\(\int \frac {\sin ^4(x)}{(a+b \sin (x))^3} \, dx\) [194]

Optimal result

Integrand size = 13, antiderivative size = 179 \[ \int \frac {\sin ^4(x)}{(a+b \sin (x))^3} \, dx=-\frac {3 a x}{b^4}+\frac {3 a^2 \left (2 a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{5/2}}-\frac {\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac {a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))} \]

[Out]

-3*a*x/b^4+3*a^2*(2*a^4-5*a^2*b^2+4*b^4)*arctan((b+a*tan(1/2*x))/(a^2-b^2)^(1/2))/b^4/(a^2-b^2)^(5/2)-1/2*(3*a
^2-2*b^2)*cos(x)/b^3/(a^2-b^2)+1/2*a^2*cos(x)*sin(x)^2/b/(a^2-b^2)/(a+b*sin(x))^2-3/2*a^3*(a^2-2*b^2)*cos(x)/b
^3/(a^2-b^2)^2/(a+b*sin(x))

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2871, 3110, 3102, 2814, 2739, 632, 210} \[ \int \frac {\sin ^4(x)}{(a+b \sin (x))^3} \, dx=\frac {a^2 \sin ^2(x) \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac {3 a^2 \left (2 a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{5/2}}-\frac {3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {3 a x}{b^4} \]

[In]

Int[Sin[x]^4/(a + b*Sin[x])^3,x]

[Out]

(-3*a*x)/b^4 + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(
5/2)) - ((3*a^2 - 2*b^2)*Cos[x])/(2*b^3*(a^2 - b^2)) + (a^2*Cos[x]*Sin[x]^2)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2
) - (3*a^3*(a^2 - 2*b^2)*Cos[x])/(2*b^3*(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 3110

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

Rubi steps \begin{align*} \text {integral}& = \frac {a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\int \frac {\sin (x) \left (2 a^2-2 a b \sin (x)-\left (3 a^2-2 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 ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\int \frac {3 a^2 b \left (a^2-2 b^2\right )+a \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right ) \sin (x)-b \left (3 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin ^2(x)}{a+b \sin (x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac {a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\int \frac {3 a^2 b^2 \left (a^2-2 b^2\right )+6 a b \left (a^2-b^2\right )^2 \sin (x)}{a+b \sin (x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2} \\ & = -\frac {3 a x}{b^4}-\frac {\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac {a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (3 a^2 \left (2 a^4-5 a^2 b^2+4 b^4\right )\right ) \int \frac {1}{a+b \sin (x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2} \\ & = -\frac {3 a x}{b^4}-\frac {\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac {a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (3 a^2 \left (2 a^4-5 a^2 b^2+4 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^4 \left (a^2-b^2\right )^2} \\ & = -\frac {3 a x}{b^4}-\frac {\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac {a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (6 a^2 \left (2 a^4-5 a^2 b^2+4 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^4 \left (a^2-b^2\right )^2} \\ & = -\frac {3 a x}{b^4}+\frac {3 a^2 \left (2 a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{5/2}}-\frac {\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac {a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.80 \[ \int \frac {\sin ^4(x)}{(a+b \sin (x))^3} \, dx=\frac {-6 a x+\frac {6 a^2 \left (2 a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-2 b \cos (x)+\frac {a^4 b \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+\frac {a^3 b \left (-5 a^2+8 b^2\right ) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}}{2 b^4} \]

[In]

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

[Out]

(-6*a*x + (6*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 2*b
*Cos[x] + (a^4*b*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (a^3*b*(-5*a^2 + 8*b^2)*Cos[x])/((a - b)^2*(a +
b)^2*(a + b*Sin[x])))/(2*b^4)

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.60

method	result	size
default	\(-\frac {2 \left (\frac {b}{1+\tan ^{2}\left (\frac {x}{2}\right )}+3 a \arctan \left (\tan \left (\frac {x}{2}\right )\right )\right )}{b^{4}}+\frac {2 a^{2} \left (\frac {-\frac {3 a \,b^{2} \left (a^{2}-2 b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (4 a^{4}+a^{2} b^{2}-14 b^{4}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {a \,b^{2} \left (13 a^{2}-22 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {a^{2} b \left (4 a^{2}-7 b^{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )}^{2}}+\frac {3 \left (2 a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{b^{4}}\)	\(286\)
risch	\(-\frac {3 a x}{b^{4}}-\frac {{\mathrm e}^{i x}}{2 b^{3}}-\frac {{\mathrm e}^{-i x}}{2 b^{3}}+\frac {i a^{3} \left (-6 i a^{3} b \,{\mathrm e}^{3 i x}+9 i a \,b^{3} {\mathrm e}^{3 i x}+14 i a^{3} b \,{\mathrm e}^{i x}-23 i a \,b^{3} {\mathrm e}^{i x}+10 a^{4} {\mathrm e}^{2 i x}-11 a^{2} b^{2} {\mathrm e}^{2 i x}-8 b^{4} {\mathrm e}^{2 i x}-5 a^{2} b^{2}+8 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right )^{2} \left (a^{2}-b^{2}\right )^{2} b^{4}}-\frac {3 a^{6} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{4}}+\frac {15 a^{4} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{2}}-\frac {6 a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {3 a^{6} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{4}}-\frac {15 a^{4} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{2}}+\frac {6 a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\)	\(624\)

[In]

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

[Out]

-2/b^4*(b/(1+tan(1/2*x)^2)+3*a*arctan(tan(1/2*x)))+2*a^2/b^4*((-3/2*a*b^2*(a^2-2*b^2)/(a^4-2*a^2*b^2+b^4)*tan(
1/2*x)^3-1/2*b*(4*a^4+a^2*b^2-14*b^4)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2-1/2*a*b^2*(13*a^2-22*b^2)/(a^4-2*a^2*b^
2+b^4)*tan(1/2*x)-1/2*a^2*b*(4*a^2-7*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+3/2*(2*a^4-
5*a^2*b^2+4*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. 441 vs. \(2 (167) = 334\).

Time = 0.36 (sec) , antiderivative size = 945, normalized size of antiderivative = 5.28 \[ \int \frac {\sin ^4(x)}{(a+b \sin (x))^3} \, dx=\left [\frac {12 \, {\left (a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right )} x \cos \left (x\right )^{2} + 4 \, {\left (a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right )} \cos \left (x\right )^{3} - 3 \, {\left (2 \, a^{8} - 3 \, a^{6} b^{2} - a^{4} b^{4} + 4 \, a^{2} b^{6} - {\left (2 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6}\right )} \cos \left (x\right )^{2} + 2 \, {\left (2 \, a^{7} b - 5 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right )} \sin \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) - 12 \, {\left (a^{9} - 2 \, a^{7} b^{2} + 2 \, a^{3} b^{6} - a b^{8}\right )} x - 2 \, {\left (6 \, a^{8} b - 15 \, a^{6} b^{3} + 7 \, a^{4} b^{5} + 4 \, a^{2} b^{7} - 2 \, b^{9}\right )} \cos \left (x\right ) - 2 \, {\left (12 \, {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} x + {\left (9 \, a^{7} b^{2} - 25 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - 4 \, a b^{8}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{4 \, {\left (a^{8} b^{4} - 2 \, a^{6} b^{6} + 2 \, a^{2} b^{10} - b^{12} - {\left (a^{6} b^{6} - 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} - b^{12}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b^{5} - 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} - a b^{11}\right )} \sin \left (x\right )\right )}}, \frac {6 \, {\left (a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right )} x \cos \left (x\right )^{2} + 2 \, {\left (a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right )} \cos \left (x\right )^{3} - 3 \, {\left (2 \, a^{8} - 3 \, a^{6} b^{2} - a^{4} b^{4} + 4 \, a^{2} b^{6} - {\left (2 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6}\right )} \cos \left (x\right )^{2} + 2 \, {\left (2 \, a^{7} b - 5 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right )} \sin \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (x\right )}\right ) - 6 \, {\left (a^{9} - 2 \, a^{7} b^{2} + 2 \, a^{3} b^{6} - a b^{8}\right )} x - {\left (6 \, a^{8} b - 15 \, a^{6} b^{3} + 7 \, a^{4} b^{5} + 4 \, a^{2} b^{7} - 2 \, b^{9}\right )} \cos \left (x\right ) - {\left (12 \, {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} x + {\left (9 \, a^{7} b^{2} - 25 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - 4 \, a b^{8}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{2 \, {\left (a^{8} b^{4} - 2 \, a^{6} b^{6} + 2 \, a^{2} b^{10} - b^{12} - {\left (a^{6} b^{6} - 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} - b^{12}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b^{5} - 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} - a b^{11}\right )} \sin \left (x\right )\right )}}\right ] \]

[In]

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

[Out]

[1/4*(12*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*x*cos(x)^2 + 4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(
x)^3 - 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 4*a^2*b^6 - (2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*cos(x)^2 + 2*(2*a^7*b
- 5*a^5*b^3 + 4*a^3*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)) - 12*(a^9 - 2*a^7*b^2
+ 2*a^3*b^6 - a*b^8)*x - 2*(6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*b^9)*cos(x) - 2*(12*(a^8*b - 3*a
^6*b^3 + 3*a^4*b^5 - a^2*b^7)*x + (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*a*b^8)*cos(x))*sin(x))/(a^8*b^4 - 2
*a^6*b^6 + 2*a^2*b^10 - b^12 - (a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*cos(x)^2 + 2*(a^7*b^5 - 3*a^5*b^7 + 3
*a^3*b^9 - a*b^11)*sin(x)), 1/2*(6*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*x*cos(x)^2 + 2*(a^6*b^3 - 3*a^4*b
^5 + 3*a^2*b^7 - b^9)*cos(x)^3 - 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 4*a^2*b^6 - (2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b
^6)*cos(x)^2 + 2*(2*a^7*b - 5*a^5*b^3 + 4*a^3*b^5)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 -
b^2)*cos(x))) - 6*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8)*x - (6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*
b^9)*cos(x) - (12*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*x + (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*a*b^8
)*cos(x))*sin(x))/(a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12 - (a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*cos(x)^
2 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*sin(x))]

Sympy [F(-1)]

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

[In]

integrate(sin(x)**4/(a+b*sin(x))**3,x)

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(sin(x)^4/(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 [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.43 \[ \int \frac {\sin ^4(x)}{(a+b \sin (x))^3} \, dx=\frac {3 \, {\left (2 \, a^{6} - 5 \, a^{4} b^{2} + 4 \, a^{2} b^{4}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \sqrt {a^{2} - b^{2}}} - \frac {3 \, a^{5} b \tan \left (\frac {1}{2} \, x\right )^{3} - 6 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 4 \, a^{6} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{4} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 14 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 13 \, a^{5} b \tan \left (\frac {1}{2} \, x\right ) - 22 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, x\right ) + 4 \, a^{6} - 7 \, a^{4} b^{2}}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2}} - \frac {3 \, a x}{b^{4}} - \frac {2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )} b^{3}} \]

[In]

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

[Out]

3*(2*a^6 - 5*a^4*b^2 + 4*a^2*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^4 - 2*a^2*b^6 + b^8)*sqrt(a^2 - b^2)) - (3*a^5*b*tan(1/2*x)^3 - 6*a^3*b^3*tan(1/2*x)^3 + 4*a^6*tan(
1/2*x)^2 + a^4*b^2*tan(1/2*x)^2 - 14*a^2*b^4*tan(1/2*x)^2 + 13*a^5*b*tan(1/2*x) - 22*a^3*b^3*tan(1/2*x) + 4*a^
6 - 7*a^4*b^2)/((a^4*b^3 - 2*a^2*b^5 + b^7)*(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)^2) - 3*a*x/b^4 - 2/((tan(1/2
*x)^2 + 1)*b^3)

Mupad [B] (verification not implemented)

Time = 15.06 (sec) , antiderivative size = 5945, normalized size of antiderivative = 33.21 \[ \int \frac {\sin ^4(x)}{(a+b \sin (x))^3} \, dx=\text {Too large to display} \]

[In]

int(sin(x)^4/(a + b*sin(x))^3,x)

[Out]

- ((a^2*(6*a^4 + 2*b^4 - 11*a^2*b^2))/(b^3*(a^2 - b^2)^2) + (3*tan(x/2)^5*(a^5 - 2*a^3*b^2))/(b^2*(a^2 - b^2)^
2) - (3*tan(x/2)^4*(4*a^2*b^4 - 2*a^6 + a^4*b^2))/(b^3*(a^2 - b^2)^2) + (2*tan(x/2)^2*(6*a^6 + 4*b^6 - 13*a^2*
b^4 - 3*a^4*b^2))/(b^3*(a^2 - b^2)^2) + (4*a*tan(x/2)^3*(6*a^4 + 2*b^4 - 11*a^2*b^2))/(b^2*(a^2 - b^2)^2) + (a
*tan(x/2)*(21*a^4 + 8*b^4 - 38*a^2*b^2))/(b^2*(a^2 - b^2)^2))/(tan(x/2)^2*(3*a^2 + 4*b^2) + tan(x/2)^4*(3*a^2
+ 4*b^2) + a^2 + a^2*tan(x/2)^6 + 4*a*b*tan(x/2) + 8*a*b*tan(x/2)^3 + 4*a*b*tan(x/2)^5) - (6*a*atan(((3*a*((8*
(36*a^4*b^11 - 144*a^6*b^9 + 216*a^8*b^7 - 144*a^10*b^5 + 36*a^12*b^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^
6*b^10 + a^8*b^8) + (8*tan(x/2)*(72*a^3*b^13 - 468*a^5*b^11 + 936*a^7*b^9 - 873*a^9*b^7 + 396*a^11*b^5 - 72*a^
13*b^3))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (a*((8*(12*a^2*b^16 - 36*a^4*b^14 + 42*a^6*
b^12 - 24*a^8*b^10 + 6*a^10*b^8))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) - (a*((8*(4*a^2*b^19
- 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*
b^8) + (8*tan(x/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 -
4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*3i)/b^4 + (8*tan(x/2)*(48*a^3*b^16 - 156*a^5*b^14 + 192*a^7*
b^12 - 108*a^9*b^10 + 24*a^11*b^8))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*3i)/b^4))/b^4 + (
3*a*((8*(36*a^4*b^11 - 144*a^6*b^9 + 216*a^8*b^7 - 144*a^10*b^5 + 36*a^12*b^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^1
2 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(72*a^3*b^13 - 468*a^5*b^11 + 936*a^7*b^9 - 873*a^9*b^7 + 396*a^11*b^5
- 72*a^13*b^3))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + (a*((8*(12*a^2*b^16 - 36*a^4*b^14 +
42*a^6*b^12 - 24*a^8*b^10 + 6*a^10*b^8))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (a*((8*(4*
a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^1
0 + a^8*b^8) + (8*tan(x/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))
/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*3i)/b^4 + (8*tan(x/2)*(48*a^3*b^16 - 156*a^5*b^14 +
192*a^7*b^12 - 108*a^9*b^10 + 24*a^11*b^8))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*3i)/b^4))
/b^4)/((16*(54*a^12 - 216*a^6*b^6 + 378*a^8*b^4 - 243*a^10*b^2))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10
+ a^8*b^8) + (16*tan(x/2)*(216*a^13 + 432*a^5*b^8 - 1404*a^7*b^6 + 1728*a^9*b^4 - 972*a^11*b^2))/(b^17 - 4*a^2
*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (a*((8*(36*a^4*b^11 - 144*a^6*b^9 + 216*a^8*b^7 - 144*a^10*b^5 +
36*a^12*b^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(72*a^3*b^13 - 468*a^5*b^1
1 + 936*a^7*b^9 - 873*a^9*b^7 + 396*a^11*b^5 - 72*a^13*b^3))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^
8*b^9) - (a*((8*(12*a^2*b^16 - 36*a^4*b^14 + 42*a^6*b^12 - 24*a^8*b^10 + 6*a^10*b^8))/(b^16 - 4*a^2*b^14 + 6*a
^4*b^12 - 4*a^6*b^10 + a^8*b^8) - (a*((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))
/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17
- 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*3i)/b^4
+ (8*tan(x/2)*(48*a^3*b^16 - 156*a^5*b^14 + 192*a^7*b^12 - 108*a^9*b^10 + 24*a^11*b^8))/(b^17 - 4*a^2*b^15 + 6
*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*3i)/b^4)*3i)/b^4 + (a*((8*(36*a^4*b^11 - 144*a^6*b^9 + 216*a^8*b^7 - 144*a^
10*b^5 + 36*a^12*b^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(72*a^3*b^13 - 46
8*a^5*b^11 + 936*a^7*b^9 - 873*a^9*b^7 + 396*a^11*b^5 - 72*a^13*b^3))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*
b^11 + a^8*b^9) + (a*((8*(12*a^2*b^16 - 36*a^4*b^14 + 42*a^6*b^12 - 24*a^8*b^10 + 6*a^10*b^8))/(b^16 - 4*a^2*b
^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (a*((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^
10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(12*a*b^21 - 56*a^3*b^19 + 104
*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))
*3i)/b^4 + (8*tan(x/2)*(48*a^3*b^16 - 156*a^5*b^14 + 192*a^7*b^12 - 108*a^9*b^10 + 24*a^11*b^8))/(b^17 - 4*a^2
*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*3i)/b^4)*3i)/b^4)))/b^4 - (a^2*atan(((a^2*(-(a + b)^5*(a - b)^5)^(
1/2)*(2*a^4 + 4*b^4 - 5*a^2*b^2)*((8*(36*a^4*b^11 - 144*a^6*b^9 + 216*a^8*b^7 - 144*a^10*b^5 + 36*a^12*b^3))/(
b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(72*a^3*b^13 - 468*a^5*b^11 + 936*a^7*b^9
- 873*a^9*b^7 + 396*a^11*b^5 - 72*a^13*b^3))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (3*a^2
*(-(a + b)^5*(a - b)^5)^(1/2)*((8*(12*a^2*b^16 - 36*a^4*b^14 + 42*a^6*b^12 - 24*a^8*b^10 + 6*a^10*b^8))/(b^16
- 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(48*a^3*b^16 - 156*a^5*b^14 + 192*a^7*b^12 - 1
08*a^9*b^10 + 24*a^11*b^8))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (3*a^2*((8*(4*a^2*b^19 -
16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^
8) + (8*tan(x/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4
*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + 4*b^4 - 5*a^2*b^2))/(2*(
b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(2*a^4 + 4*b^4 - 5*a^2*b^2))/(2*(b^14 -
5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*3i)/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*
a^6*b^8 + 5*a^8*b^6 - a^10*b^4)) + (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + 4*b^4 - 5*a^2*b^2)*((8*(36*a^4*b
^11 - 144*a^6*b^9 + 216*a^8*b^7 - 144*a^10*b^5 + 36*a^12*b^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 +
a^8*b^8) + (8*tan(x/2)*(72*a^3*b^13 - 468*a^5*b^11 + 936*a^7*b^9 - 873*a^9*b^7 + 396*a^11*b^5 - 72*a^13*b^3))/
(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + (3*a^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*(12*a^2*b^16
- 36*a^4*b^14 + 42*a^6*b^12 - 24*a^8*b^10 + 6*a^10*b^8))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b
^8) + (8*tan(x/2)*(48*a^3*b^16 - 156*a^5*b^14 + 192*a^7*b^12 - 108*a^9*b^10 + 24*a^11*b^8))/(b^17 - 4*a^2*b^15
+ 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + (3*a^2*((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a
^10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(12*a*b^21 - 56*a^3*b^19 + 10
4*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9)
)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + 4*b^4 - 5*a^2*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 +
5*a^8*b^6 - a^10*b^4)))*(2*a^4 + 4*b^4 - 5*a^2*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8
*b^6 - a^10*b^4)))*3i)/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))/((16*(54*a^1
2 - 216*a^6*b^6 + 378*a^8*b^4 - 243*a^10*b^2))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (16*t
an(x/2)*(216*a^13 + 432*a^5*b^8 - 1404*a^7*b^6 + 1728*a^9*b^4 - 972*a^11*b^2))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13
- 4*a^6*b^11 + a^8*b^9) - (3*a^2*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + 4*b^4 - 5*a^2*b^2)*((8*(36*a^4*b^11 -
144*a^6*b^9 + 216*a^8*b^7 - 144*a^10*b^5 + 36*a^12*b^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^
8) + (8*tan(x/2)*(72*a^3*b^13 - 468*a^5*b^11 + 936*a^7*b^9 - 873*a^9*b^7 + 396*a^11*b^5 - 72*a^13*b^3))/(b^17
- 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (3*a^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*(12*a^2*b^16 - 36*
a^4*b^14 + 42*a^6*b^12 - 24*a^8*b^10 + 6*a^10*b^8))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) +
(8*tan(x/2)*(48*a^3*b^16 - 156*a^5*b^14 + 192*a^7*b^12 - 108*a^9*b^10 + 24*a^11*b^8))/(b^17 - 4*a^2*b^15 + 6*a
^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (3*a^2*((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^
11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*
b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(-(a
+ b)^5*(a - b)^5)^(1/2)*(2*a^4 + 4*b^4 - 5*a^2*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8
*b^6 - a^10*b^4)))*(2*a^4 + 4*b^4 - 5*a^2*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 -
a^10*b^4))))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)) + (3*a^2*(-(a + b)^5*(
a - b)^5)^(1/2)*(2*a^4 + 4*b^4 - 5*a^2*b^2)*((8*(36*a^4*b^11 - 144*a^6*b^9 + 216*a^8*b^7 - 144*a^10*b^5 + 36*a
^12*b^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(72*a^3*b^13 - 468*a^5*b^11 +
936*a^7*b^9 - 873*a^9*b^7 + 396*a^11*b^5 - 72*a^13*b^3))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^
9) + (3*a^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*(12*a^2*b^16 - 36*a^4*b^14 + 42*a^6*b^12 - 24*a^8*b^10 + 6*a^10*b
^8))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(x/2)*(48*a^3*b^16 - 156*a^5*b^14 + 192*a
^7*b^12 - 108*a^9*b^10 + 24*a^11*b^8))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + (3*a^2*((8*(4
*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^
10 + a^8*b^8) + (8*tan(x/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11)
)/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + 4*b^4 - 5*a^2
*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(2*a^4 + 4*b^4 - 5*a^2*b^2))
/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4))))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^
10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4))))*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + 4*b^4 - 5*a^2*b^2)*3i)/(b^14
- 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)