\(\int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx\) [27]

Optimal result

Integrand size = 16, antiderivative size = 184 \[ \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {3 b \text {arctanh}(\cos (x))}{a^4}-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 a^2 \sqrt {a^2+b^2}}-\frac {2 b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^4}-\frac {\csc (x)}{a^3}-\frac {b \cos (x)-a \sin (x)}{2 a^2 (a \cos (x)+b \sin (x))^2}-\frac {2 b}{a^3 (a \cos (x)+b \sin (x))} \] Output:

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

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.05 \[ \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {\csc ^3(x) (a \cos (x)+b \sin (x)) \left (a \left (a^2+b^2\right ) \sin (x)-5 a b (a \cos (x)+b \sin (x))+\frac {6 \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right ) (a \cos (x)+b \sin (x))^2}{\sqrt {a^2+b^2}}-a \cot \left (\frac {x}{2}\right ) (a \cos (x)+b \sin (x))^2+6 b \log \left (\cos \left (\frac {x}{2}\right )\right ) (a \cos (x)+b \sin (x))^2-6 b \log \left (\sin \left (\frac {x}{2}\right )\right ) (a \cos (x)+b \sin (x))^2-a (a \cos (x)+b \sin (x))^2 \tan \left (\frac {x}{2}\right )\right )}{2 a^4 (b+a \cot (x))^3} \] Input:

Integrate[Csc[x]^2/(a*Cos[x] + b*Sin[x])^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.16, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3584, 3042, 3555, 3042, 3553, 219, 3572, 3042, 3553, 219, 3582, 3042, 3553, 219, 4257}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))^3}dx\)

\(\Big \downarrow \) 3584

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{(a \cos (x)+b \sin (x))^3}dx}{a^2}+\frac {\int \frac {\csc ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2}-\frac {2 b \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{(a \cos (x)+b \sin (x))^3}dx}{a^2}-\frac {2 b \int \frac {1}{\sin (x) (a \cos (x)+b \sin (x))^2}dx}{a^2}+\frac {\int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}\)

\(\Big \downarrow \) 3555

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \frac {1}{a \cos (x)+b \sin (x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}-\frac {2 b \int \frac {1}{\sin (x) (a \cos (x)+b \sin (x))^2}dx}{a^2}+\frac {\int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \frac {1}{a \cos (x)+b \sin (x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}-\frac {2 b \int \frac {1}{\sin (x) (a \cos (x)+b \sin (x))^2}dx}{a^2}+\frac {\int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}\)

\(\Big \downarrow \) 3553

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {\int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{2 \left (a^2+b^2\right )}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}-\frac {2 b \int \frac {1}{\sin (x) (a \cos (x)+b \sin (x))^2}dx}{a^2}+\frac {\int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 b \int \frac {1}{\sin (x) (a \cos (x)+b \sin (x))^2}dx}{a^2}+\frac {\int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}\)

\(\Big \downarrow \) 3572

\(\displaystyle \frac {\int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}-\frac {2 b \left (-\frac {b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2}+\frac {\int \csc (x)dx}{a^2}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}-\frac {2 b \left (-\frac {b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2}+\frac {\int \csc (x)dx}{a^2}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}\)

\(\Big \downarrow \) 3553

\(\displaystyle -\frac {2 b \left (\frac {b \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2}+\frac {\int \csc (x)dx}{a^2}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}+\frac {\int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 b \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}+\frac {\int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}\)

\(\Big \downarrow \) 3582

\(\displaystyle -\frac {2 b \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2}-\frac {b \int \csc (x)dx}{a^2}-\frac {\csc (x)}{a}}{a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2}-\frac {b \int \csc (x)dx}{a^2}-\frac {\csc (x)}{a}}{a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}\)

\(\Big \downarrow \) 3553

\(\displaystyle -\frac {2 b \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}+\frac {-\frac {\left (a^2+b^2\right ) \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2}-\frac {b \int \csc (x)dx}{a^2}-\frac {\csc (x)}{a}}{a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 b \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}+\frac {-\frac {b \int \csc (x)dx}{a^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2}-\frac {\csc (x)}{a}}{a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\right )}{a^2}-\frac {2 b \left (\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}-\frac {\text {arctanh}(\cos (x))}{a^2}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}+\frac {-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2}+\frac {b \text {arctanh}(\cos (x))}{a^2}-\frac {\csc (x)}{a}}{a^2}\)

Input:

Int[Csc[x]^2/(a*Cos[x] + b*Sin[x])^3,x]
 

Output:

((b*ArcTanh[Cos[x]])/a^2 - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*Sin[x])/ 
Sqrt[a^2 + b^2]])/a^2 - Csc[x]/a)/a^2 + ((a^2 + b^2)*(-1/2*ArcTanh[(b*Cos[ 
x] - a*Sin[x])/Sqrt[a^2 + b^2]]/(a^2 + b^2)^(3/2) - (b*Cos[x] - a*Sin[x])/ 
(2*(a^2 + b^2)*(a*Cos[x] + b*Sin[x])^2)))/a^2 - (2*b*(-(ArcTanh[Cos[x]]/a^ 
2) + (b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^ 
2]) + 1/(a*(a*Cos[x] + b*Sin[x]))))/a^2
 

Defintions of rubi rules used

rule 219

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3553

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x 
_Symbol] :> Simp[-d^(-1)   Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + 
d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
 

rule 3555

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x 
_Symbol] :> Simp[(b*Cos[c + d*x] - a*Sin[c + d*x])*((a*Cos[c + d*x] + b*Sin 
[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[(n + 2)/((n + 1)*(a^ 
2 + b^2))   Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && NeQ[n, -2]
 

rule 3572

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_)/si 
n[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-(a*Cos[c + d*x] + b*Sin[c + d*x]) 
^(n + 1)/(a*d*(n + 1)), x] + (Simp[1/a^2   Int[(a*Cos[c + d*x] + b*Sin[c + 
d*x])^(n + 2)/Sin[c + d*x], x], x] - Simp[b/a^2   Int[(a*Cos[c + d*x] + b*S 
in[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0 
] && LtQ[n, -1]
 

rule 3582

Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin 
[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[Sin[c + d*x]^(m + 1)/(a*d*(m + 1)) 
, x] + (-Simp[b/a^2   Int[Sin[c + d*x]^(m + 1), x], x] + Simp[(a^2 + b^2)/a 
^2   Int[Sin[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; 
 FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
 

rule 3584

Int[sin[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin 
[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a^2 + b^2)/a^2   Int[Sin[c + 
 d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^n, x], x] + (Simp[1/a^2   I 
nt[Sin[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] - Simp[ 
2*(b/a^2)   Int[Sin[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 
 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] & 
& LtQ[m, -1]
 

rule 4257

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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {\tan \left (\frac {x}{2}\right )}{2 a^{3}}-\frac {4 \left (\frac {-\frac {a \left (a^{2}+6 b^{2}\right ) \tan \left (\frac {x}{2}\right )^{3}}{4}-\frac {5 b \left (a^{2}-2 b^{2}\right ) \tan \left (\frac {x}{2}\right )^{2}}{4}-\frac {a \left (a^{2}-14 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{4}+\frac {5 a^{2} b}{4}}{\left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )^{2}}-\frac {3 \left (a^{2}+2 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{4 \sqrt {a^{2}+b^{2}}}\right )}{a^{4}}-\frac {1}{2 a^{3} \tan \left (\frac {x}{2}\right )}-\frac {3 b \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{4}}\)	\(163\)
risch	\(-\frac {i {\mathrm e}^{i x} \left (9 i a b \,{\mathrm e}^{4 i x}-3 a^{2} {\mathrm e}^{4 i x}+6 b^{2} {\mathrm e}^{4 i x}-2 a^{2} {\mathrm e}^{2 i x}-12 b^{2} {\mathrm e}^{2 i x}-9 i a b -3 a^{2}+6 b^{2}\right )}{\left ({\mathrm e}^{2 i x}-1\right ) \left (b \,{\mathrm e}^{2 i x}+i a \,{\mathrm e}^{2 i x}-b +i a \right )^{2} a^{3}}-\frac {3 \ln \left ({\mathrm e}^{i x}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}\, a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i x}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right ) b^{2}}{\sqrt {a^{2}+b^{2}}\, a^{4}}+\frac {3 \ln \left ({\mathrm e}^{i x}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}\, a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i x}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right ) b^{2}}{\sqrt {a^{2}+b^{2}}\, a^{4}}+\frac {3 b \ln \left ({\mathrm e}^{i x}+1\right )}{a^{4}}-\frac {3 b \ln \left ({\mathrm e}^{i x}-1\right )}{a^{4}}\)	\(322\)

Input:

int(csc(x)^2/(a*cos(x)+b*sin(x))^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*tan(1/2*x)/a^3-4/a^4*((-1/4*a*(a^2+6*b^2)*tan(1/2*x)^3-5/4*b*(a^2-2*b 
^2)*tan(1/2*x)^2-1/4*a*(a^2-14*b^2)*tan(1/2*x)+5/4*a^2*b)/(tan(1/2*x)^2*a- 
2*b*tan(1/2*x)-a)^2-3/4*(a^2+2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1 
/2*x)-2*b)/(a^2+b^2)^(1/2)))-1/2/a^3/tan(1/2*x)-3/a^4*b*ln(tan(1/2*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (168) = 336\).

Time = 0.13 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.52 \[ \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {2 \, a^{5} - 10 \, a^{3} b^{2} - 12 \, a b^{4} - 6 \, {\left (a^{5} - a^{3} b^{2} - 2 \, a b^{4}\right )} \cos \left (x\right )^{2} - 18 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) - 3 \, {\left (2 \, {\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (x\right )^{3} - 2 \, {\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (x\right ) - {\left (a^{2} b^{2} + 2 \, b^{4} + {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) - 6 \, {\left (2 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} - 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right ) - {\left (a^{2} b^{3} + b^{5} + {\left (a^{4} b - b^{5}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 6 \, {\left (2 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} - 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right ) - {\left (a^{2} b^{3} + b^{5} + {\left (a^{4} b - b^{5}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{4 \, {\left (2 \, {\left (a^{7} b + a^{5} b^{3}\right )} \cos \left (x\right )^{3} - 2 \, {\left (a^{7} b + a^{5} b^{3}\right )} \cos \left (x\right ) - {\left (a^{6} b^{2} + a^{4} b^{4} + {\left (a^{8} - a^{4} b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )}} \] Input:

integrate(csc(x)^2/(a*cos(x)+b*sin(x))^3,x, algorithm="fricas")
 

Output:

-1/4*(2*a^5 - 10*a^3*b^2 - 12*a*b^4 - 6*(a^5 - a^3*b^2 - 2*a*b^4)*cos(x)^2 
 - 18*(a^4*b + a^2*b^3)*cos(x)*sin(x) - 3*(2*(a^3*b + 2*a*b^3)*cos(x)^3 - 
2*(a^3*b + 2*a*b^3)*cos(x) - (a^2*b^2 + 2*b^4 + (a^4 + a^2*b^2 - 2*b^4)*co 
s(x)^2)*sin(x))*sqrt(a^2 + b^2)*log(-(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*co 
s(x)^2 - 2*a^2 - b^2 + 2*sqrt(a^2 + b^2)*(b*cos(x) - a*sin(x)))/(2*a*b*cos 
(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2)) - 6*(2*(a^3*b^2 + a*b^4)*cos(x)^ 
3 - 2*(a^3*b^2 + a*b^4)*cos(x) - (a^2*b^3 + b^5 + (a^4*b - b^5)*cos(x)^2)* 
sin(x))*log(1/2*cos(x) + 1/2) + 6*(2*(a^3*b^2 + a*b^4)*cos(x)^3 - 2*(a^3*b 
^2 + a*b^4)*cos(x) - (a^2*b^3 + b^5 + (a^4*b - b^5)*cos(x)^2)*sin(x))*log( 
-1/2*cos(x) + 1/2))/(2*(a^7*b + a^5*b^3)*cos(x)^3 - 2*(a^7*b + a^5*b^3)*co 
s(x) - (a^6*b^2 + a^4*b^4 + (a^8 - a^4*b^4)*cos(x)^2)*sin(x))
 

Sympy [F]

\[ \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\int \frac {\csc ^{2}{\left (x \right )}}{\left (a \cos {\left (x \right )} + b \sin {\left (x \right )}\right )^{3}}\, dx \] Input:

integrate(csc(x)**2/(a*cos(x)+b*sin(x))**3,x)
 

Output:

Integral(csc(x)**2/(a*cos(x) + b*sin(x))**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.50 \[ \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {a^{3} + \frac {14 \, a^{2} b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {4 \, {\left (a^{3} - 8 \, a b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {2 \, {\left (7 \, a^{2} b - 10 \, b^{3}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {{\left (a^{3} + 12 \, a b^{2}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}}{2 \, {\left (\frac {a^{6} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {4 \, a^{5} b \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {4 \, a^{5} b \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{6} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {2 \, {\left (a^{6} - 2 \, a^{4} b^{2}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} - \frac {3 \, b \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{4}} - \frac {3 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{2 \, \sqrt {a^{2} + b^{2}} a^{4}} - \frac {\sin \left (x\right )}{2 \, a^{3} {\left (\cos \left (x\right ) + 1\right )}} \] Input:

integrate(csc(x)^2/(a*cos(x)+b*sin(x))^3,x, algorithm="maxima")
 

Output:

-1/2*(a^3 + 14*a^2*b*sin(x)/(cos(x) + 1) - 4*(a^3 - 8*a*b^2)*sin(x)^2/(cos 
(x) + 1)^2 - 2*(7*a^2*b - 10*b^3)*sin(x)^3/(cos(x) + 1)^3 - (a^3 + 12*a*b^ 
2)*sin(x)^4/(cos(x) + 1)^4)/(a^6*sin(x)/(cos(x) + 1) + 4*a^5*b*sin(x)^2/(c 
os(x) + 1)^2 - 4*a^5*b*sin(x)^4/(cos(x) + 1)^4 + a^6*sin(x)^5/(cos(x) + 1) 
^5 - 2*(a^6 - 2*a^4*b^2)*sin(x)^3/(cos(x) + 1)^3) - 3*b*log(sin(x)/(cos(x) 
 + 1))/a^4 - 3/2*(a^2 + 2*b^2)*log((b - a*sin(x)/(cos(x) + 1) + sqrt(a^2 + 
 b^2))/(b - a*sin(x)/(cos(x) + 1) - sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^4 
) - 1/2*sin(x)/(a^3*(cos(x) + 1))
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {3 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{4}} - \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, a^{3}} - \frac {3 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, \sqrt {a^{2} + b^{2}} a^{4}} + \frac {6 \, b \tan \left (\frac {1}{2} \, x\right ) - a}{2 \, a^{4} \tan \left (\frac {1}{2} \, x\right )} + \frac {a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 5 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{2} - 10 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{3} \tan \left (\frac {1}{2} \, x\right ) - 14 \, a b^{2} \tan \left (\frac {1}{2} \, x\right ) - 5 \, a^{2} b}{{\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )}^{2} a^{4}} \] Input:

integrate(csc(x)^2/(a*cos(x)+b*sin(x))^3,x, algorithm="giac")
 

Output:

-3*b*log(abs(tan(1/2*x)))/a^4 - 1/2*tan(1/2*x)/a^3 - 3/2*(a^2 + 2*b^2)*log 
(abs(2*a*tan(1/2*x) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*x) - 2*b + 
2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^4) + 1/2*(6*b*tan(1/2*x) - a)/(a^4* 
tan(1/2*x)) + (a^3*tan(1/2*x)^3 + 6*a*b^2*tan(1/2*x)^3 + 5*a^2*b*tan(1/2*x 
)^2 - 10*b^3*tan(1/2*x)^2 + a^3*tan(1/2*x) - 14*a*b^2*tan(1/2*x) - 5*a^2*b 
)/((a*tan(1/2*x)^2 - 2*b*tan(1/2*x) - a)^2*a^4)
 

Mupad [B] (verification not implemented)

Time = 17.05 (sec) , antiderivative size = 813, normalized size of antiderivative = 4.42 \[ \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx =\text {Too large to display} \] Input:

int(1/(sin(x)^2*(a*cos(x) + b*sin(x))^3),x)
 

Output:

(tan(x/2)^4*(a^2 + 12*b^2) + tan(x/2)^2*(4*a^2 - 32*b^2) - a^2 - 14*a*b*ta 
n(x/2) + (2*tan(x/2)^3*(7*a^2*b - 10*b^3))/a)/(2*a^5*tan(x/2) - tan(x/2)^3 
*(4*a^5 - 8*a^3*b^2) + 2*a^5*tan(x/2)^5 + 8*a^4*b*tan(x/2)^2 - 8*a^4*b*tan 
(x/2)^4) - tan(x/2)/(2*a^3) - (3*b*log(tan(x/2)))/a^4 - (atan((((a^2 + 2*b 
^2)*(a^2 + b^2)^(1/2)*((3*a^6 + 12*a^4*b^2)/a^6 + (tan(x/2)*(12*a^4*b + 24 
*a^2*b^3))/a^5 - (3*(a^2 + 2*b^2)*(2*a^2*b + (tan(x/2)*(6*a^8 + 8*a^6*b^2) 
)/a^5)*(a^2 + b^2)^(1/2))/(2*(a^6 + a^4*b^2)))*3i)/(2*(a^6 + a^4*b^2)) + ( 
(a^2 + 2*b^2)*(a^2 + b^2)^(1/2)*((3*a^6 + 12*a^4*b^2)/a^6 + (tan(x/2)*(12* 
a^4*b + 24*a^2*b^3))/a^5 + (3*(a^2 + 2*b^2)*(2*a^2*b + (tan(x/2)*(6*a^8 + 
8*a^6*b^2))/a^5)*(a^2 + b^2)^(1/2))/(2*(a^6 + a^4*b^2)))*3i)/(2*(a^6 + a^4 
*b^2)))/((2*(9*a^2*b + 18*b^3))/a^6 - (2*tan(x/2)*(9*a^2 + 18*b^2))/a^5 - 
(3*(a^2 + 2*b^2)*(a^2 + b^2)^(1/2)*((3*a^6 + 12*a^4*b^2)/a^6 + (tan(x/2)*( 
12*a^4*b + 24*a^2*b^3))/a^5 - (3*(a^2 + 2*b^2)*(2*a^2*b + (tan(x/2)*(6*a^8 
 + 8*a^6*b^2))/a^5)*(a^2 + b^2)^(1/2))/(2*(a^6 + a^4*b^2))))/(2*(a^6 + a^4 
*b^2)) + (3*(a^2 + 2*b^2)*(a^2 + b^2)^(1/2)*((3*a^6 + 12*a^4*b^2)/a^6 + (t 
an(x/2)*(12*a^4*b + 24*a^2*b^3))/a^5 + (3*(a^2 + 2*b^2)*(2*a^2*b + (tan(x/ 
2)*(6*a^8 + 8*a^6*b^2))/a^5)*(a^2 + b^2)^(1/2))/(2*(a^6 + a^4*b^2))))/(2*( 
a^6 + a^4*b^2))))*(a^2 + 2*b^2)*(a^2 + b^2)^(1/2)*3i)/(a^6 + a^4*b^2)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 643, normalized size of antiderivative = 3.49 \[ \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx =\text {Too large to display} \] Input:

int(csc(x)^2/(a*cos(x)+b*sin(x))^3,x)
 

Output:

( - 48*sqrt(a**2 + b**2)*atan((tan(x/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos( 
x)*sin(x)**2*a**3*b**2*i - 96*sqrt(a**2 + b**2)*atan((tan(x/2)*a*i - b*i)/ 
sqrt(a**2 + b**2))*cos(x)*sin(x)**2*a*b**4*i + 24*sqrt(a**2 + b**2)*atan(( 
tan(x/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(x)**3*a**4*b*i + 24*sqrt(a**2 + 
 b**2)*atan((tan(x/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(x)**3*a**2*b**3*i 
- 48*sqrt(a**2 + b**2)*atan((tan(x/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(x) 
**3*b**5*i - 24*sqrt(a**2 + b**2)*atan((tan(x/2)*a*i - b*i)/sqrt(a**2 + b* 
*2))*sin(x)*a**4*b*i - 48*sqrt(a**2 + b**2)*atan((tan(x/2)*a*i - b*i)/sqrt 
(a**2 + b**2))*sin(x)*a**2*b**3*i - 48*cos(x)*log(tan(x/2))*sin(x)**2*a**3 
*b**3 - 48*cos(x)*log(tan(x/2))*sin(x)**2*a*b**5 + 6*cos(x)*sin(x)**2*a**5 
*b - 18*cos(x)*sin(x)**2*a**3*b**3 - 24*cos(x)*sin(x)**2*a*b**5 - 36*cos(x 
)*sin(x)*a**4*b**2 - 36*cos(x)*sin(x)*a**2*b**4 + 24*log(tan(x/2))*sin(x)* 
*3*a**4*b**2 - 24*log(tan(x/2))*sin(x)**3*b**6 - 24*log(tan(x/2))*sin(x)*a 
**4*b**2 - 24*log(tan(x/2))*sin(x)*a**2*b**4 - 3*sin(x)**3*a**6 + 12*sin(x 
)**3*a**4*b**2 + 3*sin(x)**3*a**2*b**4 - 12*sin(x)**3*b**6 + 12*sin(x)**2* 
a**5*b - 12*sin(x)**2*a**3*b**3 - 24*sin(x)**2*a*b**5 + 3*sin(x)*a**6 - 9* 
sin(x)*a**4*b**2 - 12*sin(x)*a**2*b**4 - 8*a**5*b - 8*a**3*b**3)/(8*sin(x) 
*a**4*b*(2*cos(x)*sin(x)*a**3*b + 2*cos(x)*sin(x)*a*b**3 - sin(x)**2*a**4 
+ sin(x)**2*b**4 + a**4 + a**2*b**2))