∫ csc(c+dx)___ (a−b sin4(c+dx))3 dx

3.229 $\int \frac {\csc (c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx$

Optimal. Leaf size=617 \[ -\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \left (5 \sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\sqrt [4]{b} \left (5 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {b \cos (c+d x) \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+11 a+b\right )}{32 a^2 d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

[Out]

-arctanh(cos(d*x+c))/a^3/d-1/8*b*cos(d*x+c)*(2-cos(d*x+c)^2)/a/(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)^2

-1/4*b*cos(d*x+c)*(2-cos(d*x+c)^2)/a^2/(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)-1/32*b*cos(d*x+c)*(11*a+b

-(5*a+b)*cos(d*x+c)^2)/a^2/(a-b)^2/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)-1/64*b^(1/4)*arctan(b^(1/4)*cos(d*x

+c)/(a^(1/2)-b^(1/2))^(1/2))*(5*a^(1/2)-2*b^(1/2))/a^(5/2)/d/(a^(1/2)-b^(1/2))^(5/2)-1/8*b^(1/4)*arctan(b^(1/4

)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^(5/2)/d/(a^(1/2)-b^(1/2))^(3/2)+1/8*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)

/(a^(1/2)+b^(1/2))^(1/2))/a^(5/2)/d/(a^(1/2)+b^(1/2))^(3/2)+1/64*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b

^(1/2))^(1/2))*(5*a^(1/2)+2*b^(1/2))/a^(5/2)/d/(a^(1/2)+b^(1/2))^(5/2)-1/2*b^(1/4)*arctan(b^(1/4)*cos(d*x+c)/(

a^(1/2)-b^(1/2))^(1/2))/a^3/d/(a^(1/2)-b^(1/2))^(1/2)+1/2*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))

^(1/2))/a^3/d/(a^(1/2)+b^(1/2))^(1/2)

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Rubi [A] time = 0.84, antiderivative size = 617, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 22, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.318, Rules used = {3215, 1238, 207, 1178, 1166, 205, 208} \[ -\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {b \cos (c+d x) \left (-(5 a+b) \cos ^2(c+d x)+11 a+b\right )}{32 a^2 d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \left (5 \sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\sqrt [4]{b} \left (5 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a]

- Sqrt[b])^(5/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] -

Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] - Sqr

t[b]]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(

8*a^(5/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*

a^3*Sqrt[Sqrt[a] + Sqrt[b]]*d) + ((5*Sqrt[a] + 2*Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a]

+ Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*a*(a - b)*d*(

a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a^2*(a - b)*d*(a

- b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)) - (b*Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a^2

*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]

&& PosQ[a/b]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;

FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,

b}, x] && NegQ[a/b]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di

st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +

q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^

2 - 4*a*c]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*

a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)

*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +

b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e

^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1238

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d

+ e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((Intege

rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free

Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4

)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{a^3 \left (-1+x^2\right )}+\frac {b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )^3}+\frac {b-b x^2}{a^2 \left (a-b+2 b x^2-b x^4\right )^2}+\frac {b-b x^2}{a^3 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {\operatorname {Subst}\left (\int \frac {b-b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {\operatorname {Subst}\left (\int \frac {b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int \frac {b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {-4 a b^2+2 a b^2 x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{8 a^3 (a-b) b d}+\frac {\operatorname {Subst}\left (\int \frac {-12 a b^2+10 a b^2 x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{16 a^2 (a-b) b d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^3 d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^3 d}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {4 a (13 a-b) b^3-4 a b^3 (5 a+b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{128 a^3 (a-b)^2 b^2 d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right ) d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right ) d}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {\left (\left (5 \sqrt {a}-2 \sqrt {b}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^2 d}+\frac {\left (\left (5 \sqrt {a}+2 \sqrt {b}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^2 d}\\ &=-\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [C] time = 4.28, size = 920, normalized size = 1.49 \[ \frac {\frac {512 b (\cos (3 (c+d x))-5 \cos (c+d x)) a^2}{(a-b) (-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}+\frac {32 b \cos (c+d x) (-41 a+23 b+(13 a-7 b) \cos (2 (c+d x))) a}{(a-b)^2 (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}-256 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+256 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {i b \text {RootSum}\left [b \text {$\#$1}^8-4 b \text {$\#$1}^6-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^2+b\& ,\frac {90 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+64 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-142 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-45 i a^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^6-32 i b^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^6+71 i a b \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^6-398 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-192 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+506 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+199 i a^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^4+96 i b^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^4-253 i a b \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^4+398 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+192 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-506 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-199 i a^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^2-96 i b^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^2+253 i a b \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^2-90 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-64 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+142 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+45 i a^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right )+32 i b^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right )-71 i a b \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right )}{b \text {$\#$1}^7-3 b \text {$\#$1}^5-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-b \text {$\#$1}}\& \right ]}{(a-b)^2}}{256 a^3 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((32*a*b*Cos[c + d*x]*(-41*a + 23*b + (13*a - 7*b)*Cos[2*(c + d*x)]))/((a - b)^2*(8*a - 3*b + 4*b*Cos[2*(c + d

*x)] - b*Cos[4*(c + d*x)])) + (512*a^2*b*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/((a - b)*(-8*a + 3*b - 4*b*Cos[

2*(c + d*x)] + b*Cos[4*(c + d*x)])^2) - 256*Log[Cos[(c + d*x)/2]] + 256*Log[Sin[(c + d*x)/2]] - (I*b*RootSum[b

- 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] +

142*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 64*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + (45*I)*a

^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (71*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (32*I)*b^2*Log[1 - 2*Cos

[c + d*x]*#1 + #1^2] + 398*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 506*a*b*ArcTan[Sin[c + d*x]/(Co

s[c + d*x] - #1)]*#1^2 + 192*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (199*I)*a^2*Log[1 - 2*Cos[c +

d*x]*#1 + #1^2]*#1^2 + (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (96*I)*b^2*Log[1 - 2*Cos[c + d*x]

*#1 + #1^2]*#1^2 - 398*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 506*a*b*ArcTan[Sin[c + d*x]/(Cos[c

+ d*x] - #1)]*#1^4 - 192*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (199*I)*a^2*Log[1 - 2*Cos[c + d*x

]*#1 + #1^2]*#1^4 - (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (96*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1

+ #1^2]*#1^4 + 90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 142*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x

] - #1)]*#1^6 + 64*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - (45*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 +

#1^2]*#1^6 + (71*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - (32*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*

#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b)^2)/(256*a^3*d)

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fricas [B] time = 3.46, size = 5020, normalized size = 8.14 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(4*(13*a^2*b^2 - 7*a*b^3)*cos(d*x + c)^7 - 4*(53*a^2*b^2 - 29*a*b^3)*cos(d*x + c)^5 - 4*(17*a^3*b - 78*

a^2*b^2 + 37*a*b^3)*cos(d*x + c)^3 - ((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^

3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*

a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(

3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3

+ 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 399

47241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*

b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a

^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log((4100625*a^6*b -

14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d

*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 -

22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b

^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a

^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d

^4)) - (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 122

88*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 + (a^11 - 5*a^10*b + 1

0*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 -

49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21

- 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13

*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2)

)) + ((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 -

2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d

*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 1004

5*a^2*b^3 - 5084*a*b^4 + 1024*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt

((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6

+ 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^

4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10

*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log((4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4

*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b

+ 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a

^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*

a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^1

6*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)) + (123525*a^9*b - 450359*a^

8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b

- 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b

^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*

b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*

a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*

d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))) + ((a^5*b^2 - 2*a^4*b^3 + a^3

*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3

- 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b

+ 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^

5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b

^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b

^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6

- 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5

*a^7*b^4 - a^6*b^5)*d^2))*log(-(4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504

896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3

+ 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971

086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 14745

6*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14

*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)) - (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957

*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3

- 5084*a*b^4 + 1024*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625

*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 740147

2*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a

^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a

^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))) - ((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5

*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4

+ 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*

b^4)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 - (a^11 - 5*a^10*b + 10*a^9*b

^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 4967945

2*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^

20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 -

10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(-

(4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 3

93216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5

+ 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4

+ 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*

a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^

9 + a^11*b^10)*d^4)) + (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 951

04*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 - (a^1

1 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 399

71086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147

456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^

14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4

- a^6*b^5)*d^2))) + 20*(7*a^3*b - 10*a^2*b^2 + 3*a*b^3)*cos(d*x + c) + 64*((a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x

+ c)^8 - 4*(a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^6 - 2*(a^3*b - 5*a^2*b^2 + 7*a*b^3 - 3*b^4)*cos(d*x + c)^4 +

a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 + 4*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(d*x + c)^2)*log(1/2*cos

(d*x + c) + 1/2) - 64*((a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^8 - 4*(a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^6 -

2*(a^3*b - 5*a^2*b^2 + 7*a*b^3 - 3*b^4)*cos(d*x + c)^4 + a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 + 4*(a^3*b

- 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d

*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^

3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^

5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)

________________________________________________________________________________________

giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the

root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[53,-89]Warning,

need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done

assuming [a,b]=[-57,87]Precision problem choosing root in common_EXT, current precision 14-2/d*((-8*((1-cos(c

+d*x))/(1+cos(c+d*x)))^7*a^3*b+5*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^2*b^2-5*((1-cos(c+d*x))/(1+cos(c+d*x)))^6

*a^3*b-86*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^2*b^2+64*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a*b^3+104*((1-cos(c+d

*x))/(1+cos(c+d*x)))^5*a^3*b-327*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^2*b^2+208*((1-cos(c+d*x))/(1+cos(c+d*x)))

^5*a*b^3+315*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^3*b-1074*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^2*b^2+1728*((1-c

os(c+d*x))/(1+cos(c+d*x)))^4*a*b^3-768*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*b^4+400*((1-cos(c+d*x))/(1+cos(c+d*x)

))^3*a^3*b-1161*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^2*b^2+560*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a*b^3+257*((1-

cos(c+d*x))/(1+cos(c+d*x)))^2*a^3*b-370*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^2*b^2+128*((1-cos(c+d*x))/(1+cos(c

+d*x)))^2*a*b^3+80*(1-cos(c+d*x))/(1+cos(c+d*x))*a^3*b-53*(1-cos(c+d*x))/(1+cos(c+d*x))*a^2*b^2+9*a^3*b-6*a^2*

b^2)/(16*a^5-32*a^4*b+16*a^3*b^2)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+4*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a+6

*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a+

a)^2-1/4/a^3*ln(abs(1-cos(c+d*x))/abs(1+cos(c+d*x))))-2/d/(16*a^5-32*a^4*b+16*a^3*b^2)*2/d*((-32*b^2+71*b*a-45

*a^2)/8*(c+d*x)+(32*a^5*b+135*a^5*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-32*a^4*b^2-579*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(a

-b))-32*a^4*a*b-222*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-184*a^3*b^3+729*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)

*(a-b))+32*a^3*b*a*b+840*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+312*a^2*b^4-209*a^2*b^3*sqrt(a^2-a*b+sq

rt(a*b)*(a-b))+184*a^2*b^2*a*b-910*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-128*a*b^5-52*a*b^4*sqrt(a^2

-a*b+sqrt(a*b)*(a-b))-312*a*b^3*a*b+252*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+128*b^4*a*b+64*b^4*sqrt(

a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(48*a^5-192*a^4*b+224*a^3*b^2-64*a^2*b^3-16*a*b^4)*(atan(tan(c+d*

x)/sqrt(-(16*a+sqrt(16*a*16*a+4*(8*b-8*a)*8*a))/2/(8*b-8*a)))+pi*floor((c+d*x)/pi+1/2))-(32*a^5*b-135*a^5*sqrt

(a^2-a*b+sqrt(a*b)*(-a+b))-32*a^4*b^2+579*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-32*a^4*a*b-222*a^4*sqrt(a*b)*sq

rt(a^2-a*b+sqrt(a*b)*(-a+b))-184*a^3*b^3-729*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+32*a^3*b*a*b+840*a^3*b*sqr

t(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+312*a^2*b^4+209*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+184*a^2*b^2*a*b-9

10*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-128*a*b^5+52*a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-312*a*b^

3*a*b+252*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+128*b^4*a*b+64*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(

-a+b)))*abs(a-b)/(48*a^5-192*a^4*b+224*a^3*b^2-64*a^2*b^3-16*a*b^4)*(atan(tan(c+d*x)/sqrt(-(16*a-sqrt(16*a*16*

a+4*(8*b-8*a)*8*a))/2/(8*b-8*a)))+pi*floor((c+d*x)/pi+1/2)))

________________________________________________________________________________________

maple [B] time = 0.61, size = 1139, normalized size = 1.85 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d/a^3*ln(cos(d*x+c)-1)-1/2/d/a^3*ln(1+cos(d*x+c))-13/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2*b^2/a/(a

^2-2*a*b+b^2)*cos(d*x+c)^7+7/32/d/a^2*b^3/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^7

+53/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/a/(a^2-2*a*b+b^2)*cos(d*x+c)^5*b^2-29/32/d/a^2*b^3/(b*cos(d*x

+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^5+17/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2*b/(

a^2-2*a*b+b^2)*cos(d*x+c)^3-39/16/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/a/(a^2-2*a*b+b^2)*cos(d*x+c)^3*b^2

+37/32/d/a^2*b^3/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3-35/32/d/(b*cos(d*x+c)^4-

2*b*cos(d*x+c)^2-a+b)^2/(a-b)/a*b*cos(d*x+c)+15/32/d/a^2*b^2/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a-b)*cos

(d*x+c)+45/64/d/a/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))*b-

71/64/d*b^2/a^2/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))+1/2/

d/a^3*b^3/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))-1/4/d/a/(a

^2-2*a*b+b^2)/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))*b^2+5/32/d

*b^3/a^2/(a*b)^(1/2)/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))

-45/64/d/a/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))*b+71/64/d*

b^2/a^2/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-1/2/d/a^3*b^3

/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-1/4/d/a/(a^2-2*a*b+b

^2)/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))*b^2+5/32/d*b^3/a^2/(a

*b)^(1/2)/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))

________________________________________________________________________________________

maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

mupad [B] time = 20.83, size = 12247, normalized size = 19.85 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(c + d*x)*(a - b*sin(c + d*x)^4)^3),x)

[Out]

- ((5*cos(c + d*x)*(7*a*b - 3*b^2))/(32*a^2*(a - b)) - (cos(c + d*x)^3*(17*a^2*b - 78*a*b^2 + 37*b^3))/(32*a^2

*(a - b)^2) - (cos(c + d*x)^5*(53*a*b^2 - 29*b^3))/(32*a^2*(a - b)^2) + (b*cos(c + d*x)^7*(13*a*b - 7*b^2))/(3

2*a^2*(a - b)^2))/(d*(a^2 - 2*a*b + b^2 + cos(c + d*x)^2*(4*a*b - 4*b^2) - cos(c + d*x)^4*(2*a*b - 6*b^2) - 4*

b^2*cos(c + d*x)^6 + b^2*cos(c + d*x)^8)) - (atan((((((((192*a^11*b^9 - 990*a^12*b^8 + 2050*a^13*b^7 - 2154*a^

14*b^6 + 1158*a^15*b^5 - 256*a^16*b^4)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d

*x)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 1476395008*a^16*b^

5 - 268435456*a^17*b^4))/(2097152*a^3*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) + (cos(c

+ d*x)*(75497472*a^6*b^9 - 337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(1

048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) - (12*a^5*b^9 - (4311*a^6*b^8)/64 + (307

961*a^7*b^7)/2048 - (1290253*a^8*b^6)/8192 + (546059*a^9*b^5)/8192)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^

3 + 6*a^12*b^2)))*1i)/(2*a^3) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*

b^6 + 8247825*a^4*b^5)*1i)/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/a^3 - ((((((192*a^1

1*b^9 - 990*a^12*b^8 + 2050*a^13*b^7 - 2154*a^14*b^6 + 1158*a^15*b^5 - 256*a^16*b^4)/(2*(a^14 - 4*a^13*b + a^1

0*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) + (cos(c + d*x)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*

b^7 - 3221225472*a^15*b^6 + 1476395008*a^16*b^5 - 268435456*a^17*b^4))/(2097152*a^3*(a^12 - 4*a^11*b + a^8*b^4

- 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) - (cos(c + d*x)*(75497472*a^6*b^9 - 337215488*a^7*b^8 + 592748544*a^8*b^7

- 489406464*a^9*b^6 + 163684352*a^10*b^5))/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2

*a^3) - (12*a^5*b^9 - (4311*a^6*b^8)/64 + (307961*a^7*b^7)/2048 - (1290253*a^8*b^6)/8192 + (546059*a^9*b^5)/81

92)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*1i)/(2*a^3) + (cos(c + d*x)*(3145728*b^9 - 144

17920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6 + 8247825*a^4*b^5)*1i)/(1048576*(a^12 - 4*a^11*b + a^8*b^4 -

4*a^9*b^3 + 6*a^10*b^2)))/a^3)/((((((192*a^11*b^9 - 990*a^12*b^8 + 2050*a^13*b^7 - 2154*a^14*b^6 + 1158*a^15*

b^5 - 256*a^16*b^4)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d*x)*(402653184*a^12

*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 1476395008*a^16*b^5 - 268435456*a^17*

b^4))/(2097152*a^3*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) + (cos(c + d*x)*(75497472*a^

6*b^9 - 337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(1048576*(a^12 - 4*a^

11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) - (12*a^5*b^9 - (4311*a^6*b^8)/64 + (307961*a^7*b^7)/2048 -

(1290253*a^8*b^6)/8192 + (546059*a^9*b^5)/8192)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))/(

2*a^3) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6 + 8247825*a^4*b^5))

/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/a^3 + (((((192*a^11*b^9 - 990*a^12*b^8 + 2050

*a^13*b^7 - 2154*a^14*b^6 + 1158*a^15*b^5 - 256*a^16*b^4)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12

*b^2)) + (cos(c + d*x)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 +

1476395008*a^16*b^5 - 268435456*a^17*b^4))/(2097152*a^3*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2))

)/(2*a^3) - (cos(c + d*x)*(75497472*a^6*b^9 - 337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 1636

84352*a^10*b^5))/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) - (12*a^5*b^9 - (4311

*a^6*b^8)/64 + (307961*a^7*b^7)/2048 - (1290253*a^8*b^6)/8192 + (546059*a^9*b^5)/8192)/(2*(a^14 - 4*a^13*b + a

^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))/(2*a^3) + (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7

- 23076232*a^3*b^6 + 8247825*a^4*b^5))/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/a^3 + (

(90009*a*b^7)/32768 - (81*b^8)/128 - (271845*a^2*b^6)/65536 + (1184625*a^3*b^5)/524288)/(a^14 - 4*a^13*b + a^1

0*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*1i)/(a^3*d) - (atan(((((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 214958

0800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b + a^

10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d*x)*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*

a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(

a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 1

0*a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 14

76395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(202

5*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 93

06*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^

16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(75497472*a^6*b^9 - 3

37215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a^8

*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*

b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 44

29*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2

) - (12582912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(1048576*

(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) -

3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a

^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b

^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6

+ 8247825*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2)

+ 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b

^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^

5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2)*1i - (((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 21495

80800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b + a

^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) + (cos(c + d*x)*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465

*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*

(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 -

10*a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 1

476395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(20

25*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9

306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a

^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(75497472*a^6*b^9 -

337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a^

8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6

*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4

429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/

2) - (12582912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(1048576

*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2)

- 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*

a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*

b^3 - 10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6

+ 8247825*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2)

+ 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*

b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b

^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2)*1i)/((((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 2149

580800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b +

a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d*x)*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 346

5*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b

*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 -

10*a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 +

1476395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2

025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 +

9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*

a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(75497472*a^6*b^9 -

337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a

^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^

6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) +

4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1

/2) - (12582912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(104857

6*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2)

- 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694

*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14

*b^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^

6 + 8247825*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2

) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a

*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*

b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 214958

0800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b + a^

10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) + (cos(c + d*x)*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*

a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(

a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 1

0*a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 14

76395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(202

5*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 93

06*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^

16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(75497472*a^6*b^9 - 3