3.3.1 \(\int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx\) [201]

Optimal. Leaf size=241 \[ -\frac {b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{5/2}}-\frac {\left (a^2+12 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^5}+\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{2 a^4 \left (a^2-b^2\right )^2}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \]

[Out]

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

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Rubi [A]
time = 0.59, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2881, 3134, 3080, 3855, 2739, 632, 210} \begin {gather*} -\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\left (a^2+12 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^5}+\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{2 a^4 \left (a^2-b^2\right )^2}-\frac {b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{5/2}}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a + b*Sin[x])^3,x]

[Out]

-((b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(5/2))) - ((a
^2 + 12*b^2)*ArcTanh[Cos[x]])/(2*a^5) + (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Cot[x])/(2*a^4*(a^2 - b^2)^2) - ((a^4
- 10*a^2*b^2 + 6*b^4)*Cot[x]*Csc[x])/(2*a^3*(a^2 - b^2)^2) - (b^2*Cot[x]*Csc[x])/(2*a*(a^2 - b^2)*(a + b*Sin[
x])^2) - (b^2*(7*a^2 - 4*b^2)*Cot[x]*Csc[x])/(2*a^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 2881

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2
- b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])
^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m +
n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||
!(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3080

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3134

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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx &=-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {\int \frac {\csc ^3(x) \left (2 \left (a^2-2 b^2\right )-2 a b \sin (x)+3 b^2 \sin ^2(x)\right )}{(a+b \sin (x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {\csc ^3(x) \left (2 \left (a^4-10 a^2 b^2+6 b^4\right )-a b \left (4 a^2-b^2\right ) \sin (x)+2 b^2 \left (7 a^2-4 b^2\right ) \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {\csc ^2(x) \left (-6 b \left (2 a^4-7 a^2 b^2+4 b^4\right )+2 a \left (a^4+4 a^2 b^2-2 b^4\right ) \sin (x)+2 b \left (a^4-10 a^2 b^2+6 b^4\right ) \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{2 a^4 \left (a^2-b^2\right )^2}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {\csc (x) \left (2 \left (a^2-b^2\right )^2 \left (a^2+12 b^2\right )+2 a b \left (a^4-10 a^2 b^2+6 b^4\right ) \sin (x)\right )}{a+b \sin (x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{2 a^4 \left (a^2-b^2\right )^2}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (a^2+12 b^2\right ) \int \csc (x) \, dx}{2 a^5}-\frac {\left (b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right )\right ) \int \frac {1}{a+b \sin (x)} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (a^2+12 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^5}+\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{2 a^4 \left (a^2-b^2\right )^2}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (b^3 \left (20 a^4-29 a^2 b^2+12 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 )}{a^5 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (a^2+12 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^5}+\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{2 a^4 \left (a^2-b^2\right )^2}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (2 b^3 \left (20 a^4-29 a^2 b^2+12 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 )}{a^5 \left (a^2-b^2\right )^2}\\ &=-\frac {b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{5/2}}-\frac {\left (a^2+12 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^5}+\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{2 a^4 \left (a^2-b^2\right )^2}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}\\ \end {align*}

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Mathematica [A]
time = 1.26, size = 220, normalized size = 0.91 \begin {gather*} \frac {-\frac {8 b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \tan ^{-1}\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 \cot \left (\frac {x}{2}\right )-a^2 \csc ^2\left (\frac {x}{2}\right )-4 \left (a^2+12 b^2\right ) \log \left (\cos \left (\frac {x}{2}\right )\right )+4 \left (a^2+12 b^2\right ) \log \left (\sin \left (\frac {x}{2}\right )\right )+a^2 \sec ^2\left (\frac {x}{2}\right )-\frac {4 a^2 b^4 \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+\frac {12 a b^4 \left (-3 a^2+2 b^2\right ) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-12 a b \tan \left (\frac {x}{2}\right )}{8 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 12*a*b*C
ot[x/2] - a^2*Csc[x/2]^2 - 4*(a^2 + 12*b^2)*Log[Cos[x/2]] + 4*(a^2 + 12*b^2)*Log[Sin[x/2]] + a^2*Sec[x/2]^2 -
(4*a^2*b^4*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (12*a*b^4*(-3*a^2 + 2*b^2)*Cos[x])/((a - b)^2*(a + b)^
2*(a + b*Sin[x])) - 12*a*b*Tan[x/2])/(8*a^5)

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Maple [A]
time = 0.54, size = 329, normalized size = 1.37

method	result	size
default	\(\frac {\frac {a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-6 b \tan \left (\frac {x}{2}\right )}{4 a^{4}}-\frac {1}{8 a^{3} \tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (2 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {x}{2}\right )}-\frac {4 b^{3} \left (\frac {\frac {a \,b^{2} \left (11 a^{2}-8 b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4 a^{4}-8 a^{2} b^{2}+4 b^{4}}+\frac {b \left (10 a^{4}+13 a^{2} b^{2}-14 b^{4}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4 a^{4}-8 a^{2} b^{2}+4 b^{4}}+\frac {a \,b^{2} \left (29 a^{2}-20 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{4 a^{4}-8 a^{2} b^{2}+4 b^{4}}+\frac {a^{2} b \left (10 a^{2}-7 b^{2}\right )}{4 a^{4}-8 a^{2} b^{2}+4 b^{4}}}{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )^{2}}+\frac {\left (20 a^{4}-29 a^{2} b^{2}+12 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{a^{5}}\)	\(329\)
risch	\(\frac {5 i a^{2} b^{5} {\mathrm e}^{6 i x}+24 i a^{6} b \,{\mathrm e}^{4 i x}+40 i a^{4} b^{3} {\mathrm e}^{2 i x}+12 i b^{7}+4 a^{7} {\mathrm e}^{5 i x}+4 a^{7} {\mathrm e}^{3 i x}-4 i a^{6} b \,{\mathrm e}^{6 i x}+20 i a^{4} b^{3} {\mathrm e}^{6 i x}-20 i a^{6} b \,{\mathrm e}^{2 i x}-66 i a^{4} b^{3} {\mathrm e}^{4 i x}-15 i a^{2} b^{5} {\mathrm e}^{4 i x}+31 i b^{5} a^{2} {\mathrm e}^{2 i x}-90 a^{3} b^{4} {\mathrm e}^{5 i x}+54 a \,b^{6} {\mathrm e}^{5 i x}+6 i a^{4} b^{3}-21 i a^{2} b^{5}-74 a^{3} b^{4} {\mathrm e}^{i x}+42 a \,b^{6} {\mathrm e}^{i x}+23 a^{5} b^{2} {\mathrm e}^{i x}-36 i b^{7} {\mathrm e}^{2 i x}-a^{5} b^{2} {\mathrm e}^{7 i x}+10 a^{3} b^{4} {\mathrm e}^{7 i x}-6 a \,b^{6} {\mathrm e}^{7 i x}+17 a^{5} b^{2} {\mathrm e}^{5 i x}+36 i b^{7} {\mathrm e}^{4 i x}-12 i b^{7} {\mathrm e}^{6 i x}-55 a^{5} b^{2} {\mathrm e}^{3 i x}+162 a^{3} b^{4} {\mathrm e}^{3 i x}-90 a \,b^{6} {\mathrm e}^{3 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2} \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} a^{4}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{3}}+\frac {6 \ln \left ({\mathrm e}^{i x}-1\right ) b^{2}}{a^{5}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{3}}-\frac {6 \ln \left ({\mathrm e}^{i x}+1\right ) b^{2}}{a^{5}}-\frac {10 i b^{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} a}+\frac {29 i b^{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} a^{3}}-\frac {6 i b^{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} a^{5}}+\frac {10 i b^{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} a}-\frac {29 i b^{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} a^{3}}+\frac {6 i b^{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} a^{5}}\)	\(922\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a^4*(1/2*a*tan(1/2*x)^2-6*b*tan(1/2*x))-1/8/a^3/tan(1/2*x)^2+1/4/a^5*(2*a^2+24*b^2)*ln(tan(1/2*x))+3/2/a^4
*b/tan(1/2*x)-4/a^5*b^3*((1/4*a*b^2*(11*a^2-8*b^2)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^3+1/4*b*(10*a^4+13*a^2*b^2-1
4*b^4)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2+1/4*a*b^2*(29*a^2-20*b^2)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)+1/4*a^2*b*(10
*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+1/4*(20*a^4-29*a^2*b^2+12*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)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 971 vs. \(2 (225) = 450\).
time = 1.46, size = 2005, normalized size = 8.32 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*(11*a^8*b^2 - 43*a^6*b^4 + 50*a^4*b^6 - 18*a^2*b^8)*cos(x)^3 + (20*a^6*b^3 - 9*a^4*b^5 - 17*a^2*b^7 +
12*b^9 + (20*a^4*b^5 - 29*a^2*b^7 + 12*b^9)*cos(x)^4 - (20*a^6*b^3 + 11*a^4*b^5 - 46*a^2*b^7 + 24*b^9)*cos(x)
^2 + 2*(20*a^5*b^4 - 29*a^3*b^6 + 12*a*b^8 - (20*a^5*b^4 - 29*a^3*b^6 + 12*a*b^8)*cos(x)^2)*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*(a^10 - 14*a^8*b^2 + 46*a^6*b^4 - 51*a^4*b^6 + 18*a^2*b^8)*
cos(x) + (a^10 + 10*a^8*b^2 - 24*a^6*b^4 + 2*a^4*b^6 + 23*a^2*b^8 - 12*b^10 + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^
6 + 35*a^2*b^8 - 12*b^10)*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^6*b^4 - 31*a^4*b^6 + 58*a^2*b^8 - 24*b^10)*cos(
x)^2 + 2*(a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9 - (a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^
7 - 12*a*b^9)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - (a^10 + 10*a^8*b^2 - 24*a^6*b^4 + 2*a^4*b^6 + 23*a^2*b
^8 - 12*b^10 + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^6 + 35*a^2*b^8 - 12*b^10)*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^
6*b^4 - 31*a^4*b^6 + 58*a^2*b^8 - 24*b^10)*cos(x)^2 + 2*(a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^
9 - (a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2) + 2*(3*(
2*a^7*b^3 - 9*a^5*b^5 + 11*a^3*b^7 - 4*a*b^9)*cos(x)^3 - (4*a^9*b - 6*a^7*b^3 - 15*a^5*b^5 + 29*a^3*b^7 - 12*a
*b^9)*cos(x))*sin(x))/(a^13 - 2*a^11*b^2 + 2*a^7*b^6 - a^5*b^8 + (a^11*b^2 - 3*a^9*b^4 + 3*a^7*b^6 - a^5*b^8)*
cos(x)^4 - (a^13 - a^11*b^2 - 3*a^9*b^4 + 5*a^7*b^6 - 2*a^5*b^8)*cos(x)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5
- a^6*b^7 - (a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*cos(x)^2)*sin(x)), -1/4*(2*(11*a^8*b^2 - 43*a^6*b^4 +
50*a^4*b^6 - 18*a^2*b^8)*cos(x)^3 - 2*(20*a^6*b^3 - 9*a^4*b^5 - 17*a^2*b^7 + 12*b^9 + (20*a^4*b^5 - 29*a^2*b^
7 + 12*b^9)*cos(x)^4 - (20*a^6*b^3 + 11*a^4*b^5 - 46*a^2*b^7 + 24*b^9)*cos(x)^2 + 2*(20*a^5*b^4 - 29*a^3*b^6 +
12*a*b^8 - (20*a^5*b^4 - 29*a^3*b^6 + 12*a*b^8)*cos(x)^2)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqr
t(a^2 - b^2)*cos(x))) + 2*(a^10 - 14*a^8*b^2 + 46*a^6*b^4 - 51*a^4*b^6 + 18*a^2*b^8)*cos(x) + (a^10 + 10*a^8*b
^2 - 24*a^6*b^4 + 2*a^4*b^6 + 23*a^2*b^8 - 12*b^10 + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^6 + 35*a^2*b^8 - 12*b^10)
*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^6*b^4 - 31*a^4*b^6 + 58*a^2*b^8 - 24*b^10)*cos(x)^2 + 2*(a^9*b + 9*a^7*b
^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9 - (a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*cos(x)^2)*s
in(x))*log(1/2*cos(x) + 1/2) - (a^10 + 10*a^8*b^2 - 24*a^6*b^4 + 2*a^4*b^6 + 23*a^2*b^8 - 12*b^10 + (a^8*b^2 +
9*a^6*b^4 - 33*a^4*b^6 + 35*a^2*b^8 - 12*b^10)*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^6*b^4 - 31*a^4*b^6 + 58*a
^2*b^8 - 24*b^10)*cos(x)^2 + 2*(a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9 - (a^9*b + 9*a^7*b^3 -
33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2) + 2*(3*(2*a^7*b^3 - 9*a^5*b^5 + 1
1*a^3*b^7 - 4*a*b^9)*cos(x)^3 - (4*a^9*b - 6*a^7*b^3 - 15*a^5*b^5 + 29*a^3*b^7 - 12*a*b^9)*cos(x))*sin(x))/(a^
13 - 2*a^11*b^2 + 2*a^7*b^6 - a^5*b^8 + (a^11*b^2 - 3*a^9*b^4 + 3*a^7*b^6 - a^5*b^8)*cos(x)^4 - (a^13 - a^11*b
^2 - 3*a^9*b^4 + 5*a^7*b^6 - 2*a^5*b^8)*cos(x)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7 - (a^12*b - 3*
a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*cos(x)^2)*sin(x))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{3}{\left (x \right )}}{\left (a + b \sin {\left (x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(x)**3/(a + b*sin(x))**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (225) = 450\).
time = 0.49, size = 514, normalized size = 2.13 \begin {gather*} -\frac {{\left (20 \, a^{4} b^{3} - 29 \, a^{2} b^{5} + 12 \, b^{7}\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^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {2 \, a^{8} \tan \left (\frac {1}{2} \, x\right )^{6} + 20 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, x\right )^{6} - 46 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, x\right )^{6} + 24 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, x\right )^{6} - 4 \, a^{7} b \tan \left (\frac {1}{2} \, x\right )^{5} + 104 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, x\right )^{5} - 108 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, x\right )^{5} + 32 \, a b^{7} \tan \left (\frac {1}{2} \, x\right )^{5} + 5 \, a^{8} \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 165 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, x\right )^{4} - 80 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, x\right )^{4} - 16 \, b^{8} \tan \left (\frac {1}{2} \, x\right )^{4} - 12 \, a^{7} b \tan \left (\frac {1}{2} \, x\right )^{3} + 72 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 124 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, x\right )^{3} - 112 \, a b^{7} \tan \left (\frac {1}{2} \, x\right )^{3} + 4 \, a^{8} \tan \left (\frac {1}{2} \, x\right )^{2} - 28 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 124 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} - 76 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a^{7} b \tan \left (\frac {1}{2} \, x\right ) + 16 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, x\right ) - 8 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, x\right ) + a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}}{8 \, {\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, x\right )^{2} + a \tan \left (\frac {1}{2} \, x\right )\right )}^{2}} + \frac {{\left (a^{2} + 12 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{5}} + \frac {a^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20*a^4*b^3 - 29*a^2*b^5 + 12*b^7)*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^
2)))/((a^9 - 2*a^7*b^2 + a^5*b^4)*sqrt(a^2 - b^2)) - 1/8*(2*a^8*tan(1/2*x)^6 + 20*a^6*b^2*tan(1/2*x)^6 - 46*a^
4*b^4*tan(1/2*x)^6 + 24*a^2*b^6*tan(1/2*x)^6 - 4*a^7*b*tan(1/2*x)^5 + 104*a^5*b^3*tan(1/2*x)^5 - 108*a^3*b^5*t
an(1/2*x)^5 + 32*a*b^7*tan(1/2*x)^5 + 5*a^8*tan(1/2*x)^4 - 2*a^6*b^2*tan(1/2*x)^4 + 165*a^4*b^4*tan(1/2*x)^4 -
80*a^2*b^6*tan(1/2*x)^4 - 16*b^8*tan(1/2*x)^4 - 12*a^7*b*tan(1/2*x)^3 + 72*a^5*b^3*tan(1/2*x)^3 + 124*a^3*b^5
*tan(1/2*x)^3 - 112*a*b^7*tan(1/2*x)^3 + 4*a^8*tan(1/2*x)^2 - 28*a^6*b^2*tan(1/2*x)^2 + 124*a^4*b^4*tan(1/2*x)
^2 - 76*a^2*b^6*tan(1/2*x)^2 - 8*a^7*b*tan(1/2*x) + 16*a^5*b^3*tan(1/2*x) - 8*a^3*b^5*tan(1/2*x) + a^8 - 2*a^6
*b^2 + a^4*b^4)/((a^9 - 2*a^7*b^2 + a^5*b^4)*(a*tan(1/2*x)^3 + 2*b*tan(1/2*x)^2 + a*tan(1/2*x))^2) + 1/2*(a^2
+ 12*b^2)*log(abs(tan(1/2*x)))/a^5 + 1/8*(a^3*tan(1/2*x)^2 - 12*a^2*b*tan(1/2*x))/a^6

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Mupad [B]
time = 9.28, size = 2405, normalized size = 9.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^3*(a + b*sin(x))^3),x)

[Out]

(4*a^2*b*tan(x/2) - a^3/2 + (tan(x/2)^2*(50*a*b^6 - a^7 - 85*a^3*b^4 + 24*a^5*b^2))/(a^4 + b^4 - 2*a^2*b^2) +
(2*tan(x/2)^5*(3*a^6*b + 16*b^7 - 19*a^2*b^5 - 6*a^4*b^3))/(a^4 + b^4 - 2*a^2*b^2) + (2*tan(x/2)^3*(5*a^6*b +
52*b^7 - 77*a^2*b^5 + 2*a^4*b^3))/(a^4 + b^4 - 2*a^2*b^2) - (tan(x/2)^4*(a^8 - 112*b^8 + 56*a^2*b^6 + 177*a^4*
b^4 - 50*a^6*b^2))/(2*a*(a^4 + b^4 - 2*a^2*b^2)))/(tan(x/2)^4*(8*a^6 + 16*a^4*b^2) + 4*a^6*tan(x/2)^2 + 4*a^6*
tan(x/2)^6 + 16*a^5*b*tan(x/2)^3 + 16*a^5*b*tan(x/2)^5) + tan(x/2)^2/(8*a^3) + (log(tan(x/2))*(a^2 + 12*b^2))/
(2*a^5) - (3*b*tan(x/2))/(2*a^4) + (b^3*atan(((b^3*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2)
*((a^11*b + 24*a^5*b^7 - 52*a^7*b^5 + 30*a^9*b^3)/(a^12 + a^8*b^4 - 2*a^10*b^2) - (tan(x/2)*(a^15 - 48*a^3*b^1
2 + 212*a^5*b^10 - 363*a^7*b^8 + 290*a^9*b^6 - 98*a^11*b^4 + 6*a^13*b^2))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^11
*b^4 - 4*a^13*b^2) + (b^3*((2*a^14*b + 2*a^10*b^5 - 4*a^12*b^3)/(a^12 + a^8*b^4 - 2*a^10*b^2) - (tan(x/2)*(6*a
^18 - 8*a^8*b^10 + 38*a^10*b^8 - 72*a^12*b^6 + 68*a^14*b^4 - 32*a^16*b^2))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^1
1*b^4 - 4*a^13*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2))/(2*(a^15 - a^5*b^10 + 5*a^7*
b^8 - 10*a^9*b^6 + 10*a^11*b^4 - 5*a^13*b^2)))*1i)/(2*(a^15 - a^5*b^10 + 5*a^7*b^8 - 10*a^9*b^6 + 10*a^11*b^4
- 5*a^13*b^2)) - (b^3*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2)*((tan(x/2)*(a^15 - 48*a^3*b^
12 + 212*a^5*b^10 - 363*a^7*b^8 + 290*a^9*b^6 - 98*a^11*b^4 + 6*a^13*b^2))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^1
1*b^4 - 4*a^13*b^2) - (a^11*b + 24*a^5*b^7 - 52*a^7*b^5 + 30*a^9*b^3)/(a^12 + a^8*b^4 - 2*a^10*b^2) + (b^3*((2
*a^14*b + 2*a^10*b^5 - 4*a^12*b^3)/(a^12 + a^8*b^4 - 2*a^10*b^2) - (tan(x/2)*(6*a^18 - 8*a^8*b^10 + 38*a^10*b^
8 - 72*a^12*b^6 + 68*a^14*b^4 - 32*a^16*b^2))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^11*b^4 - 4*a^13*b^2))*(-(a + b
)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2))/(2*(a^15 - a^5*b^10 + 5*a^7*b^8 - 10*a^9*b^6 + 10*a^11*b^
4 - 5*a^13*b^2)))*1i)/(2*(a^15 - a^5*b^10 + 5*a^7*b^8 - 10*a^9*b^6 + 10*a^11*b^4 - 5*a^13*b^2)))/((144*b^9 - 3
36*a^2*b^7 + 211*a^4*b^5 + 20*a^6*b^3)/(a^12 + a^8*b^4 - 2*a^10*b^2) + (2*tan(x/2)*(72*b^12 - 294*a^2*b^10 + 4
22*a^4*b^8 - 229*a^6*b^6 + 20*a^8*b^4))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^11*b^4 - 4*a^13*b^2) + (b^3*(-(a + b
)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2)*((a^11*b + 24*a^5*b^7 - 52*a^7*b^5 + 30*a^9*b^3)/(a^12 + a
^8*b^4 - 2*a^10*b^2) - (tan(x/2)*(a^15 - 48*a^3*b^12 + 212*a^5*b^10 - 363*a^7*b^8 + 290*a^9*b^6 - 98*a^11*b^4
+ 6*a^13*b^2))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^11*b^4 - 4*a^13*b^2) + (b^3*((2*a^14*b + 2*a^10*b^5 - 4*a^12*
b^3)/(a^12 + a^8*b^4 - 2*a^10*b^2) - (tan(x/2)*(6*a^18 - 8*a^8*b^10 + 38*a^10*b^8 - 72*a^12*b^6 + 68*a^14*b^4
- 32*a^16*b^2))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^11*b^4 - 4*a^13*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 +
12*b^4 - 29*a^2*b^2))/(2*(a^15 - a^5*b^10 + 5*a^7*b^8 - 10*a^9*b^6 + 10*a^11*b^4 - 5*a^13*b^2))))/(2*(a^15 -
a^5*b^10 + 5*a^7*b^8 - 10*a^9*b^6 + 10*a^11*b^4 - 5*a^13*b^2)) + (b^3*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 + 1
2*b^4 - 29*a^2*b^2)*((tan(x/2)*(a^15 - 48*a^3*b^12 + 212*a^5*b^10 - 363*a^7*b^8 + 290*a^9*b^6 - 98*a^11*b^4 +
6*a^13*b^2))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^11*b^4 - 4*a^13*b^2) - (a^11*b + 24*a^5*b^7 - 52*a^7*b^5 + 30*a
^9*b^3)/(a^12 + a^8*b^4 - 2*a^10*b^2) + (b^3*((2*a^14*b + 2*a^10*b^5 - 4*a^12*b^3)/(a^12 + a^8*b^4 - 2*a^10*b^
2) - (tan(x/2)*(6*a^18 - 8*a^8*b^10 + 38*a^10*b^8 - 72*a^12*b^6 + 68*a^14*b^4 - 32*a^16*b^2))/(a^15 + a^7*b^8
- 4*a^9*b^6 + 6*a^11*b^4 - 4*a^13*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2))/(2*(a^15
- a^5*b^10 + 5*a^7*b^8 - 10*a^9*b^6 + 10*a^11*b^4 - 5*a^13*b^2))))/(2*(a^15 - a^5*b^10 + 5*a^7*b^8 - 10*a^9*b^
6 + 10*a^11*b^4 - 5*a^13*b^2))))*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2)*1i)/(a^15 - a^5*b
^10 + 5*a^7*b^8 - 10*a^9*b^6 + 10*a^11*b^4 - 5*a^13*b^2)

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