a^4 - 17*a^2*b^2)/(3*b^5) - (20*tan(c/2 + (d*x)/2)^7*(a*b^2 - 5*a^3))/b^4 - (2*tan(c/2 + (d*x)/2)^9*(a*
b^2 - 5*a^3))/b^4 - (4*tan(c/2 + (d*x)/2)^5*(17*a*b^2 - 60*a^3))/b^4 - (4*tan(c/2 + (d*x)/2)^3*(53*a*b^2 - 165
*a^3))/(3*b^4) + (tan(c/2 + (d*x)/2)^8*(20*a^4 - 6*b^4 + 21*a^2*b^2))/b^5 + (2*tan(c/2 + (d*x)/2)^6*(40*a^4 -
17*b^4 + 42*a^2*b^2))/b^5 + (2*tan(c/2 + (d*x)/2)^2*(120*a^4 - 25*b^4 + 46*a^2*b^2))/(3*b^5) + (2*tan(c/2 + (d
*x)/2)^4*(180*a^4 - 51*b^4 + 149*a^2*b^2))/(3*b^5) - (2*tan(c/2 + (d*x)/2)*(31*a*b^2 - 105*a^3))/(3*b^4))/(d*(
tan(c/2 + (d*x)/2)^2*(5*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^8*(5*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^4*(10*a^2 + 1
2*b^2) + tan(c/2 + (d*x)/2)^6*(10*a^2 + 12*b^2) + a^2*tan(c/2 + (d*x)/2)^10 + a^2 + 16*a*b*tan(c/2 + (d*x)/2)^
3 + 24*a*b*tan(c/2 + (d*x)/2)^5 + 16*a*b*tan(c/2 + (d*x)/2)^7 + 4*a*b*tan(c/2 + (d*x)/2)^9 + 4*a*b*tan(c/2 + (
d*x)/2))) - (a*atan((280*a^4*tan(c/2 + (d*x)/2))/(280*a^4 - 288*a^2*b^2 + (800*a^6)/b^2) - (288*a^2*tan(c/2 +
(d*x)/2))/((280*a^4)/b^2 - 288*a^2 + (800*a^6)/b^4) + (800*a^6*tan(c/2 + (d*x)/2))/(800*a^6 - 288*a^2*b^4 + 28
0*a^4*b^2))*(20*a^2 - 9*b^2))/(b^6*d) - (atan((((-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^2*b^2)/2)*((8*(
81*a^4*b^9 - 360*a^6*b^7 + 400*a^8*b^5))/b^14 - (8*tan(c/2 + (d*x)/2)*(4*a*b^13 - 238*a^3*b^11 + 1242*a^5*b^9
- 1920*a^7*b^7 + 800*a^9*b^5))/b^15 + ((-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^2*b^2)/2)*((8*tan(c/2 +
(d*x)/2)*(8*a*b^16 - 76*a^3*b^14 + 80*a^5*b^12))/b^15 - (8*(14*a^2*b^14 - 20*a^4*b^12))/b^14 + ((-(a + b)*(a -
b))^(1/2)*(32*a^2*b^3 + (8*tan(c/2 + (d*x)/2)*(12*a*b^19 - 8*a^3*b^17))/b^15)*(10*a^4 + b^4 - (19*a^2*b^2)/2)
)/(b^8 - a^2*b^6)))/(b^8 - a^2*b^6))*1i)/(b^8 - a^2*b^6) + ((-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^2*b
^2)/2)*((8*(81*a^4*b^9 - 360*a^6*b^7 + 400*a^8*b^5))/b^14 - (8*tan(c/2 + (d*x)/2)*(4*a*b^13 - 238*a^3*b^11 + 1
242*a^5*b^9 - 1920*a^7*b^7 + 800*a^9*b^5))/b^15 + ((-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^2*b^2)/2)*((
8*(14*a^2*b^14 - 20*a^4*b^12))/b^14 - (8*tan(c/2 + (d*x)/2)*(8*a*b^16 - 76*a^3*b^14 + 80*a^5*b^12))/b^15 + ((-
(a + b)*(a - b))^(1/2)*(32*a^2*b^3 + (8*tan(c/2 + (d*x)/2)*(12*a*b^19 - 8*a^3*b^17))/b^15)*(10*a^4 + b^4 - (19
*a^2*b^2)/2))/(b^8 - a^2*b^6)))/(b^8 - a^2*b^6))*1i)/(b^8 - a^2*b^6))/((16*(2000*a^10 + 18*a^2*b^8 - 301*a^4*b
^6 + 1615*a^6*b^4 - 3200*a^8*b^2))/b^14 + (16*tan(c/2 + (d*x)/2)*(8000*a^11 + 162*a^3*b^8 - 2259*a^5*b^6 + 926
0*a^7*b^4 - 14800*a^9*b^2))/b^15 + ((-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^2*b^2)/2)*((8*(81*a^4*b^9 -
360*a^6*b^7 + 400*a^8*b^5))/b^14 - (8*tan(c/2 + (d*x)/2)*(4*a*b^13 - 238*a^3*b^11 + 1242*a^5*b^9 - 1920*a^7*b
^7 + 800*a^9*b^5))/b^15 + ((-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^2*b^2)/2)*((8*tan(c/2 + (d*x)/2)*(8*
a*b^16 - 76*a^3*b^14 + 80*a^5*b^12))/b^15 - (8*(14*a^2*b^14 - 20*a^4*b^12))/b^14 + ((-(a + b)*(a - b))^(1/2)*(
32*a^2*b^3 + (8*tan(c/2 + (d*x)/2)*(12*a*b^19 - 8*a^3*b^17))/b^15)*(10*a^4 + b^4 - (19*a^2*b^2)/2))/(b^8 - a^2
*b^6)))/(b^8 - a^2*b^6)))/(b^8 - a^2*b^6) - ((-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^2*b^2)/2)*((8*(81*
a^4*b^9 - 360*a^6*b^7 + 400*a^8*b^5))/b^14 - (8*tan(c/2 + (d*x)/2)*(4*a*b^13 - 238*a^3*b^11 + 1242*a^5*b^9 - 1
920*a^7*b^7 + 800*a^9*b^5))/b^15 + ((-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^2*b^2)/2)*((8*(14*a^2*b^14
- 20*a^4*b^12))/b^14 - (8*tan(c/2 + (d*x)/2)*(8*a*b^16 - 76*a^3*b^14 + 80*a^5*b^12))/b^15 + ((-(a + b)*(a - b)
)^(1/2)*(32*a^2*b^3 + (8*tan(c/2 + (d*x)/2)*(12*a*b^19 - 8*a^3*b^17))/b^15)*(10*a^4 + b^4 - (19*a^2*b^2)/2))/(
b^8 - a^2*b^6)))/(b^8 - a^2*b^6)))/(b^8 - a^2*b^6)))*(-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^2*b^2)/2)*
2i)/(d*(b^8 - a^2*b^6))