\(\int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx\) [199]

Optimal result

Integrand size = 11, antiderivative size = 145 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2}}-\frac {\text {arctanh}(\cos (x))}{a^3}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \]

[Out]

-b*(6*a^4-5*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*x))/(a^2-b^2)^(1/2))/a^3/(a^2-b^2)^(5/2)-arctanh(cos(x))/a^3-1/
2*b^2*cos(x)/a/(a^2-b^2)/(a+b*sin(x))^2-1/2*b^2*(5*a^2-2*b^2)*cos(x)/a^2/(a^2-b^2)^2/(a+b*sin(x))

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {2881, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=-\frac {\text {arctanh}(\cos (x))}{a^3}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2}} \]

[In]

Int[Csc[x]/(a + b*Sin[x])^3,x]

[Out]

-((b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2))) - ArcTanh[
Cos[x]]/a^3 - (b^2*Cos[x])/(2*a*(a^2 - b^2)*(a + b*Sin[x])^2) - (b^2*(5*a^2 - 2*b^2)*Cos[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*} \text {integral}& = -\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {\int \frac {\csc (x) \left (2 \left (a^2-b^2\right )-2 a b \sin (x)+b^2 \sin ^2(x)\right )}{(a+b \sin (x))^2} \, dx}{2 a \left (a^2-b^2\right )} \\ & = -\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (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-a b \left (4 a^2-b^2\right ) \sin (x)\right )}{a+b \sin (x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2} \\ & = -\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \csc (x) \, dx}{a^3}-\frac {\left (b \left (6 a^4-5 a^2 b^2+2 b^4\right )\right ) \int \frac {1}{a+b \sin (x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2} \\ & = -\frac {\text {arctanh}(\cos (x))}{a^3}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (b \left (6 a^4-5 a^2 b^2+2 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^3 \left (a^2-b^2\right )^2} \\ & = -\frac {\text {arctanh}(\cos (x))}{a^3}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (2 b \left (6 a^4-5 a^2 b^2+2 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^3 \left (a^2-b^2\right )^2} \\ & = -\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2}}-\frac {\text {arctanh}(\cos (x))}{a^3}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=-\frac {\frac {2 b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+2 \log \left (\cos \left (\frac {x}{2}\right )\right )-2 \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {a b^2 \cos (x) \left (6 a^3-3 a b^2+b \left (5 a^2-2 b^2\right ) \sin (x)\right )}{(a-b)^2 (a+b)^2 (a+b \sin (x))^2}}{2 a^3} \]

[In]

Integrate[Csc[x]/(a + b*Sin[x])^3,x]

[Out]

-1/2*((2*b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 2*Log[Cos
[x/2]] - 2*Log[Sin[x/2]] + (a*b^2*Cos[x]*(6*a^3 - 3*a*b^2 + b*(5*a^2 - 2*b^2)*Sin[x]))/((a - b)^2*(a + b)^2*(a
+ b*Sin[x])^2))/a^3

Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.86

method	result	size
default	\(-\frac {2 b \left (\frac {\frac {a \,b^{2} \left (7 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}+\frac {3 b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {a \,b^{2} \left (17 a^{2}-8 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}+\frac {3 a^{2} b \left (2 a^{2}-b^{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )}^{2}}+\frac {\left (6 a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{a^{3}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\)	\(270\)
risch	\(\frac {i b \left (-4 i a^{3} b \,{\mathrm e}^{3 i x}+i a \,b^{3} {\mathrm e}^{3 i x}+16 i a^{3} b \,{\mathrm e}^{i x}-7 i a \,b^{3} {\mathrm e}^{i x}+10 a^{4} {\mathrm e}^{2 i x}+a^{2} b^{2} {\mathrm e}^{2 i x}-2 b^{4} {\mathrm e}^{2 i x}-5 a^{2} b^{2}+2 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right )^{2} a^{2} \left (a^{2}-b^{2}\right )^{2}}+\frac {3 i a b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {5 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a}+\frac {i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}-\frac {3 i a b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {5 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a}-\frac {i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a^{3}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a^{3}}\)	\(623\)

[In]

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

[Out]

-2*b/a^3*((1/2*a*b^2*(7*a^2-4*b^2)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^3+3/2*b*(2*a^4+3*a^2*b^2-2*b^4)/(a^4-2*a^2*b
^2+b^4)*tan(1/2*x)^2+1/2*a*b^2*(17*a^2-8*b^2)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)+3/2*a^2*b*(2*a^2-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+1/2*(6*a^4-5*a^2*b^2+2*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)))+1/a^3*ln(tan(1/2*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (135) = 270\).

Time = 0.83 (sec) , antiderivative size = 1027, normalized size of antiderivative = 7.08 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=\int \frac {\csc {\left (x \right )}}{\left (a + b \sin {\left (x \right )}\right )^{3}}\, dx \]

[In]

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

[Out]

Integral(csc(x)/(a + b*sin(x))**3, x)

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.70 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=-\frac {{\left (6 \, a^{4} b - 5 \, a^{2} b^{3} + 2 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {7 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} - 4 \, a b^{5} \tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} - 6 \, b^{6} \tan \left (\frac {1}{2} \, x\right )^{2} + 17 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, x\right ) - 8 \, a b^{5} \tan \left (\frac {1}{2} \, x\right ) + 6 \, a^{4} b^{2} - 3 \, a^{2} b^{4}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2}} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} \]

[In]

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

[Out]

-(6*a^4*b - 5*a^2*b^3 + 2*b^5)*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))/
((a^7 - 2*a^5*b^2 + a^3*b^4)*sqrt(a^2 - b^2)) - (7*a^3*b^3*tan(1/2*x)^3 - 4*a*b^5*tan(1/2*x)^3 + 6*a^4*b^2*tan
(1/2*x)^2 + 9*a^2*b^4*tan(1/2*x)^2 - 6*b^6*tan(1/2*x)^2 + 17*a^3*b^3*tan(1/2*x) - 8*a*b^5*tan(1/2*x) + 6*a^4*b
^2 - 3*a^2*b^4)/((a^7 - 2*a^5*b^2 + a^3*b^4)*(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)^2) + log(abs(tan(1/2*x)))/a
^3

Mupad [B] (verification not implemented)

Time = 11.48 (sec) , antiderivative size = 2191, normalized size of antiderivative = 15.11 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=\text {Too large to display} \]

[In]

int(1/(sin(x)*(a + b*sin(x))^3),x)

[Out]

((3*(b^4 - 2*a^2*b^2))/(a*(a^4 + b^4 - 2*a^2*b^2)) - (3*tan(x/2)^2*(3*a^2*b^4 - 2*b^6 + 2*a^4*b^2))/(a^3*(a^4
+ b^4 - 2*a^2*b^2)) + (tan(x/2)*(8*b^5 - 17*a^2*b^3))/(a^2*(a^4 + b^4 - 2*a^2*b^2)) + (b*tan(x/2)^3*(4*b^4 - 7
*a^2*b^2))/(a^2*(a^4 + b^4 - 2*a^2*b^2)))/(tan(x/2)^2*(2*a^2 + 4*b^2) + a^2 + a^2*tan(x/2)^4 + 4*a*b*tan(x/2)
+ 4*a*b*tan(x/2)^3) + log(tan(x/2))/a^3 + (b*atan(((b*(-(a + b)^5*(a - b)^5)^(1/2)*((8*a^7*b + 4*a^3*b^5 - 9*a
^5*b^3)/(a^8 + a^4*b^4 - 2*a^6*b^2) + (tan(x/2)*(8*a*b^10 - 2*a^11 - 36*a^3*b^8 + 68*a^5*b^6 - 62*a^7*b^4 + 24
*a^9*b^2))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) - (b*((2*a^10*b + 2*a^6*b^5 - 4*a^8*b^3)/(a^8
+ a^4*b^4 - 2*a^6*b^2) - (tan(x/2)*(6*a^14 - 8*a^4*b^10 + 38*a^6*b^8 - 72*a^8*b^6 + 68*a^10*b^4 - 32*a^12*b^2)
)/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(6*a^4 + 2*b^4 - 5*a^2*b^
2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*(6*a^4 + 2*b^4 - 5*a^2*b^2)*1i)/
(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)) + (b*(-(a + b)^5*(a - b)^5)^(1/2)*((8
*a^7*b + 4*a^3*b^5 - 9*a^5*b^3)/(a^8 + a^4*b^4 - 2*a^6*b^2) + (tan(x/2)*(8*a*b^10 - 2*a^11 - 36*a^3*b^8 + 68*a
^5*b^6 - 62*a^7*b^4 + 24*a^9*b^2))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) + (b*((2*a^10*b + 2*a^
6*b^5 - 4*a^8*b^3)/(a^8 + a^4*b^4 - 2*a^6*b^2) - (tan(x/2)*(6*a^14 - 8*a^4*b^10 + 38*a^6*b^8 - 72*a^8*b^6 + 68
*a^10*b^4 - 32*a^12*b^2))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(
6*a^4 + 2*b^4 - 5*a^2*b^2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*(6*a^4 +
2*b^4 - 5*a^2*b^2)*1i)/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))/((2*(6*a^4*b
+ 2*b^5 - 5*a^2*b^3))/(a^8 + a^4*b^4 - 2*a^6*b^2) + (2*tan(x/2)*(2*b^8 - 13*a^2*b^6 + 26*a^4*b^4 - 24*a^6*b^2
))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) - (b*(-(a + b)^5*(a - b)^5)^(1/2)*((8*a^7*b + 4*a^3*b^
5 - 9*a^5*b^3)/(a^8 + a^4*b^4 - 2*a^6*b^2) + (tan(x/2)*(8*a*b^10 - 2*a^11 - 36*a^3*b^8 + 68*a^5*b^6 - 62*a^7*b
^4 + 24*a^9*b^2))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) - (b*((2*a^10*b + 2*a^6*b^5 - 4*a^8*b^3
)/(a^8 + a^4*b^4 - 2*a^6*b^2) - (tan(x/2)*(6*a^14 - 8*a^4*b^10 + 38*a^6*b^8 - 72*a^8*b^6 + 68*a^10*b^4 - 32*a^
12*b^2))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(6*a^4 + 2*b^4 - 5
*a^2*b^2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*(6*a^4 + 2*b^4 - 5*a^2*b^
2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)) + (b*(-(a + b)^5*(a - b)^5)^(1/2)
*((8*a^7*b + 4*a^3*b^5 - 9*a^5*b^3)/(a^8 + a^4*b^4 - 2*a^6*b^2) + (tan(x/2)*(8*a*b^10 - 2*a^11 - 36*a^3*b^8 +
68*a^5*b^6 - 62*a^7*b^4 + 24*a^9*b^2))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) + (b*((2*a^10*b +
2*a^6*b^5 - 4*a^8*b^3)/(a^8 + a^4*b^4 - 2*a^6*b^2) - (tan(x/2)*(6*a^14 - 8*a^4*b^10 + 38*a^6*b^8 - 72*a^8*b^6
+ 68*a^10*b^4 - 32*a^12*b^2))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2))*(-(a + b)^5*(a - b)^5)^(1/
2)*(6*a^4 + 2*b^4 - 5*a^2*b^2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*(6*a
^4 + 2*b^4 - 5*a^2*b^2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))))*(-(a + b)^
5*(a - b)^5)^(1/2)*(6*a^4 + 2*b^4 - 5*a^2*b^2)*1i)/(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*
a^11*b^2)