3.201 \(\int \frac{\csc ^3(x)}{(a+b \sin (x))^3} \, dx\)

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

[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]))

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Rubi [A]  time = 0.873067, antiderivative size = 241, normalized size of antiderivative = 1., 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 = {2802, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac{b^3 \left (-29 a^2 b^2+20 a^4+12 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{5/2}}+\frac{3 b \left (-7 a^2 b^2+2 a^4+4 b^4\right ) \cot (x)}{2 a^4 \left (a^2-b^2\right )^2}-\frac{\left (a^2+12 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^5}-\frac{\left (-10 a^2 b^2+a^4+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac{b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac{b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2} \]

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 2802

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(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 3055

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] + Dis
t[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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

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 3770

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

Rule 2660

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 618

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 1.88916, size = 220, normalized size = 0.91 \[ \frac{-\frac{8 b^3 \left (-29 a^2 b^2+20 a^4+12 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+4 \left (a^2+12 b^2\right ) \log \left (\sin \left (\frac{x}{2}\right )\right )-4 \left (a^2+12 b^2\right ) \log \left (\cos \left (\frac{x}{2}\right )\right )+\frac{12 a b^4 \left (2 b^2-3 a^2\right ) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-\frac{4 a^2 b^4 \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}-a^2 \csc ^2\left (\frac{x}{2}\right )+a^2 \sec ^2\left (\frac{x}{2}\right )-12 a b \tan \left (\frac{x}{2}\right )+12 a b \cot \left (\frac{x}{2}\right )}{8 a^5} \]

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 [B]  time = 0.088, size = 686, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a^3*tan(1/2*x)^2-3/2/a^4*tan(1/2*x)*b-11/a^2*b^5/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*t
an(1/2*x)^3+8/a^4*b^7/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^3-10/a*b^4/(tan(1/2*x
)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2-13/a^3*b^6/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4
-2*a^2*b^2+b^4)*tan(1/2*x)^2+14/a^5*b^8/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2-2
9/a^2*b^5/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)+20/a^4*b^7/(tan(1/2*x)^2*a+2*tan(
1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)-10/a*b^4/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)+
7/a^3*b^6/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)-20/a*b^3/(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))+29/a^3*b^5/(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))-12/a^5*b^7/(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))-1/8/a^3/tan(1/2*x)^2+1/2/a^3*ln(tan(1/2*x))+6/a^5*ln(tan(1/2*x))*b^2+3/2*b/a^4/tan(1/2*x
)

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

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

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Fricas [B]  time = 15.1431, size = 4528, normalized size = 18.79 \begin{align*} \text{result too large to display} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{3}{\left (x \right )}}{\left (a + b \sin{\left (x \right )}\right )^{3}}\, dx \end{align*}

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]  time = 1.64417, size = 694, normalized size = 2.88 \begin{align*} \text{result too large to display} \end{align*}

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