\(\int \frac {1}{(a+b \csc ^2(c+d x))^4} \, dx\) [8]

Optimal result

Integrand size = 14, antiderivative size = 204 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\frac {x}{a^4}+\frac {\sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b}}\right )}{16 a^4 (a+b)^{7/2} d}+\frac {b \cot (c+d x)}{6 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^3}+\frac {b (11 a+6 b) \cot (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{16 a^3 (a+b)^3 d \left (a+b+b \cot ^2(c+d x)\right )} \] Output:

x/a^4+1/16*b^(1/2)*(35*a^3+70*a^2*b+56*a*b^2+16*b^3)*arctan(b^(1/2)*cot(d* 
x+c)/(a+b)^(1/2))/a^4/(a+b)^(7/2)/d+1/6*b*cot(d*x+c)/a/(a+b)/d/(a+b+b*cot( 
d*x+c)^2)^3+1/24*b*(11*a+6*b)*cot(d*x+c)/a^2/(a+b)^2/d/(a+b+b*cot(d*x+c)^2 
)^2+1/16*b*(19*a^2+22*a*b+8*b^2)*cot(d*x+c)/a^3/(a+b)^3/d/(a+b+b*cot(d*x+c 
)^2)
 

Mathematica [A] (verified)

Time = 8.45 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\frac {(-a-2 b+a \cos (2 (c+d x))) \csc ^8(c+d x) \left (\frac {3 \sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {b}}\right ) (a+2 b-a \cos (2 (c+d x)))^3}{(a+b)^{7/2}}+48 (c+d x) (-a-2 b+a \cos (2 (c+d x)))^3-\frac {32 a b^3 \sin (2 (c+d x))}{a+b}-\frac {a b \left (87 a^2+116 a b+44 b^2\right ) (a+2 b-a \cos (2 (c+d x)))^2 \sin (2 (c+d x))}{(a+b)^3}-\frac {4 a b^2 (19 a+14 b) (-a-2 b+a \cos (2 (c+d x))) \sin (2 (c+d x))}{(a+b)^2}\right )}{768 a^4 d \left (a+b \csc ^2(c+d x)\right )^4} \] Input:

Integrate[(a + b*Csc[c + d*x]^2)^(-4),x]
 

Output:

((-a - 2*b + a*Cos[2*(c + d*x)])*Csc[c + d*x]^8*((3*Sqrt[b]*(35*a^3 + 70*a 
^2*b + 56*a*b^2 + 16*b^3)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[b]]*(a + 
2*b - a*Cos[2*(c + d*x)])^3)/(a + b)^(7/2) + 48*(c + d*x)*(-a - 2*b + a*Co 
s[2*(c + d*x)])^3 - (32*a*b^3*Sin[2*(c + d*x)])/(a + b) - (a*b*(87*a^2 + 1 
16*a*b + 44*b^2)*(a + 2*b - a*Cos[2*(c + d*x)])^2*Sin[2*(c + d*x)])/(a + b 
)^3 - (4*a*b^2*(19*a + 14*b)*(-a - 2*b + a*Cos[2*(c + d*x)])*Sin[2*(c + d* 
x)])/(a + b)^2))/(768*a^4*d*(a + b*Csc[c + d*x]^2)^4)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3042, 4616, 316, 402, 27, 402, 397, 216, 218}

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.

\(\displaystyle \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a+b \sec \left (c+d x+\frac {\pi }{2}\right )^2\right )^4}dx\)

\(\Big \downarrow \) 4616

\(\displaystyle -\frac {\int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a+b\right )^4}d\cot (c+d x)}{d}\)

\(\Big \downarrow \) 316

\(\displaystyle -\frac {\frac {\int \frac {-5 b \cot ^2(c+d x)+6 a+b}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a+b\right )^3}d\cot (c+d x)}{6 a (a+b)}-\frac {b \cot (c+d x)}{6 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^3}}{d}\)

\(\Big \downarrow \) 402

\(\displaystyle -\frac {\frac {\frac {\int \frac {3 \left (8 a^2+5 b a+2 b^2-b (11 a+6 b) \cot ^2(c+d x)\right )}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a+b\right )^2}d\cot (c+d x)}{4 a (a+b)}-\frac {b (11 a+6 b) \cot (c+d x)}{4 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \cot (c+d x)}{6 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^3}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {3 \int \frac {8 a^2+5 b a+2 b^2-b (11 a+6 b) \cot ^2(c+d x)}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a+b\right )^2}d\cot (c+d x)}{4 a (a+b)}-\frac {b (11 a+6 b) \cot (c+d x)}{4 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \cot (c+d x)}{6 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^3}}{d}\)

\(\Big \downarrow \) 402

\(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\int \frac {16 a^3+29 b a^2+26 b^2 a+8 b^3-b \left (19 a^2+22 b a+8 b^2\right ) \cot ^2(c+d x)}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a+b\right )}d\cot (c+d x)}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{2 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \cot (c+d x)}{4 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \cot (c+d x)}{6 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^3}}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \int \frac {1}{\cot ^2(c+d x)+1}d\cot (c+d x)}{a}-\frac {b \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \int \frac {1}{b \cot ^2(c+d x)+a+b}d\cot (c+d x)}{a}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{2 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \cot (c+d x)}{4 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \cot (c+d x)}{6 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^3}}{d}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \arctan (\cot (c+d x))}{a}-\frac {b \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \int \frac {1}{b \cot ^2(c+d x)+a+b}d\cot (c+d x)}{a}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{2 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \cot (c+d x)}{4 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \cot (c+d x)}{6 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^3}}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \arctan (\cot (c+d x))}{a}-\frac {\sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{2 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \cot (c+d x)}{4 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \cot (c+d x)}{6 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^3}}{d}\)

Input:

Int[(a + b*Csc[c + d*x]^2)^(-4),x]
 

Output:

-((-1/6*(b*Cot[c + d*x])/(a*(a + b)*(a + b + b*Cot[c + d*x]^2)^3) + (-1/4* 
(b*(11*a + 6*b)*Cot[c + d*x])/(a*(a + b)*(a + b + b*Cot[c + d*x]^2)^2) + ( 
3*(((16*(a + b)^3*ArcTan[Cot[c + d*x]])/a - (Sqrt[b]*(35*a^3 + 70*a^2*b + 
56*a*b^2 + 16*b^3)*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + 
 b]))/(2*a*(a + b)) - (b*(19*a^2 + 22*a*b + 8*b^2)*Cot[c + d*x])/(2*a*(a + 
 b)*(a + b + b*Cot[c + d*x]^2))))/(4*a*(a + b)))/(6*a*(a + b)))/d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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

rule 218

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

rule 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 3042

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

rule 4616

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = 
FreeFactors[Tan[e + f*x], x]}, Simp[ff/f   Subst[Int[(a + b + b*ff^2*x^2)^p 
/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] 
&& NeQ[a + b, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(\frac {-\frac {b \left (\frac {-\frac {\left (29 a^{2}+26 b a +8 b^{2}\right ) a \tan \left (d x +c \right )^{5}}{16 \left (a +b \right )}-\frac {\left (17 a^{2}+18 b a +6 b^{2}\right ) a b \tan \left (d x +c \right )^{3}}{6 \left (a^{2}+2 b a +b^{2}\right )}-\frac {b^{2} a \left (19 a^{2}+22 b a +8 b^{2}\right ) \tan \left (d x +c \right )}{16 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}}{\left (\tan \left (d x +c \right )^{2} a +\tan \left (d x +c \right )^{2} b +b \right )^{3}}+\frac {\left (35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {b \left (a +b \right )}}\right )}{16 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \sqrt {b \left (a +b \right )}}\right )}{a^{4}}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a^{4}}}{d}\)	\(240\)
default	\(\frac {-\frac {b \left (\frac {-\frac {\left (29 a^{2}+26 b a +8 b^{2}\right ) a \tan \left (d x +c \right )^{5}}{16 \left (a +b \right )}-\frac {\left (17 a^{2}+18 b a +6 b^{2}\right ) a b \tan \left (d x +c \right )^{3}}{6 \left (a^{2}+2 b a +b^{2}\right )}-\frac {b^{2} a \left (19 a^{2}+22 b a +8 b^{2}\right ) \tan \left (d x +c \right )}{16 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}}{\left (\tan \left (d x +c \right )^{2} a +\tan \left (d x +c \right )^{2} b +b \right )^{3}}+\frac {\left (35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {b \left (a +b \right )}}\right )}{16 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \sqrt {b \left (a +b \right )}}\right )}{a^{4}}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a^{4}}}{d}\)	\(240\)
risch	\(\text {Expression too large to display}\)	\(900\)

Input:

int(1/(a+b*csc(d*x+c)^2)^4,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-b/a^4*((-1/16*(29*a^2+26*a*b+8*b^2)*a/(a+b)*tan(d*x+c)^5-1/6*(17*a^2 
+18*a*b+6*b^2)*a*b/(a^2+2*a*b+b^2)*tan(d*x+c)^3-1/16*b^2*a*(19*a^2+22*a*b+ 
8*b^2)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(d*x+c))/(tan(d*x+c)^2*a+tan(d*x+c)^2* 
b+b)^3+1/16*(35*a^3+70*a^2*b+56*a*b^2+16*b^3)/(a^3+3*a^2*b+3*a*b^2+b^3)/(b 
*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(b*(a+b))^(1/2)))+1/a^4*arctan(tan(d 
*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (188) = 376\).

Time = 0.17 (sec) , antiderivative size = 1592, normalized size of antiderivative = 7.80 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*csc(d*x+c)^2)^4,x, algorithm="fricas")
 

Output:

[1/192*(192*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*x*cos(d*x + c)^6 - 576 
*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cos(d*x + c)^4 + 57 
6*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*d*x*cos(d* 
x + c)^2 - 192*(a^6 + 6*a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*a 
*b^5 + b^6)*d*x + 3*((35*a^6 + 70*a^5*b + 56*a^4*b^2 + 16*a^3*b^3)*cos(d*x 
 + c)^6 - 35*a^6 - 175*a^5*b - 371*a^4*b^2 - 429*a^3*b^3 - 286*a^2*b^4 - 1 
04*a*b^5 - 16*b^6 - 3*(35*a^6 + 105*a^5*b + 126*a^4*b^2 + 72*a^3*b^3 + 16* 
a^2*b^4)*cos(d*x + c)^4 + 3*(35*a^6 + 140*a^5*b + 231*a^4*b^2 + 198*a^3*b^ 
3 + 88*a^2*b^4 + 16*a*b^5)*cos(d*x + c)^2)*sqrt(-b/(a + b))*log(((a^2 + 8* 
a*b + 8*b^2)*cos(d*x + c)^4 - 2*(a^2 + 5*a*b + 4*b^2)*cos(d*x + c)^2 + 4*( 
(a^2 + 3*a*b + 2*b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*s 
qrt(-b/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(a^2*cos(d*x + c)^4 - 2* 
(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*((87*a^5*b + 116*a^4* 
b^2 + 44*a^3*b^3)*cos(d*x + c)^5 - 2*(87*a^5*b + 184*a^4*b^2 + 127*a^3*b^3 
 + 30*a^2*b^4)*cos(d*x + c)^3 + 3*(29*a^5*b + 84*a^4*b^2 + 89*a^3*b^3 + 42 
*a^2*b^4 + 8*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b 
^2 + a^7*b^3)*d*cos(d*x + c)^6 - 3*(a^10 + 4*a^9*b + 6*a^8*b^2 + 4*a^7*b^3 
 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 
 + 5*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 - (a^10 + 6*a^9*b + 15*a^8*b^2 + 
20*a^7*b^3 + 15*a^6*b^4 + 6*a^5*b^5 + a^4*b^6)*d), 1/96*(96*(a^6 + 3*a^...
 

Sympy [F]

\[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\int \frac {1}{\left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{4}}\, dx \] Input:

integrate(1/(a+b*csc(d*x+c)**2)**4,x)
 

Output:

Integral((a + b*csc(c + d*x)**2)**(-4), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (188) = 376\).

Time = 0.12 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=-\frac {\frac {3 \, {\left (35 \, a^{3} b + 70 \, a^{2} b^{2} + 56 \, a b^{3} + 16 \, b^{4}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt {{\left (a + b\right )} b}} - \frac {3 \, {\left (29 \, a^{4} b + 84 \, a^{3} b^{2} + 89 \, a^{2} b^{3} + 42 \, a b^{4} + 8 \, b^{5}\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (17 \, a^{3} b^{2} + 35 \, a^{2} b^{3} + 24 \, a b^{4} + 6 \, b^{5}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (19 \, a^{2} b^{3} + 22 \, a b^{4} + 8 \, b^{5}\right )} \tan \left (d x + c\right )}{a^{6} b^{3} + 3 \, a^{5} b^{4} + 3 \, a^{4} b^{5} + a^{3} b^{6} + {\left (a^{9} + 6 \, a^{8} b + 15 \, a^{7} b^{2} + 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} + 6 \, a^{4} b^{5} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (a^{8} b + 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} + 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{2}} - \frac {48 \, {\left (d x + c\right )}}{a^{4}}}{48 \, d} \] Input:

integrate(1/(a+b*csc(d*x+c)^2)^4,x, algorithm="maxima")
 

Output:

-1/48*(3*(35*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*arctan((a + b)*tan(d* 
x + c)/sqrt((a + b)*b))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt((a + b 
)*b)) - (3*(29*a^4*b + 84*a^3*b^2 + 89*a^2*b^3 + 42*a*b^4 + 8*b^5)*tan(d*x 
 + c)^5 + 8*(17*a^3*b^2 + 35*a^2*b^3 + 24*a*b^4 + 6*b^5)*tan(d*x + c)^3 + 
3*(19*a^2*b^3 + 22*a*b^4 + 8*b^5)*tan(d*x + c))/(a^6*b^3 + 3*a^5*b^4 + 3*a 
^4*b^5 + a^3*b^6 + (a^9 + 6*a^8*b + 15*a^7*b^2 + 20*a^6*b^3 + 15*a^5*b^4 + 
 6*a^4*b^5 + a^3*b^6)*tan(d*x + c)^6 + 3*(a^8*b + 5*a^7*b^2 + 10*a^6*b^3 + 
 10*a^5*b^4 + 5*a^4*b^5 + a^3*b^6)*tan(d*x + c)^4 + 3*(a^7*b^2 + 4*a^6*b^3 
 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*tan(d*x + c)^2) - 48*(d*x + c)/a^4)/d
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.74 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=-\frac {\frac {3 \, {\left (35 \, a^{3} b + 70 \, a^{2} b^{2} + 56 \, a b^{3} + 16 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt {a b + b^{2}}} - \frac {87 \, a^{4} b \tan \left (d x + c\right )^{5} + 252 \, a^{3} b^{2} \tan \left (d x + c\right )^{5} + 267 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 126 \, a b^{4} \tan \left (d x + c\right )^{5} + 24 \, b^{5} \tan \left (d x + c\right )^{5} + 136 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 280 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} + 192 \, a b^{4} \tan \left (d x + c\right )^{3} + 48 \, b^{5} \tan \left (d x + c\right )^{3} + 57 \, a^{2} b^{3} \tan \left (d x + c\right ) + 66 \, a b^{4} \tan \left (d x + c\right ) + 24 \, b^{5} \tan \left (d x + c\right )}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + b\right )}^{3}} - \frac {48 \, {\left (d x + c\right )}}{a^{4}}}{48 \, d} \] Input:

integrate(1/(a+b*csc(d*x+c)^2)^4,x, algorithm="giac")
 

Output:

-1/48*(3*(35*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*(pi*floor((d*x + c)/p 
i + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a* 
b + b^2)))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt(a*b + b^2)) - (87*a 
^4*b*tan(d*x + c)^5 + 252*a^3*b^2*tan(d*x + c)^5 + 267*a^2*b^3*tan(d*x + c 
)^5 + 126*a*b^4*tan(d*x + c)^5 + 24*b^5*tan(d*x + c)^5 + 136*a^3*b^2*tan(d 
*x + c)^3 + 280*a^2*b^3*tan(d*x + c)^3 + 192*a*b^4*tan(d*x + c)^3 + 48*b^5 
*tan(d*x + c)^3 + 57*a^2*b^3*tan(d*x + c) + 66*a*b^4*tan(d*x + c) + 24*b^5 
*tan(d*x + c))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*(a*tan(d*x + c)^2 + 
b*tan(d*x + c)^2 + b)^3) - 48*(d*x + c)/a^4)/d
 

Mupad [B] (verification not implemented)

Time = 19.92 (sec) , antiderivative size = 4447, normalized size of antiderivative = 21.80 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\text {Too large to display} \] Input:

int(1/(a + b/sin(c + d*x)^2)^4,x)
 

Output:

((tan(c + d*x)^5*(26*a*b^2 + 29*a^2*b + 8*b^3))/(16*(a^3*b + a^4)) + (tan( 
c + d*x)^3*(18*a*b^3 + 6*b^4 + 17*a^2*b^2))/(6*(2*a^4*b + a^5 + a^3*b^2)) 
+ (b^2*tan(c + d*x)*(22*a*b^2 + 19*a^2*b + 8*b^3))/(16*(3*a^5*b + a^6 + a^ 
3*b^3 + 3*a^4*b^2)))/(d*(tan(c + d*x)^6*(3*a*b^2 + 3*a^2*b + a^3 + b^3) + 
tan(c + d*x)^2*(3*a*b^2 + 3*b^3) + tan(c + d*x)^4*(6*a*b^2 + 3*a^2*b + 3*b 
^3) + b^3)) - atan(((((((19*a^14*b)/4 + 2*a^8*b^7 + (27*a^9*b^6)/2 + (155* 
a^10*b^5)/4 + 60*a^11*b^4 + (105*a^12*b^3)/2 + (49*a^13*b^2)/2)*1i)/(2*(5* 
a^13*b + a^14 + a^9*b^5 + 5*a^10*b^4 + 10*a^11*b^3 + 10*a^12*b^2)) - (tan( 
c + d*x)*(9216*a^15*b + 1024*a^16 + 2048*a^8*b^8 + 15360*a^9*b^7 + 50176*a 
^10*b^6 + 93184*a^11*b^5 + 107520*a^12*b^4 + 78848*a^13*b^3 + 35840*a^14*b 
^2))/(512*a^4*(5*a^10*b + a^11 + a^6*b^5 + 5*a^7*b^4 + 10*a^8*b^3 + 10*a^9 
*b^2)))/(2*a^4) + (tan(c + d*x)*(3840*a*b^7 + 2048*a^7*b + 256*a^8 + 512*b 
^8 + 12544*a^2*b^6 + 23296*a^3*b^5 + 26740*a^4*b^4 + 19236*a^5*b^3 + 8393* 
a^6*b^2))/(256*(5*a^10*b + a^11 + a^6*b^5 + 5*a^7*b^4 + 10*a^8*b^3 + 10*a^ 
9*b^2)))/a^4 - (((((19*a^14*b)/4 + 2*a^8*b^7 + (27*a^9*b^6)/2 + (155*a^10* 
b^5)/4 + 60*a^11*b^4 + (105*a^12*b^3)/2 + (49*a^13*b^2)/2)*1i)/(2*(5*a^13* 
b + a^14 + a^9*b^5 + 5*a^10*b^4 + 10*a^11*b^3 + 10*a^12*b^2)) + (tan(c + d 
*x)*(9216*a^15*b + 1024*a^16 + 2048*a^8*b^8 + 15360*a^9*b^7 + 50176*a^10*b 
^6 + 93184*a^11*b^5 + 107520*a^12*b^4 + 78848*a^13*b^3 + 35840*a^14*b^2))/ 
(512*a^4*(5*a^10*b + a^11 + a^6*b^5 + 5*a^7*b^4 + 10*a^8*b^3 + 10*a^9*b...
 

Reduce [B] (verification not implemented)

Time = 1.28 (sec) , antiderivative size = 7813, normalized size of antiderivative = 38.30 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx =\text {Too large to display} \] Input:

int(1/(a+b*csc(d*x+c)^2)^4,x)
 

Output:

(210*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*ata 
n((tan((c + d*x)/2)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*si 
n(c + d*x)**6*a**6 + 420*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a 
 + b) + 2*a + b)*atan((tan((c + d*x)/2)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a 
+ b) + 2*a + b)))*sin(c + d*x)**6*a**5*b + 336*sqrt(b)*sqrt(a)*sqrt(a + b) 
*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((tan((c + d*x)/2)*b)/(sqrt(b)* 
sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*sin(c + d*x)**6*a**4*b**2 + 96*sqr 
t(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((tan(( 
c + d*x)/2)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*sin(c + d* 
x)**6*a**3*b**3 + 630*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + 
b) + 2*a + b)*atan((tan((c + d*x)/2)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b 
) + 2*a + b)))*sin(c + d*x)**4*a**5*b + 1260*sqrt(b)*sqrt(a)*sqrt(a + b)*s 
qrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((tan((c + d*x)/2)*b)/(sqrt(b)*sq 
rt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*sin(c + d*x)**4*a**4*b**2 + 1008*sqr 
t(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((tan(( 
c + d*x)/2)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*sin(c + d* 
x)**4*a**3*b**3 + 288*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + 
b) + 2*a + b)*atan((tan((c + d*x)/2)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b 
) + 2*a + b)))*sin(c + d*x)**4*a**2*b**4 + 630*sqrt(b)*sqrt(a)*sqrt(a + b) 
*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((tan((c + d*x)/2)*b)/(sqrt(...