\(\int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1085]

Optimal result

Integrand size = 29, antiderivative size = 266 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\left (12 a^2-b^2\right ) x}{2 b^5}+\frac {a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{3/2} d}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \]

[Out]

-1/2*(12*a^2-b^2)*x/b^5+a*(12*a^4-19*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^5/(a^2-
b^2)^(3/2)/d-1/2*a*(12*a^2-11*b^2)*cos(d*x+c)/b^4/(a^2-b^2)/d+1/2*(6*a^2-5*b^2)*cos(d*x+c)*sin(d*x+c)/b^3/(a^2
-b^2)/d-1/2*cos(d*x+c)*sin(d*x+c)^3/b/d/(a+b*sin(d*x+c))^2-1/2*(4*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)^2/b^2/(a^2-
b^2)/d/(a+b*sin(d*x+c))

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2968, 3127, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {x \left (12 a^2-b^2\right )}{2 b^5}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 d \left (a^2-b^2\right )}+\frac {\left (6 a^2-5 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left (a^2-b^2\right )}+\frac {a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d \left (a^2-b^2\right )^{3/2}}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2} \]

[In]

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

[Out]

-1/2*((12*a^2 - b^2)*x)/b^5 + (a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]
])/(b^5*(a^2 - b^2)^(3/2)*d) - (a*(12*a^2 - 11*b^2)*Cos[c + d*x])/(2*b^4*(a^2 - b^2)*d) + ((6*a^2 - 5*b^2)*Cos
[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(2*b*d*(a + b*Sin[c + d*x])^2) -
((4*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(2*b^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 2968

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f,
m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])

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 3127

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Si
n[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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n
+ 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(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, 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}& = \int \frac {\sin ^3(c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx \\ & = -\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\int \frac {\sin ^2(c+d x) \left (-3 \left (a^2-b^2\right )+4 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )} \\ & = -\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\sin (c+d x) \left (2 \left (4 a^4-7 a^2 b^2+3 b^4\right )-a b \left (a^2-b^2\right ) \sin (c+d x)-2 \left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {-2 a \left (6 a^4-11 a^2 b^2+5 b^4\right )+2 b \left (2 a^4-3 a^2 b^2+b^4\right ) \sin (c+d x)+2 a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2} \\ & = -\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {-2 a b \left (6 a^4-11 a^2 b^2+5 b^4\right )-2 \left (a^2-b^2\right )^2 \left (12 a^2-b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (12 a^2-b^2\right ) x}{2 b^5}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\left (a \left (12 a^4-19 a^2 b^2+6 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )} \\ & = -\frac {\left (12 a^2-b^2\right ) x}{2 b^5}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\left (a \left (12 a^4-19 a^2 b^2+6 b^4\right )\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^5 \left (a^2-b^2\right ) d} \\ & = -\frac {\left (12 a^2-b^2\right ) x}{2 b^5}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {\left (2 a \left (12 a^4-19 a^2 b^2+6 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^5 \left (a^2-b^2\right ) d} \\ & = -\frac {\left (12 a^2-b^2\right ) x}{2 b^5}+\frac {a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{3/2} d}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.44 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {4 a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {-4 a b \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x) \sin (c+d x)-2 b^2 \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x) \sin ^2(c+d x)+\cos (c+d x) \left (-24 a^5 b+22 a^3 b^3-8 a b^3 \left (a^2-b^2\right ) \sin ^2(c+d x)+2 b^4 \left (a^2-b^2\right ) \sin ^3(c+d x)\right )-a^2 \left (2 \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x)+\left (18 a^2 b^2-17 b^4\right ) \sin (2 (c+d x))\right )}{(a+b \sin (c+d x))^2}}{4 (a-b) b^5 (a+b) d} \]

[In]

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

[Out]

((4*a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (-4*a*
b*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Sin[c + d*x] - 2*b^2*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Sin[c + d*x
]^2 + Cos[c + d*x]*(-24*a^5*b + 22*a^3*b^3 - 8*a*b^3*(a^2 - b^2)*Sin[c + d*x]^2 + 2*b^4*(a^2 - b^2)*Sin[c + d*
x]^3) - a^2*(2*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x) + (18*a^2*b^2 - 17*b^4)*Sin[2*(c + d*x)]))/(a + b*Sin[c +
d*x])^2)/(4*(a - b)*b^5*(a + b)*d)

Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2}+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+3 a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (12 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}+\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {b \left (6 a^{4}+7 a^{2} b^{2}-10 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a \,b^{2} \left (19 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a^{2} b \left (6 a^{2}-5 b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (12 a^{4}-19 a^{2} b^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{5}}}{d}\)	\(349\)
default	\(\frac {-\frac {2 \left (\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2}+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+3 a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (12 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}+\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {b \left (6 a^{4}+7 a^{2} b^{2}-10 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a \,b^{2} \left (19 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a^{2} b \left (6 a^{2}-5 b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (12 a^{4}-19 a^{2} b^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{5}}}{d}\)	\(349\)
risch	\(-\frac {6 x \,a^{2}}{b^{5}}+\frac {x}{2 b^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{3} d}-\frac {3 a \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{4} d}-\frac {3 a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {i a^{2} \left (-8 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+7 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+20 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-17 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+14 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-5 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-7 a^{2} b^{2}+6 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} \left (a^{2}-b^{2}\right ) d \,b^{5}}+\frac {6 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{5}}-\frac {19 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}-\frac {6 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{5}}+\frac {19 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}-\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}\)	\(773\)

[In]

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

[Out]

1/d*(-2/b^5*((1/2*tan(1/2*d*x+1/2*c)^3*b^2+3*tan(1/2*d*x+1/2*c)^2*a*b-1/2*tan(1/2*d*x+1/2*c)*b^2+3*a*b)/(1+tan
(1/2*d*x+1/2*c)^2)^2+1/2*(12*a^2-b^2)*arctan(tan(1/2*d*x+1/2*c)))+2*a/b^5*((-1/2*a*b^2*(5*a^2-4*b^2)/(a^2-b^2)
*tan(1/2*d*x+1/2*c)^3-1/2*b*(6*a^4+7*a^2*b^2-10*b^4)/(a^2-b^2)*tan(1/2*d*x+1/2*c)^2-1/2*a*b^2*(19*a^2-16*b^2)/
(a^2-b^2)*tan(1/2*d*x+1/2*c)-1/2*a^2*b*(6*a^2-5*b^2)/(a^2-b^2))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)
+a)^2+1/2*(12*a^4-19*a^2*b^2+6*b^4)/(a^2-b^2)^(3/2)*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 = 0.36 (sec) , antiderivative size = 1058, normalized size of antiderivative = 3.98 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/4*(2*(12*a^6*b^2 - 25*a^4*b^4 + 14*a^2*b^6 - b^8)*d*x*cos(d*x + c)^2 + 8*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos
(d*x + c)^3 - 2*(12*a^8 - 13*a^6*b^2 - 11*a^4*b^4 + 13*a^2*b^6 - b^8)*d*x + (12*a^7 - 7*a^5*b^2 - 13*a^3*b^4 +
6*a*b^6 - (12*a^5*b^2 - 19*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^2 + 2*(12*a^6*b - 19*a^4*b^3 + 6*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)) - 2*(
12*a^7*b - 19*a^5*b^3 + 3*a^3*b^5 + 4*a*b^7)*cos(d*x + c) - 2*((a^4*b^4 - 2*a^2*b^6 + b^8)*cos(d*x + c)^3 + 2*
(12*a^7*b - 25*a^5*b^3 + 14*a^3*b^5 - a*b^7)*d*x + (18*a^6*b^2 - 36*a^4*b^4 + 19*a^2*b^6 - b^8)*cos(d*x + c))*
sin(d*x + c))/((a^4*b^7 - 2*a^2*b^9 + b^11)*d*cos(d*x + c)^2 - 2*(a^5*b^6 - 2*a^3*b^8 + a*b^10)*d*sin(d*x + c)
- (a^6*b^5 - a^4*b^7 - a^2*b^9 + b^11)*d), -1/2*((12*a^6*b^2 - 25*a^4*b^4 + 14*a^2*b^6 - b^8)*d*x*cos(d*x + c
)^2 + 4*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c)^3 - (12*a^8 - 13*a^6*b^2 - 11*a^4*b^4 + 13*a^2*b^6 - b^8)*d
*x - (12*a^7 - 7*a^5*b^2 - 13*a^3*b^4 + 6*a*b^6 - (12*a^5*b^2 - 19*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^2 + 2*(12*a
^6*b - 19*a^4*b^3 + 6*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))) - (12*a^7*b - 19*a^5*b^3 + 3*a^3*b^5 + 4*a*b^7)*cos(d*x + c) - ((a^4*b^4 - 2*a^2*b^6 + b^8)*cos(d*
x + c)^3 + 2*(12*a^7*b - 25*a^5*b^3 + 14*a^3*b^5 - a*b^7)*d*x + (18*a^6*b^2 - 36*a^4*b^4 + 19*a^2*b^6 - b^8)*c
os(d*x + c))*sin(d*x + c))/((a^4*b^7 - 2*a^2*b^9 + b^11)*d*cos(d*x + c)^2 - 2*(a^5*b^6 - 2*a^3*b^8 + a*b^10)*d
*sin(d*x + c) - (a^6*b^5 - a^4*b^7 - a^2*b^9 + b^11)*d)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^2*sin(d*x+c)^3/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (251) = 502\).

Time = 0.37 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.01 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (12 \, a^{5} - 19 \, a^{3} b^{2} + 6 \, a 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 )}}{{\left (a^{2} b^{5} - b^{7}\right )} \sqrt {a^{2} - b^{2}}} - \frac {2 \, {\left (6 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 14 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 54 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 15 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 44 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 87 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 42 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 39 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, a^{5} - 11 \, a^{3} b^{2}\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, 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}} - \frac {{\left (12 \, a^{2} - b^{2}\right )} {\left (d x + c\right )}}{b^{5}}}{2 \, d} \]

[In]

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

[Out]

1/2*(2*(12*a^5 - 19*a^3*b^2 + 6*a*b^4)*(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)))/((a^2*b^5 - b^7)*sqrt(a^2 - b^2)) - 2*(6*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 5*a^2*b^3*ta
n(1/2*d*x + 1/2*c)^7 + 12*a^5*tan(1/2*d*x + 1/2*c)^6 + 5*a^3*b^2*tan(1/2*d*x + 1/2*c)^6 - 14*a*b^4*tan(1/2*d*x
+ 1/2*c)^6 + 54*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 45*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 4*b^5*tan(1/2*d*x + 1/2*c)
^5 + 36*a^5*tan(1/2*d*x + 1/2*c)^4 + 15*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 - 44*a*b^4*tan(1/2*d*x + 1/2*c)^4 + 90*
a^4*b*tan(1/2*d*x + 1/2*c)^3 - 87*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 4*b^5*tan(1/2*d*x + 1/2*c)^3 + 36*a^5*tan(1
/2*d*x + 1/2*c)^2 - a^3*b^2*tan(1/2*d*x + 1/2*c)^2 - 30*a*b^4*tan(1/2*d*x + 1/2*c)^2 + 42*a^4*b*tan(1/2*d*x +
1/2*c) - 39*a^2*b^3*tan(1/2*d*x + 1/2*c) + 12*a^5 - 11*a^3*b^2)/((a^2*b^4 - b^6)*(a*tan(1/2*d*x + 1/2*c)^4 + 2
*b*tan(1/2*d*x + 1/2*c)^3 + 2*a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2) - (12*a^2 - b^2)*(d*
x + c)/b^5)/d

Mupad [B] (verification not implemented)

Time = 20.07 (sec) , antiderivative size = 4943, normalized size of antiderivative = 18.58 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- ((12*a^5 - 11*a^3*b^2)/(b^4*(a^2 - b^2)) + (tan(c/2 + (d*x)/2)^7*(6*a^4 - 5*a^2*b^2))/(b^3*(a^2 - b^2)) - (t
an(c/2 + (d*x)/2)^5*(4*b^4 - 54*a^4 + 45*a^2*b^2))/(b^3*(a^2 - b^2)) + (tan(c/2 + (d*x)/2)^3*(90*a^4 + 4*b^4 -
87*a^2*b^2))/(b^3*(a^2 - b^2)) + (tan(c/2 + (d*x)/2)^6*(12*a^5 - 14*a*b^4 + 5*a^3*b^2))/(b^4*(a^2 - b^2)) - (
tan(c/2 + (d*x)/2)^2*(30*a*b^4 - 36*a^5 + a^3*b^2))/(b^4*(a^2 - b^2)) + (3*tan(c/2 + (d*x)/2)*(14*a^4 - 13*a^2
*b^2))/(b^3*(a^2 - b^2)) - (tan(c/2 + (d*x)/2)^4*(11*a*b^2 - 12*a^3)*(3*a^2 + 4*b^2))/(b^4*(a^2 - b^2)))/(d*(t
an(c/2 + (d*x)/2)^2*(4*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^6*(4*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^4*(6*a^2 + 8*b
^2) + a^2*tan(c/2 + (d*x)/2)^8 + a^2 + 12*a*b*tan(c/2 + (d*x)/2)^3 + 12*a*b*tan(c/2 + (d*x)/2)^5 + 4*a*b*tan(c
/2 + (d*x)/2)^7 + 4*a*b*tan(c/2 + (d*x)/2))) - (atan((((a^2*12i - b^2*1i)*((4*(2*a^2*b^12 - 52*a^4*b^10 + 386*
a^6*b^8 - 624*a^8*b^6 + 288*a^10*b^4))/(b^15 - 2*a^2*b^13 + a^4*b^11) - ((a^2*12i - b^2*1i)*((4*(4*a*b^16 - 36
*a^3*b^14 + 56*a^5*b^12 - 24*a^7*b^10))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(24*a^2*b^16 -
100*a^4*b^14 + 124*a^6*b^12 - 48*a^8*b^10))/(b^16 - 2*a^2*b^14 + a^4*b^12) - ((a^2*12i - b^2*1i)*((4*(8*a^2*b^
18 - 16*a^4*b^16 + 8*a^6*b^14))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(12*a*b^20 - 32*a^3*b^1
8 + 28*a^5*b^16 - 8*a^7*b^14))/(b^16 - 2*a^2*b^14 + a^4*b^12)))/(2*b^5)))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(2*a
*b^14 - 89*a^3*b^12 + 640*a^5*b^10 - 1322*a^7*b^8 + 1056*a^9*b^6 - 288*a^11*b^4))/(b^16 - 2*a^2*b^14 + a^4*b^1
2))*1i)/(2*b^5) + ((a^2*12i - b^2*1i)*((4*(2*a^2*b^12 - 52*a^4*b^10 + 386*a^6*b^8 - 624*a^8*b^6 + 288*a^10*b^4
))/(b^15 - 2*a^2*b^13 + a^4*b^11) + ((a^2*12i - b^2*1i)*((4*(4*a*b^16 - 36*a^3*b^14 + 56*a^5*b^12 - 24*a^7*b^1
0))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(24*a^2*b^16 - 100*a^4*b^14 + 124*a^6*b^12 - 48*a^8
*b^10))/(b^16 - 2*a^2*b^14 + a^4*b^12) + ((a^2*12i - b^2*1i)*((4*(8*a^2*b^18 - 16*a^4*b^16 + 8*a^6*b^14))/(b^1
5 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(12*a*b^20 - 32*a^3*b^18 + 28*a^5*b^16 - 8*a^7*b^14))/(b^16
- 2*a^2*b^14 + a^4*b^12)))/(2*b^5)))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(2*a*b^14 - 89*a^3*b^12 + 640*a^5*b^10 -
1322*a^7*b^8 + 1056*a^9*b^6 - 288*a^11*b^4))/(b^16 - 2*a^2*b^14 + a^4*b^12))*1i)/(2*b^5))/((8*(864*a^11 + 30*
a^3*b^8 - 491*a^5*b^6 + 1746*a^7*b^4 - 2160*a^9*b^2))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (16*tan(c/2 + (d*x)/2)*
(1728*a^12 - 6*a^2*b^10 + 169*a^4*b^8 - 1495*a^6*b^6 + 4356*a^8*b^4 - 4752*a^10*b^2))/(b^16 - 2*a^2*b^14 + a^4
*b^12) + ((a^2*12i - b^2*1i)*((4*(2*a^2*b^12 - 52*a^4*b^10 + 386*a^6*b^8 - 624*a^8*b^6 + 288*a^10*b^4))/(b^15
- 2*a^2*b^13 + a^4*b^11) - ((a^2*12i - b^2*1i)*((4*(4*a*b^16 - 36*a^3*b^14 + 56*a^5*b^12 - 24*a^7*b^10))/(b^15
- 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(24*a^2*b^16 - 100*a^4*b^14 + 124*a^6*b^12 - 48*a^8*b^10))/(
b^16 - 2*a^2*b^14 + a^4*b^12) - ((a^2*12i - b^2*1i)*((4*(8*a^2*b^18 - 16*a^4*b^16 + 8*a^6*b^14))/(b^15 - 2*a^2
*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(12*a*b^20 - 32*a^3*b^18 + 28*a^5*b^16 - 8*a^7*b^14))/(b^16 - 2*a^2*
b^14 + a^4*b^12)))/(2*b^5)))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(2*a*b^14 - 89*a^3*b^12 + 640*a^5*b^10 - 1322*a^7
*b^8 + 1056*a^9*b^6 - 288*a^11*b^4))/(b^16 - 2*a^2*b^14 + a^4*b^12)))/(2*b^5) - ((a^2*12i - b^2*1i)*((4*(2*a^2
*b^12 - 52*a^4*b^10 + 386*a^6*b^8 - 624*a^8*b^6 + 288*a^10*b^4))/(b^15 - 2*a^2*b^13 + a^4*b^11) + ((a^2*12i -
b^2*1i)*((4*(4*a*b^16 - 36*a^3*b^14 + 56*a^5*b^12 - 24*a^7*b^10))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2
+ (d*x)/2)*(24*a^2*b^16 - 100*a^4*b^14 + 124*a^6*b^12 - 48*a^8*b^10))/(b^16 - 2*a^2*b^14 + a^4*b^12) + ((a^2*1
2i - b^2*1i)*((4*(8*a^2*b^18 - 16*a^4*b^16 + 8*a^6*b^14))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/
2)*(12*a*b^20 - 32*a^3*b^18 + 28*a^5*b^16 - 8*a^7*b^14))/(b^16 - 2*a^2*b^14 + a^4*b^12)))/(2*b^5)))/(2*b^5) +
(8*tan(c/2 + (d*x)/2)*(2*a*b^14 - 89*a^3*b^12 + 640*a^5*b^10 - 1322*a^7*b^8 + 1056*a^9*b^6 - 288*a^11*b^4))/(b
^16 - 2*a^2*b^14 + a^4*b^12)))/(2*b^5)))*(a^2*12i - b^2*1i)*1i)/(b^5*d) - (a*atan(((a*(-(a + b)^3*(a - b)^3)^(
1/2)*((4*(2*a^2*b^12 - 52*a^4*b^10 + 386*a^6*b^8 - 624*a^8*b^6 + 288*a^10*b^4))/(b^15 - 2*a^2*b^13 + a^4*b^11)
+ (8*tan(c/2 + (d*x)/2)*(2*a*b^14 - 89*a^3*b^12 + 640*a^5*b^10 - 1322*a^7*b^8 + 1056*a^9*b^6 - 288*a^11*b^4))
/(b^16 - 2*a^2*b^14 + a^4*b^12) - (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(4*a*b^16 - 36*a^3*b^14 + 56*a^5*b^12 -
24*a^7*b^10))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(24*a^2*b^16 - 100*a^4*b^14 + 124*a^6*b^1
2 - 48*a^8*b^10))/(b^16 - 2*a^2*b^14 + a^4*b^12) - (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(8*a^2*b^18 - 16*a^4*b^
16 + 8*a^6*b^14))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(12*a*b^20 - 32*a^3*b^18 + 28*a^5*b^1
6 - 8*a^7*b^14))/(b^16 - 2*a^2*b^14 + a^4*b^12))*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b
^7 - a^6*b^5)))*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(12*a^4 + 6*b^4 -
19*a^2*b^2)*1i)/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(2*a^2*b^1
2 - 52*a^4*b^10 + 386*a^6*b^8 - 624*a^8*b^6 + 288*a^10*b^4))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*
x)/2)*(2*a*b^14 - 89*a^3*b^12 + 640*a^5*b^10 - 1322*a^7*b^8 + 1056*a^9*b^6 - 288*a^11*b^4))/(b^16 - 2*a^2*b^14
+ a^4*b^12) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(4*a*b^16 - 36*a^3*b^14 + 56*a^5*b^12 - 24*a^7*b^10))/(b^15
- 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(24*a^2*b^16 - 100*a^4*b^14 + 124*a^6*b^12 - 48*a^8*b^10))/(
b^16 - 2*a^2*b^14 + a^4*b^12) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(8*a^2*b^18 - 16*a^4*b^16 + 8*a^6*b^14))/(
b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(12*a*b^20 - 32*a^3*b^18 + 28*a^5*b^16 - 8*a^7*b^14))/(b
^16 - 2*a^2*b^14 + a^4*b^12))*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(12
*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(12*a^4 + 6*b^4 - 19*a^2*b^2)*1i)/(2
*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))/((8*(864*a^11 + 30*a^3*b^8 - 491*a^5*b^6 + 1746*a^7*b^4 - 2160*a^9
*b^2))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (16*tan(c/2 + (d*x)/2)*(1728*a^12 - 6*a^2*b^10 + 169*a^4*b^8 - 1495*a^
6*b^6 + 4356*a^8*b^4 - 4752*a^10*b^2))/(b^16 - 2*a^2*b^14 + a^4*b^12) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(2
*a^2*b^12 - 52*a^4*b^10 + 386*a^6*b^8 - 624*a^8*b^6 + 288*a^10*b^4))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c
/2 + (d*x)/2)*(2*a*b^14 - 89*a^3*b^12 + 640*a^5*b^10 - 1322*a^7*b^8 + 1056*a^9*b^6 - 288*a^11*b^4))/(b^16 - 2*
a^2*b^14 + a^4*b^12) - (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(4*a*b^16 - 36*a^3*b^14 + 56*a^5*b^12 - 24*a^7*b^10
))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(24*a^2*b^16 - 100*a^4*b^14 + 124*a^6*b^12 - 48*a^8*
b^10))/(b^16 - 2*a^2*b^14 + a^4*b^12) - (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(8*a^2*b^18 - 16*a^4*b^16 + 8*a^6*
b^14))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(12*a*b^20 - 32*a^3*b^18 + 28*a^5*b^16 - 8*a^7*b
^14))/(b^16 - 2*a^2*b^14 + a^4*b^12))*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^
5)))*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(12*a^4 + 6*b^4 - 19*a^2*b^2
))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)) - (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(2*a^2*b^12 - 52*a^4*b^1
0 + 386*a^6*b^8 - 624*a^8*b^6 + 288*a^10*b^4))/(b^15 - 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(2*a*b^1
4 - 89*a^3*b^12 + 640*a^5*b^10 - 1322*a^7*b^8 + 1056*a^9*b^6 - 288*a^11*b^4))/(b^16 - 2*a^2*b^14 + a^4*b^12) +
(a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(4*a*b^16 - 36*a^3*b^14 + 56*a^5*b^12 - 24*a^7*b^10))/(b^15 - 2*a^2*b^13
+ a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(24*a^2*b^16 - 100*a^4*b^14 + 124*a^6*b^12 - 48*a^8*b^10))/(b^16 - 2*a^2*b
^14 + a^4*b^12) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((4*(8*a^2*b^18 - 16*a^4*b^16 + 8*a^6*b^14))/(b^15 - 2*a^2*b
^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(12*a*b^20 - 32*a^3*b^18 + 28*a^5*b^16 - 8*a^7*b^14))/(b^16 - 2*a^2*b^
14 + a^4*b^12))*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(12*a^4 + 6*b^4 -
19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^
9 + 3*a^4*b^7 - a^6*b^5))))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2)*1i)/(d*(b^11 - 3*a^2*b^
9 + 3*a^4*b^7 - a^6*b^5))