\(\int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx\) [193]

Optimal result

Integrand size = 13, antiderivative size = 243 \[ \int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx=\frac {\left (12 a^2+b^2\right ) x}{2 b^5}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2}}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \]

[Out]

1/2*(12*a^2+b^2)*x/b^5-a^3*(12*a^4-29*a^2*b^2+20*b^4)*arctan((b+a*tan(1/2*x))/(a^2-b^2)^(1/2))/b^5/(a^2-b^2)^(
5/2)+3/2*a*(4*a^4-7*a^2*b^2+2*b^4)*cos(x)/b^4/(a^2-b^2)^2-1/2*(6*a^4-10*a^2*b^2+b^4)*cos(x)*sin(x)/b^3/(a^2-b^
2)^2+1/2*a^2*cos(x)*sin(x)^3/b/(a^2-b^2)/(a+b*sin(x))^2+1/2*a^2*(4*a^2-7*b^2)*cos(x)*sin(x)^2/b^2/(a^2-b^2)^2/
(a+b*sin(x))

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {2871, 3126, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx=\frac {a^2 \sin ^3(x) \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin ^2(x) \cos (x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {x \left (12 a^2+b^2\right )}{2 b^5}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (x) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2}} \]

[In]

Int[Sin[x]^5/(a + b*Sin[x])^3,x]

[Out]

((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^
5*(a^2 - b^2)^(5/2)) + (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[x])/(2*b^4*(a^2 - b^2)^2) - ((6*a^4 - 10*a^2*b^2 +
b^4)*Cos[x]*Sin[x])/(2*b^3*(a^2 - b^2)^2) + (a^2*Cos[x]*Sin[x]^3)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) + (a^2*(
4*a^2 - 7*b^2)*Cos[x]*Sin[x]^2)/(2*b^2*(a^2 - b^2)^2*(a + b*Sin[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 2871

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((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 - 3)*(c + d*Sin[e
+ f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 +
c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 +
d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || 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 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 {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\int \frac {\sin ^2(x) \left (3 a^2-2 a b \sin (x)-2 \left (2 a^2-b^2\right ) \sin ^2(x)\right )}{(a+b \sin (x))^2} \, dx}{2 b \left (a^2-b^2\right )} \\ & = \frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {\sin (x) \left (-2 a^2 \left (4 a^2-7 b^2\right )+a b \left (a^2-4 b^2\right ) \sin (x)+2 \left (6 a^4-10 a^2 b^2+b^4\right ) \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {2 a \left (6 a^4-10 a^2 b^2+b^4\right )-2 b \left (2 a^4-4 a^2 b^2-b^4\right ) \sin (x)-6 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin ^2(x)}{a+b \sin (x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2} \\ & = \frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {2 a b \left (6 a^4-10 a^2 b^2+b^4\right )+2 \left (a^2-b^2\right )^2 \left (12 a^2+b^2\right ) \sin (x)}{a+b \sin (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 {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \int \frac {1}{a+b \sin (x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2} \\ & = \frac {\left (12 a^2+b^2\right ) x}{2 b^5}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^5 \left (a^2-b^2\right )^2} \\ & = \frac {\left (12 a^2+b^2\right ) x}{2 b^5}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (2 a^3 \left (12 a^4-29 a^2 b^2+20 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 {x}{2}\right )\right )}{b^5 \left (a^2-b^2\right )^2} \\ & = \frac {\left (12 a^2+b^2\right ) x}{2 b^5}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2}}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.08 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx=\frac {2 \left (12 a^2+b^2\right ) x-\frac {4 a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+12 a b \cos (x)-\frac {2 a^5 b \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+\frac {2 a^4 b \left (7 a^2-10 b^2\right ) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-b^2 \sin (2 x)}{4 b^5} \]

[In]

Integrate[Sin[x]^5/(a + b*Sin[x])^3,x]

[Out]

(2*(12*a^2 + b^2)*x - (4*a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b
^2)^(5/2) + 12*a*b*Cos[x] - (2*a^5*b*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (2*a^4*b*(7*a^2 - 10*b^2)*Co
s[x])/((a - b)^2*(a + b)^2*(a + b*Sin[x])) - b^2*Sin[2*x])/(4*b^5)

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.36

method	result	size
default	\(\frac {\frac {4 \left (\frac {b^{2} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 a b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {b^{2} \tan \left (\frac {x}{2}\right )}{4}+\frac {3 a b}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}+\left (12 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{b^{5}}-\frac {4 a^{3} \left (\frac {-\frac {a \,b^{2} \left (5 a^{2}-8 b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {3 b \left (2 a^{4}+a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {a \,b^{2} \left (19 a^{2}-28 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {3 a^{2} b \left (2 a^{2}-3 b^{2}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )}^{2}}+\frac {\left (12 a^{4}-29 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{b^{5}}\)	\(330\)
risch	\(\frac {6 x \,a^{2}}{b^{5}}+\frac {x}{2 b^{3}}+\frac {i {\mathrm e}^{2 i x}}{8 b^{3}}+\frac {3 a \,{\mathrm e}^{i x}}{2 b^{4}}+\frac {3 a \,{\mathrm e}^{-i x}}{2 b^{4}}-\frac {i {\mathrm e}^{-2 i x}}{8 b^{3}}-\frac {i a^{4} \left (-8 i a^{3} b \,{\mathrm e}^{3 i x}+11 i a \,b^{3} {\mathrm e}^{3 i x}+20 i a^{3} b \,{\mathrm e}^{i x}-29 i a \,b^{3} {\mathrm e}^{i x}+14 a^{4} {\mathrm e}^{2 i x}-13 a^{2} b^{2} {\mathrm e}^{2 i x}-10 b^{4} {\mathrm e}^{2 i x}-7 a^{2} b^{2}+10 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right )^{2} \left (a^{2}-b^{2}\right )^{2} b^{5}}-\frac {6 i a^{7} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{5}}+\frac {29 i a^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{3}}-\frac {10 i a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b}+\frac {6 i a^{7} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{5}}-\frac {29 i a^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{3}}+\frac {10 i a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b}\)	\(668\)

[In]

int(sin(x)^5/(a+b*sin(x))^3,x,method=_RETURNVERBOSE)

[Out]

4/b^5*((1/4*b^2*tan(1/2*x)^3+3/2*a*b*tan(1/2*x)^2-1/4*b^2*tan(1/2*x)+3/2*a*b)/(1+tan(1/2*x)^2)^2+1/4*(12*a^2+b
^2)*arctan(tan(1/2*x)))-4*a^3/b^5*((-1/4*a*b^2*(5*a^2-8*b^2)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^3-3/4*b*(2*a^4+a^2
*b^2-6*b^4)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2-1/4*a*b^2*(19*a^2-28*b^2)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)-3/4*a^2*
b*(2*a^2-3*b^2)/(a^4-2*a^2*b^2+b^4))/(a*tan(1/2*x)^2+2*b*tan(1/2*x)+a)^2+1/4*(12*a^4-29*a^2*b^2+20*b^4)/(a^4-2
*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*x)+2*b)/(a^2-b^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (227) = 454\).

Time = 0.40 (sec) , antiderivative size = 1090, normalized size of antiderivative = 4.49 \[ \int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(sin(x)^5/(a+b*sin(x))^3,x, algorithm="fricas")

[Out]

[-1/4*(2*(12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*x*cos(x)^2 + 8*(a^7*b^3 - 3*a^5*b^5 + 3*a^3
*b^7 - a*b^9)*cos(x)^3 + (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*b^6 - (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6
)*cos(x)^2 + 2*(12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*sin(x))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(x)^2 - 2*
a*b*sin(x) - a^2 - b^2 - 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 -
b^2)) - 2*(12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8 - b^10)*x - 2*(12*a^9*b - 29*a^7*b^3 +
15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*cos(x) - 2*((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cos(x)^3 + 2*(12*a^9*b
- 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*x + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*
cos(x))*sin(x))/(a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13 - (a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*cos(x)^2
+ 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*sin(x)), -1/2*((12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b
^8 - b^10)*x*cos(x)^2 + 4*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cos(x)^3 - (12*a^9 - 17*a^7*b^2 - 9*a^5*b^
4 + 20*a^3*b^6 - (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*cos(x)^2 + 2*(12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*sin(
x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*
a^4*b^6 - 10*a^2*b^8 - b^10)*x - (12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*cos(x) - ((a^6*b^4
- 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cos(x)^3 + 2*(12*a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*x + (18
*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*cos(x))*sin(x))/(a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^
13 - (a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*cos(x)^2 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*sin(x)
)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx=\text {Timed out} \]

[In]

integrate(sin(x)**5/(a+b*sin(x))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(sin(x)^5/(a+b*sin(x))^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. 516 vs. \(2 (227) = 454\).

Time = 0.31 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.12 \[ \int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx=-\frac {{\left (12 \, a^{7} - 29 \, a^{5} b^{2} + 20 \, a^{3} b^{4}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} \sqrt {a^{2} - b^{2}}} + \frac {6 \, a^{6} b \tan \left (\frac {1}{2} \, x\right )^{7} - 10 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x\right )^{7} + a^{2} b^{5} \tan \left (\frac {1}{2} \, x\right )^{7} + 12 \, a^{7} \tan \left (\frac {1}{2} \, x\right )^{6} - 5 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, x\right )^{6} - 20 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x\right )^{6} + 4 \, a b^{6} \tan \left (\frac {1}{2} \, x\right )^{6} + 54 \, a^{6} b \tan \left (\frac {1}{2} \, x\right )^{5} - 90 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x\right )^{5} + 17 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, x\right )^{5} + 4 \, b^{7} \tan \left (\frac {1}{2} \, x\right )^{5} + 36 \, a^{7} \tan \left (\frac {1}{2} \, x\right )^{4} - 15 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} - 66 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x\right )^{4} + 24 \, a b^{6} \tan \left (\frac {1}{2} \, x\right )^{4} + 90 \, a^{6} b \tan \left (\frac {1}{2} \, x\right )^{3} - 162 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 55 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, x\right )^{3} - 4 \, b^{7} \tan \left (\frac {1}{2} \, x\right )^{3} + 36 \, a^{7} \tan \left (\frac {1}{2} \, x\right )^{2} - 31 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 40 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 20 \, a b^{6} \tan \left (\frac {1}{2} \, x\right )^{2} + 42 \, a^{6} b \tan \left (\frac {1}{2} \, x\right ) - 74 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x\right ) + 23 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, x\right ) + 12 \, a^{7} - 21 \, a^{5} b^{2} + 6 \, a^{3} b^{4}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2}} + \frac {{\left (12 \, a^{2} + b^{2}\right )} x}{2 \, b^{5}} \]

[In]

integrate(sin(x)^5/(a+b*sin(x))^3,x, algorithm="giac")

[Out]

-(12*a^7 - 29*a^5*b^2 + 20*a^3*b^4)*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^
2)))/((a^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(a^2 - b^2)) + (6*a^6*b*tan(1/2*x)^7 - 10*a^4*b^3*tan(1/2*x)^7 + a^2*b^5
*tan(1/2*x)^7 + 12*a^7*tan(1/2*x)^6 - 5*a^5*b^2*tan(1/2*x)^6 - 20*a^3*b^4*tan(1/2*x)^6 + 4*a*b^6*tan(1/2*x)^6
+ 54*a^6*b*tan(1/2*x)^5 - 90*a^4*b^3*tan(1/2*x)^5 + 17*a^2*b^5*tan(1/2*x)^5 + 4*b^7*tan(1/2*x)^5 + 36*a^7*tan(
1/2*x)^4 - 15*a^5*b^2*tan(1/2*x)^4 - 66*a^3*b^4*tan(1/2*x)^4 + 24*a*b^6*tan(1/2*x)^4 + 90*a^6*b*tan(1/2*x)^3 -
162*a^4*b^3*tan(1/2*x)^3 + 55*a^2*b^5*tan(1/2*x)^3 - 4*b^7*tan(1/2*x)^3 + 36*a^7*tan(1/2*x)^2 - 31*a^5*b^2*ta
n(1/2*x)^2 - 40*a^3*b^4*tan(1/2*x)^2 + 20*a*b^6*tan(1/2*x)^2 + 42*a^6*b*tan(1/2*x) - 74*a^4*b^3*tan(1/2*x) + 2
3*a^2*b^5*tan(1/2*x) + 12*a^7 - 21*a^5*b^2 + 6*a^3*b^4)/((a^4*b^4 - 2*a^2*b^6 + b^8)*(a*tan(1/2*x)^4 + 2*b*tan
(1/2*x)^3 + 2*a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)^2) + 1/2*(12*a^2 + b^2)*x/b^5

Mupad [B] (verification not implemented)

Time = 15.55 (sec) , antiderivative size = 6640, normalized size of antiderivative = 27.33 \[ \int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx=\text {Too large to display} \]

[In]

int(sin(x)^5/(a + b*sin(x))^3,x)

[Out]

((3*(4*a^7 + 2*a^3*b^4 - 7*a^5*b^2))/(b^4*(a^4 + b^4 - 2*a^2*b^2)) + (tan(x/2)^7*(6*a^6 + a^2*b^4 - 10*a^4*b^2
))/(b^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(x/2)^5*(54*a^6 + 4*b^6 + 17*a^2*b^4 - 90*a^4*b^2))/(b^3*(a^4 + b^4 - 2
*a^2*b^2)) + (tan(x/2)^3*(90*a^6 - 4*b^6 + 55*a^2*b^4 - 162*a^4*b^2))/(b^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(x/2
)^6*(4*a*b^6 + 12*a^7 - 20*a^3*b^4 - 5*a^5*b^2))/(b^4*(a^4 + b^4 - 2*a^2*b^2)) + (tan(x/2)^2*(20*a*b^6 + 36*a^
7 - 40*a^3*b^4 - 31*a^5*b^2))/(b^4*(a^4 + b^4 - 2*a^2*b^2)) + (tan(x/2)*(42*a^6 + 23*a^2*b^4 - 74*a^4*b^2))/(b
^3*(a^4 + b^4 - 2*a^2*b^2)) + (3*tan(x/2)^4*(3*a^2 + 4*b^2)*(2*a*b^4 + 4*a^5 - 7*a^3*b^2))/(b^4*(a^4 + b^4 - 2
*a^2*b^2)))/(tan(x/2)^2*(4*a^2 + 4*b^2) + tan(x/2)^6*(4*a^2 + 4*b^2) + tan(x/2)^4*(6*a^2 + 8*b^2) + a^2 + a^2*
tan(x/2)^8 + 4*a*b*tan(x/2) + 12*a*b*tan(x/2)^3 + 12*a*b*tan(x/2)^5 + 4*a*b*tan(x/2)^7) + (atan((((a^2*12i + b
^2*1i)*((4*(2*a^2*b^16 + 40*a^4*b^14 + 108*a^6*b^12 - 872*a^8*b^10 + 1538*a^10*b^8 - 1104*a^12*b^6 + 288*a^14*
b^4))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - ((a^2*12i + b^2*1i)*((4*(4*a*b^20 + 28*a^3*b^
18 - 120*a^5*b^16 + 164*a^7*b^14 - 100*a^9*b^12 + 24*a^11*b^10))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13
+ a^8*b^11) - (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 +
6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a
^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12))*(a^2*12i + b^2*1i))/(2*b^5)
+ (8*tan(x/2)*(80*a^4*b^18 - 276*a^6*b^16 + 360*a^8*b^14 - 212*a^10*b^12 + 48*a^12*b^10))/(b^20 - 4*a^2*b^18 +
6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12)))/(2*b^5) + (8*tan(x/2)*(2*a*b^18 + 39*a^3*b^16 + 88*a^5*b^14 - 1326*a^7*
b^12 + 3134*a^9*b^10 - 3194*a^11*b^8 + 1536*a^13*b^6 - 288*a^15*b^4))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*
b^14 + a^8*b^12))*1i)/(2*b^5) + ((a^2*12i + b^2*1i)*((4*(2*a^2*b^16 + 40*a^4*b^14 + 108*a^6*b^12 - 872*a^8*b^1
0 + 1538*a^10*b^8 - 1104*a^12*b^6 + 288*a^14*b^4))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) +
((a^2*12i + b^2*1i)*((4*(4*a*b^20 + 28*a^3*b^18 - 120*a^5*b^16 + 164*a^7*b^14 - 100*a^9*b^12 + 24*a^11*b^10))/
(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a
^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(12*a*b^24 - 56
*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b
^14 + a^8*b^12))*(a^2*12i + b^2*1i))/(2*b^5) + (8*tan(x/2)*(80*a^4*b^18 - 276*a^6*b^16 + 360*a^8*b^14 - 212*a^
10*b^12 + 48*a^12*b^10))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12)))/(2*b^5) + (8*tan(x/2)*(2*a
*b^18 + 39*a^3*b^16 + 88*a^5*b^14 - 1326*a^7*b^12 + 3134*a^9*b^10 - 3194*a^11*b^8 + 1536*a^13*b^6 - 288*a^15*b
^4))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12))*1i)/(2*b^5))/((8*(864*a^15 + 20*a^5*b^10 + 11*a
^7*b^8 - 2326*a^9*b^6 + 4770*a^11*b^4 - 3456*a^13*b^2))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^1
1) - ((a^2*12i + b^2*1i)*((4*(2*a^2*b^16 + 40*a^4*b^14 + 108*a^6*b^12 - 872*a^8*b^10 + 1538*a^10*b^8 - 1104*a^
12*b^6 + 288*a^14*b^4))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - ((a^2*12i + b^2*1i)*((4*(4*
a*b^20 + 28*a^3*b^18 - 120*a^5*b^16 + 164*a^7*b^14 - 100*a^9*b^12 + 24*a^11*b^10))/(b^19 - 4*a^2*b^17 + 6*a^4*
b^15 - 4*a^6*b^13 + a^8*b^11) - (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^
19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 -
96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12))*(a^2*12i +
b^2*1i))/(2*b^5) + (8*tan(x/2)*(80*a^4*b^18 - 276*a^6*b^16 + 360*a^8*b^14 - 212*a^10*b^12 + 48*a^12*b^10))/(b
^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12)))/(2*b^5) + (8*tan(x/2)*(2*a*b^18 + 39*a^3*b^16 + 88*a^
5*b^14 - 1326*a^7*b^12 + 3134*a^9*b^10 - 3194*a^11*b^8 + 1536*a^13*b^6 - 288*a^15*b^4))/(b^20 - 4*a^2*b^18 + 6
*a^4*b^16 - 4*a^6*b^14 + a^8*b^12)))/(2*b^5) + ((a^2*12i + b^2*1i)*((4*(2*a^2*b^16 + 40*a^4*b^14 + 108*a^6*b^1
2 - 872*a^8*b^10 + 1538*a^10*b^8 - 1104*a^12*b^6 + 288*a^14*b^4))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13
+ a^8*b^11) + ((a^2*12i + b^2*1i)*((4*(4*a*b^20 + 28*a^3*b^18 - 120*a^5*b^16 + 164*a^7*b^14 - 100*a^9*b^12 +
24*a^11*b^10))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*
a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*
(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4
*b^16 - 4*a^6*b^14 + a^8*b^12))*(a^2*12i + b^2*1i))/(2*b^5) + (8*tan(x/2)*(80*a^4*b^18 - 276*a^6*b^16 + 360*a^
8*b^14 - 212*a^10*b^12 + 48*a^12*b^10))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12)))/(2*b^5) + (
8*tan(x/2)*(2*a*b^18 + 39*a^3*b^16 + 88*a^5*b^14 - 1326*a^7*b^12 + 3134*a^9*b^10 - 3194*a^11*b^8 + 1536*a^13*b
^6 - 288*a^15*b^4))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12)))/(2*b^5) + (16*tan(x/2)*(1728*a^
16 + 20*a^4*b^12 + 411*a^6*b^10 + 1314*a^8*b^8 - 7829*a^10*b^6 + 11700*a^12*b^4 - 7344*a^14*b^2))/(b^20 - 4*a^
2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12)))*(a^2*12i + b^2*1i)*1i)/b^5 + (a^3*atan(((a^3*(-(a + b)^5*(a - b
)^5)^(1/2)*((4*(2*a^2*b^16 + 40*a^4*b^14 + 108*a^6*b^12 - 872*a^8*b^10 + 1538*a^10*b^8 - 1104*a^12*b^6 + 288*a
^14*b^4))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(2*a*b^18 + 39*a^3*b^16 + 88*
a^5*b^14 - 1326*a^7*b^12 + 3134*a^9*b^10 - 3194*a^11*b^8 + 1536*a^13*b^6 - 288*a^15*b^4))/(b^20 - 4*a^2*b^18 +
6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) - (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4 + 20*b^4 - 29*a^2*b^2)*((4*(4
*a*b^20 + 28*a^3*b^18 - 120*a^5*b^16 + 164*a^7*b^14 - 100*a^9*b^12 + 24*a^11*b^10))/(b^19 - 4*a^2*b^17 + 6*a^4
*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(80*a^4*b^18 - 276*a^6*b^16 + 360*a^8*b^14 - 212*a^10*b^12 + 48*a
^12*b^10))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) - (a^3*((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*
a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*
(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4
*b^16 - 4*a^6*b^14 + a^8*b^12))*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4 + 20*b^4 - 29*a^2*b^2))/(2*(b^15 - 5*a^2*
b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5))))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 +
5*a^8*b^7 - a^10*b^5)))*(12*a^4 + 20*b^4 - 29*a^2*b^2)*1i)/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 +
5*a^8*b^7 - a^10*b^5)) + (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*((4*(2*a^2*b^16 + 40*a^4*b^14 + 108*a^6*b^12 - 872*
a^8*b^10 + 1538*a^10*b^8 - 1104*a^12*b^6 + 288*a^14*b^4))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b
^11) + (8*tan(x/2)*(2*a*b^18 + 39*a^3*b^16 + 88*a^5*b^14 - 1326*a^7*b^12 + 3134*a^9*b^10 - 3194*a^11*b^8 + 153
6*a^13*b^6 - 288*a^15*b^4))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + (a^3*(-(a + b)^5*(a - b
)^5)^(1/2)*(12*a^4 + 20*b^4 - 29*a^2*b^2)*((4*(4*a*b^20 + 28*a^3*b^18 - 120*a^5*b^16 + 164*a^7*b^14 - 100*a^9*
b^12 + 24*a^11*b^10))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(80*a^4*b^18 - 27
6*a^6*b^16 + 360*a^8*b^14 - 212*a^10*b^12 + 48*a^12*b^10))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*
b^12) + (a^3*((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*
a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9
*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12))*(-(a + b)^5*(a - b)^5)^(1/2)*(
12*a^4 + 20*b^4 - 29*a^2*b^2))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5))))/(2*
(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*a^4 + 20*b^4 - 29*a^2*b^2)*1i)/(2*
(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))/((8*(864*a^15 + 20*a^5*b^10 + 11*a^7*b
^8 - 2326*a^9*b^6 + 4770*a^11*b^4 - 3456*a^13*b^2))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) +
(16*tan(x/2)*(1728*a^16 + 20*a^4*b^12 + 411*a^6*b^10 + 1314*a^8*b^8 - 7829*a^10*b^6 + 11700*a^12*b^4 - 7344*a
^14*b^2))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) - (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*((4*(2*
a^2*b^16 + 40*a^4*b^14 + 108*a^6*b^12 - 872*a^8*b^10 + 1538*a^10*b^8 - 1104*a^12*b^6 + 288*a^14*b^4))/(b^19 -
4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(2*a*b^18 + 39*a^3*b^16 + 88*a^5*b^14 - 1326*a^
7*b^12 + 3134*a^9*b^10 - 3194*a^11*b^8 + 1536*a^13*b^6 - 288*a^15*b^4))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^
6*b^14 + a^8*b^12) - (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4 + 20*b^4 - 29*a^2*b^2)*((4*(4*a*b^20 + 28*a^3*b
^18 - 120*a^5*b^16 + 164*a^7*b^14 - 100*a^9*b^12 + 24*a^11*b^10))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13
+ a^8*b^11) + (8*tan(x/2)*(80*a^4*b^18 - 276*a^6*b^16 + 360*a^8*b^14 - 212*a^10*b^12 + 48*a^12*b^10))/(b^20 -
4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) - (a^3*((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*
b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(12*a*b^24 - 56*a^
3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14
+ a^8*b^12))*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4 + 20*b^4 - 29*a^2*b^2))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11
- 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5))))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b
^5)))*(12*a^4 + 20*b^4 - 29*a^2*b^2))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)
) + (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*((4*(2*a^2*b^16 + 40*a^4*b^14 + 108*a^6*b^12 - 872*a^8*b^10 + 1538*a^10*
b^8 - 1104*a^12*b^6 + 288*a^14*b^4))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(2
*a*b^18 + 39*a^3*b^16 + 88*a^5*b^14 - 1326*a^7*b^12 + 3134*a^9*b^10 - 3194*a^11*b^8 + 1536*a^13*b^6 - 288*a^15
*b^4))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + (a^3*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4 +
20*b^4 - 29*a^2*b^2)*((4*(4*a*b^20 + 28*a^3*b^18 - 120*a^5*b^16 + 164*a^7*b^14 - 100*a^9*b^12 + 24*a^11*b^10))
/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(x/2)*(80*a^4*b^18 - 276*a^6*b^16 + 360*a^8*
b^14 - 212*a^10*b^12 + 48*a^12*b^10))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + (a^3*((4*(8*a
^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13
+ a^8*b^11) + (8*tan(x/2)*(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))
/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12))*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4 + 20*b^4 - 29*
a^2*b^2))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5))))/(2*(b^15 - 5*a^2*b^13 +
10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*a^4 + 20*b^4 - 29*a^2*b^2))/(2*(b^15 - 5*a^2*b^13 + 10*
a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5))))*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4 + 20*b^4 - 29*a^2*b^2)*1
i)/(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)