Integrand size = 14, antiderivative size = 204 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\frac {x}{a^4}+\frac {\sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b}}\right )}{16 a^4 (a+b)^{7/2} d}+\frac {b \cot (c+d x)}{6 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^3}+\frac {b (11 a+6 b) \cot (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{16 a^3 (a+b)^3 d \left (a+b+b \cot ^2(c+d x)\right )} \]
x/a^4+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)+1/16*(35*a^3+70*a^2*b+56* a*b^2+16*b^3)*arctan(cot(d*x+c)*b^(1/2)/(a+b)^(1/2))*b^(1/2)/a^4/(a+b)^(7/ 2)/d
Time = 9.48 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\frac {(-a-2 b+a \cos (2 (c+d x))) \csc ^8(c+d x) \left (\frac {3 \sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {b}}\right ) (a+2 b-a \cos (2 (c+d x)))^3}{(a+b)^{7/2}}+48 (c+d x) (-a-2 b+a \cos (2 (c+d x)))^3-\frac {32 a b^3 \sin (2 (c+d x))}{a+b}-\frac {a b \left (87 a^2+116 a b+44 b^2\right ) (a+2 b-a \cos (2 (c+d x)))^2 \sin (2 (c+d x))}{(a+b)^3}-\frac {4 a b^2 (19 a+14 b) (-a-2 b+a \cos (2 (c+d x))) \sin (2 (c+d x))}{(a+b)^2}\right )}{768 a^4 d \left (a+b \csc ^2(c+d x)\right )^4} \]
((-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)
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 |
\(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \arctan (\cot (c+d x))}{a}-\frac {\sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{2 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \cot (c+d x)}{4 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \cot (c+d x)}{6 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^3}}{d}\) |
-((-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)
3.1.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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]
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]
Time = 3.06 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a^{4}}-\frac {b \left (\frac {-\frac {\left (29 a^{2}+26 a b +8 b^{2}\right ) a \tan \left (d x +c \right )^{5}}{16 \left (a +b \right )}-\frac {\left (17 a^{2}+18 a b +6 b^{2}\right ) a b \tan \left (d x +c \right )^{3}}{6 \left (a^{2}+2 a b +b^{2}\right )}-\frac {b^{2} a \left (19 a^{2}+22 a b +8 b^{2}\right ) \tan \left (d x +c \right )}{16 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}}{\left (a \tan \left (d x +c \right )^{2}+b \tan \left (d x +c \right )^{2}+b \right )^{3}}+\frac {\left (35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{16 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{4}}}{d}\) | \(240\) |
default | \(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a^{4}}-\frac {b \left (\frac {-\frac {\left (29 a^{2}+26 a b +8 b^{2}\right ) a \tan \left (d x +c \right )^{5}}{16 \left (a +b \right )}-\frac {\left (17 a^{2}+18 a b +6 b^{2}\right ) a b \tan \left (d x +c \right )^{3}}{6 \left (a^{2}+2 a b +b^{2}\right )}-\frac {b^{2} a \left (19 a^{2}+22 a b +8 b^{2}\right ) \tan \left (d x +c \right )}{16 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}}{\left (a \tan \left (d x +c \right )^{2}+b \tan \left (d x +c \right )^{2}+b \right )^{3}}+\frac {\left (35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{16 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{4}}}{d}\) | \(240\) |
risch | \(\text {Expression too large to display}\) | \(900\) |
1/d*(1/a^4*arctan(tan(d*x+c))-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))/(a*ta n(d*x+c)^2+b*tan(d*x+c)^2+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)/((a+b)*b)^(1/2)*arctan((a+b)*tan(d*x+c)/((a+b)*b)^(1 /2))))
Leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (188) = 376\).
Time = 0.34 (sec) , antiderivative size = 1592, normalized size of antiderivative = 7.80 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\text {Too large to display} \]
[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^...
\[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\int \frac {1}{\left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{4}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (188) = 376\).
Time = 0.31 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=-\frac {\frac {3 \, {\left (35 \, a^{3} b + 70 \, a^{2} b^{2} + 56 \, a b^{3} + 16 \, b^{4}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt {{\left (a + b\right )} b}} - \frac {3 \, {\left (29 \, a^{4} b + 84 \, a^{3} b^{2} + 89 \, a^{2} b^{3} + 42 \, a b^{4} + 8 \, b^{5}\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (17 \, a^{3} b^{2} + 35 \, a^{2} b^{3} + 24 \, a b^{4} + 6 \, b^{5}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (19 \, a^{2} b^{3} + 22 \, a b^{4} + 8 \, b^{5}\right )} \tan \left (d x + c\right )}{a^{6} b^{3} + 3 \, a^{5} b^{4} + 3 \, a^{4} b^{5} + a^{3} b^{6} + {\left (a^{9} + 6 \, a^{8} b + 15 \, a^{7} b^{2} + 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} + 6 \, a^{4} b^{5} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (a^{8} b + 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} + 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{2}} - \frac {48 \, {\left (d x + c\right )}}{a^{4}}}{48 \, d} \]
-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
Time = 0.31 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.74 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=-\frac {\frac {3 \, {\left (35 \, a^{3} b + 70 \, a^{2} b^{2} + 56 \, a b^{3} + 16 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt {a b + b^{2}}} - \frac {87 \, a^{4} b \tan \left (d x + c\right )^{5} + 252 \, a^{3} b^{2} \tan \left (d x + c\right )^{5} + 267 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 126 \, a b^{4} \tan \left (d x + c\right )^{5} + 24 \, b^{5} \tan \left (d x + c\right )^{5} + 136 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 280 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} + 192 \, a b^{4} \tan \left (d x + c\right )^{3} + 48 \, b^{5} \tan \left (d x + c\right )^{3} + 57 \, a^{2} b^{3} \tan \left (d x + c\right ) + 66 \, a b^{4} \tan \left (d x + c\right ) + 24 \, b^{5} \tan \left (d x + c\right )}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + b\right )}^{3}} - \frac {48 \, {\left (d x + c\right )}}{a^{4}}}{48 \, d} \]
-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
Time = 23.99 (sec) , antiderivative size = 4447, normalized size of antiderivative = 21.80 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^4} \, dx=\text {Too large to display} \]
((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...