Integrand size = 21, antiderivative size = 205 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\left (a^2+10 b^2\right ) \cot (c+d x)}{a^6 d}+\frac {2 b \cot ^2(c+d x)}{a^5 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}-\frac {4 b \left (a^2+5 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac {4 b \left (a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}-\frac {b \left (a^2+b^2\right )}{3 a^4 d (a+b \tan (c+d x))^3}-\frac {b \left (a^2+2 b^2\right )}{a^5 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2+10 b^2\right )}{a^6 d (a+b \tan (c+d x))} \]
-(a^2+10*b^2)*cot(d*x+c)/a^6/d+2*b*cot(d*x+c)^2/a^5/d-1/3*cot(d*x+c)^3/a^4 /d-4*b*(a^2+5*b^2)*ln(tan(d*x+c))/a^7/d+4*b*(a^2+5*b^2)*ln(a+b*tan(d*x+c)) /a^7/d-1/3*b*(a^2+b^2)/a^4/d/(a+b*tan(d*x+c))^3-b*(a^2+2*b^2)/a^5/d/(a+b*t an(d*x+c))^2-b*(3*a^2+10*b^2)/a^6/d/(a+b*tan(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(528\) vs. \(2(205)=410\).
Time = 2.88 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.58 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (-192 b \left (a^2+5 b^2\right ) \log (\sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3+192 b \left (a^2+5 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3-\frac {\csc ^3(c+d x) \left (8 a^8-4 a^6 b^2-50 a^4 b^4-190 a^2 b^6-150 b^8+3 \left (3 a^8+10 a^6 b^2+45 a^4 b^4+115 a^2 b^6+75 b^8\right ) \cos (2 (c+d x))+6 \left (2 a^6 b^2-17 a^4 b^4-35 a^2 b^6-15 b^8\right ) \cos (4 (c+d x))-a^8 \cos (6 (c+d x))-22 a^6 b^2 \cos (6 (c+d x))+17 a^4 b^4 \cos (6 (c+d x))+55 a^2 b^6 \cos (6 (c+d x))+15 b^8 \cos (6 (c+d x))-3 a^7 b \sin (2 (c+d x))+3 a^5 b^3 \sin (2 (c+d x))-75 a^3 b^5 \sin (2 (c+d x))-75 a b^7 \sin (2 (c+d x))-6 a^7 b \sin (4 (c+d x))+84 a^5 b^3 \sin (4 (c+d x))+156 a^3 b^5 \sin (4 (c+d x))+60 a b^7 \sin (4 (c+d x))-3 a^7 b \sin (6 (c+d x))-65 a^5 b^3 \sin (6 (c+d x))-79 a^3 b^5 \sin (6 (c+d x))-15 a b^7 \sin (6 (c+d x))\right )}{a^2+b^2}\right )}{48 a^7 d (a+b \tan (c+d x))^4} \]
(Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])*(-192*b*(a^2 + 5*b^2)*Lo g[Sin[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 + 192*b*(a^2 + 5*b^2)* Log[a*Cos[c + d*x] + b*Sin[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 - (Csc[c + d*x]^3*(8*a^8 - 4*a^6*b^2 - 50*a^4*b^4 - 190*a^2*b^6 - 150*b^8 + 3*(3*a^8 + 10*a^6*b^2 + 45*a^4*b^4 + 115*a^2*b^6 + 75*b^8)*Cos[2*(c + d*x )] + 6*(2*a^6*b^2 - 17*a^4*b^4 - 35*a^2*b^6 - 15*b^8)*Cos[4*(c + d*x)] - a ^8*Cos[6*(c + d*x)] - 22*a^6*b^2*Cos[6*(c + d*x)] + 17*a^4*b^4*Cos[6*(c + d*x)] + 55*a^2*b^6*Cos[6*(c + d*x)] + 15*b^8*Cos[6*(c + d*x)] - 3*a^7*b*Si n[2*(c + d*x)] + 3*a^5*b^3*Sin[2*(c + d*x)] - 75*a^3*b^5*Sin[2*(c + d*x)] - 75*a*b^7*Sin[2*(c + d*x)] - 6*a^7*b*Sin[4*(c + d*x)] + 84*a^5*b^3*Sin[4* (c + d*x)] + 156*a^3*b^5*Sin[4*(c + d*x)] + 60*a*b^7*Sin[4*(c + d*x)] - 3* a^7*b*Sin[6*(c + d*x)] - 65*a^5*b^3*Sin[6*(c + d*x)] - 79*a^3*b^5*Sin[6*(c + d*x)] - 15*a*b^7*Sin[6*(c + d*x)]))/(a^2 + b^2)))/(48*a^7*d*(a + b*Tan[ c + d*x])^4)
Time = 0.41 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3999, 522, 2009}
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.
| \(\displaystyle \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^4 (a+b \tan (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3999 |
| \(\displaystyle \frac {b \int \frac {\cot ^4(c+d x) \left (\tan ^2(c+d x) b^2+b^2\right )}{b^4 (a+b \tan (c+d x))^4}d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {b \int \left (\frac {\cot ^4(c+d x)}{a^4 b^2}-\frac {4 \cot ^3(c+d x)}{a^5 b}+\frac {\left (a^2+10 b^2\right ) \cot ^2(c+d x)}{a^6 b^2}-\frac {4 \left (a^2+5 b^2\right ) \cot (c+d x)}{a^7 b}+\frac {4 \left (a^2+5 b^2\right )}{a^7 (a+b \tan (c+d x))}+\frac {3 a^2+10 b^2}{a^6 (a+b \tan (c+d x))^2}+\frac {2 \left (a^2+2 b^2\right )}{a^5 (a+b \tan (c+d x))^3}+\frac {a^2+b^2}{a^4 (a+b \tan (c+d x))^4}\right )d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (\frac {2 \cot ^2(c+d x)}{a^5}-\frac {\cot ^3(c+d x)}{3 a^4 b}-\frac {4 \left (a^2+5 b^2\right ) \log (b \tan (c+d x))}{a^7}+\frac {4 \left (a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{a^7}-\frac {3 a^2+10 b^2}{a^6 (a+b \tan (c+d x))}-\frac {\left (a^2+10 b^2\right ) \cot (c+d x)}{a^6 b}-\frac {a^2+2 b^2}{a^5 (a+b \tan (c+d x))^2}-\frac {a^2+b^2}{3 a^4 (a+b \tan (c+d x))^3}\right )}{d}\) |
(b*(-(((a^2 + 10*b^2)*Cot[c + d*x])/(a^6*b)) + (2*Cot[c + d*x]^2)/a^5 - Co t[c + d*x]^3/(3*a^4*b) - (4*(a^2 + 5*b^2)*Log[b*Tan[c + d*x]])/a^7 + (4*(a ^2 + 5*b^2)*Log[a + b*Tan[c + d*x]])/a^7 - (a^2 + b^2)/(3*a^4*(a + b*Tan[c + d*x])^3) - (a^2 + 2*b^2)/(a^5*(a + b*Tan[c + d*x])^2) - (3*a^2 + 10*b^2 )/(a^6*(a + b*Tan[c + d*x]))))/d
3.1.76.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[b/f Subst[Int[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/2]
Time = 4.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {-\frac {b \left (3 a^{2}+10 b^{2}\right )}{a^{6} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{2}+b^{2}\right ) b}{3 a^{4} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (a^{2}+2 b^{2}\right )}{a^{5} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b \left (a^{2}+5 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{7}}-\frac {1}{3 a^{4} \tan \left (d x +c \right )^{3}}-\frac {a^{2}+10 b^{2}}{a^{6} \tan \left (d x +c \right )}+\frac {2 b}{a^{5} \tan \left (d x +c \right )^{2}}-\frac {4 b \left (a^{2}+5 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{7}}}{d}\) | \(184\) |
| default | \(\frac {-\frac {b \left (3 a^{2}+10 b^{2}\right )}{a^{6} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{2}+b^{2}\right ) b}{3 a^{4} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (a^{2}+2 b^{2}\right )}{a^{5} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b \left (a^{2}+5 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{7}}-\frac {1}{3 a^{4} \tan \left (d x +c \right )^{3}}-\frac {a^{2}+10 b^{2}}{a^{6} \tan \left (d x +c \right )}+\frac {2 b}{a^{5} \tan \left (d x +c \right )^{2}}-\frac {4 b \left (a^{2}+5 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{7}}}{d}\) | \(184\) |
| risch | \(-\frac {4 i \left (i a^{7}-30 b^{7}-26 a^{4} b^{3}-56 b^{5} a^{2}-a^{6} b +32 i a^{5} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+280 i a^{3} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-300 i a \,b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-300 i a^{3} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-48 i a^{5} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-96 i a^{3} b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+60 i a^{3} b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+30 i a^{5} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+120 i a \,b^{6} {\mathrm e}^{10 i \left (d x +c \right )}-450 i a \,b^{6} {\mathrm e}^{8 i \left (d x +c \right )}+600 i a \,b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-24 i a^{5} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+150 b^{7} {\mathrm e}^{2 i \left (d x +c \right )}-300 b^{7} {\mathrm e}^{4 i \left (d x +c \right )}-150 b^{7} {\mathrm e}^{8 i \left (d x +c \right )}+300 b^{7} {\mathrm e}^{6 i \left (d x +c \right )}+30 b^{7} {\mathrm e}^{10 i \left (d x +c \right )}+420 a^{2} b^{5} {\mathrm e}^{8 i \left (d x +c \right )}-48 a^{4} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-300 a^{2} b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+270 a^{2} b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-12 a^{6} b \,{\mathrm e}^{4 i \left (d x +c \right )}-160 a^{2} b^{5} {\mathrm e}^{6 i \left (d x +c \right )}+114 a^{4} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-174 a^{2} b^{5} {\mathrm e}^{10 i \left (d x +c \right )}+30 i a \,b^{6}+56 i b^{4} a^{3}+26 i a^{5} b^{2}+6 a^{6} b \,{\mathrm e}^{10 i \left (d x +c \right )}-6 a^{4} b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-3 a^{6} b \,{\mathrm e}^{8 i \left (d x +c \right )}-124 a^{4} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-8 i a^{7} {\mathrm e}^{6 i \left (d x +c \right )}-6 i a^{7} {\mathrm e}^{4 i \left (d x +c \right )}-3 i a^{7} {\mathrm e}^{8 i \left (d x +c \right )}+90 a^{4} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-22 a^{6} b \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} \left (i a +b \right )^{2} a^{6} d}-\frac {4 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{5} d}-\frac {20 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{7} d}+\frac {4 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{5} d}+\frac {20 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{7} d}\) | \(800\) |
1/d*(-b*(3*a^2+10*b^2)/a^6/(a+b*tan(d*x+c))-1/3*(a^2+b^2)*b/a^4/(a+b*tan(d *x+c))^3-b*(a^2+2*b^2)/a^5/(a+b*tan(d*x+c))^2+4*b*(a^2+5*b^2)/a^7*ln(a+b*t an(d*x+c))-1/3/a^4/tan(d*x+c)^3-(a^2+10*b^2)/a^6/tan(d*x+c)+2/a^5*b/tan(d* x+c)^2-4*b*(a^2+5*b^2)/a^7*ln(tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 1235 vs. \(2 (201) = 402\).
Time = 0.32 (sec) , antiderivative size = 1235, normalized size of antiderivative = 6.02 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Too large to display} \]
1/3*(19*a^6*b^4 + 51*a^4*b^6 + 30*a^2*b^8 + 2*(a^10 + 23*a^8*b^2 - 22*a^6* b^4 - 138*a^4*b^6 - 90*a^2*b^8)*cos(d*x + c)^6 - 3*(a^10 + 25*a^8*b^2 - 46 *a^6*b^4 - 206*a^4*b^6 - 130*a^2*b^8)*cos(d*x + c)^4 + 3*(9*a^8*b^2 - 38*a ^6*b^4 - 131*a^4*b^6 - 80*a^2*b^8)*cos(d*x + c)^2 + 6*(a^6*b^4 + 7*a^4*b^6 + 11*a^2*b^8 + 5*b^10 + (3*a^8*b^2 + 20*a^6*b^4 + 26*a^4*b^6 + 4*a^2*b^8 - 5*b^10)*cos(d*x + c)^6 - 3*(2*a^8*b^2 + 13*a^6*b^4 + 15*a^4*b^6 - a^2*b^ 8 - 5*b^10)*cos(d*x + c)^4 + 3*(a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 - 6*a^2*b^ 8 - 5*b^10)*cos(d*x + c)^2 - ((a^9*b + 4*a^7*b^3 - 10*a^5*b^5 - 28*a^3*b^7 - 15*a*b^9)*cos(d*x + c)^5 - (a^9*b + a^7*b^3 - 31*a^5*b^5 - 61*a^3*b^7 - 30*a*b^9)*cos(d*x + c)^3 - 3*(a^7*b^3 + 7*a^5*b^5 + 11*a^3*b^7 + 5*a*b^9) *cos(d*x + c))*sin(d*x + c))*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) - 6*(a^6*b^4 + 7*a^4*b^6 + 11*a^2*b^8 + 5*b^10 + (3*a^8*b^2 + 20*a^6*b^4 + 26*a^4*b^6 + 4*a^2*b^8 - 5*b^10)*cos(d*x + c)^ 6 - 3*(2*a^8*b^2 + 13*a^6*b^4 + 15*a^4*b^6 - a^2*b^8 - 5*b^10)*cos(d*x + c )^4 + 3*(a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 - 6*a^2*b^8 - 5*b^10)*cos(d*x + c )^2 - ((a^9*b + 4*a^7*b^3 - 10*a^5*b^5 - 28*a^3*b^7 - 15*a*b^9)*cos(d*x + c)^5 - (a^9*b + a^7*b^3 - 31*a^5*b^5 - 61*a^3*b^7 - 30*a*b^9)*cos(d*x + c) ^3 - 3*(a^7*b^3 + 7*a^5*b^5 + 11*a^3*b^7 + 5*a*b^9)*cos(d*x + c))*sin(d*x + c))*log(-1/4*cos(d*x + c)^2 + 1/4) + (2*(3*a^9*b + 77*a^7*b^3 + 142*a^5* b^5 + 34*a^3*b^7 - 30*a*b^9)*cos(d*x + c)^5 - (3*a^9*b + 193*a^7*b^3 + ...
\[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\int \frac {\csc ^{4}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{4}}\, dx \]
Time = 0.40 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.11 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \, a^{4} b \tan \left (d x + c\right ) - 12 \, {\left (a^{2} b^{3} + 5 \, b^{5}\right )} \tan \left (d x + c\right )^{5} - a^{5} - 30 \, {\left (a^{3} b^{2} + 5 \, a b^{4}\right )} \tan \left (d x + c\right )^{4} - 22 \, {\left (a^{4} b + 5 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )^{3} - 3 \, {\left (a^{5} + 5 \, a^{3} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{6} b^{3} \tan \left (d x + c\right )^{6} + 3 \, a^{7} b^{2} \tan \left (d x + c\right )^{5} + 3 \, a^{8} b \tan \left (d x + c\right )^{4} + a^{9} \tan \left (d x + c\right )^{3}} + \frac {12 \, {\left (a^{2} b + 5 \, b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{7}} - \frac {12 \, {\left (a^{2} b + 5 \, b^{3}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{7}}}{3 \, d} \]
1/3*((3*a^4*b*tan(d*x + c) - 12*(a^2*b^3 + 5*b^5)*tan(d*x + c)^5 - a^5 - 3 0*(a^3*b^2 + 5*a*b^4)*tan(d*x + c)^4 - 22*(a^4*b + 5*a^2*b^3)*tan(d*x + c) ^3 - 3*(a^5 + 5*a^3*b^2)*tan(d*x + c)^2)/(a^6*b^3*tan(d*x + c)^6 + 3*a^7*b ^2*tan(d*x + c)^5 + 3*a^8*b*tan(d*x + c)^4 + a^9*tan(d*x + c)^3) + 12*(a^2 *b + 5*b^3)*log(b*tan(d*x + c) + a)/a^7 - 12*(a^2*b + 5*b^3)*log(tan(d*x + c))/a^7)/d
Time = 0.70 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {12 \, {\left (a^{2} b + 5 \, b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{7}} - \frac {12 \, {\left (a^{2} b^{2} + 5 \, b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{7} b} + \frac {12 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 60 \, b^{5} \tan \left (d x + c\right )^{5} + 30 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} + 150 \, a b^{4} \tan \left (d x + c\right )^{4} + 22 \, a^{4} b \tan \left (d x + c\right )^{3} + 110 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} + 3 \, a^{5} \tan \left (d x + c\right )^{2} + 15 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} - 3 \, a^{4} b \tan \left (d x + c\right ) + a^{5}}{{\left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right )\right )}^{3} a^{6}}}{3 \, d} \]
-1/3*(12*(a^2*b + 5*b^3)*log(abs(tan(d*x + c)))/a^7 - 12*(a^2*b^2 + 5*b^4) *log(abs(b*tan(d*x + c) + a))/(a^7*b) + (12*a^2*b^3*tan(d*x + c)^5 + 60*b^ 5*tan(d*x + c)^5 + 30*a^3*b^2*tan(d*x + c)^4 + 150*a*b^4*tan(d*x + c)^4 + 22*a^4*b*tan(d*x + c)^3 + 110*a^2*b^3*tan(d*x + c)^3 + 3*a^5*tan(d*x + c)^ 2 + 15*a^3*b^2*tan(d*x + c)^2 - 3*a^4*b*tan(d*x + c) + a^5)/((b*tan(d*x + c)^2 + a*tan(d*x + c))^3*a^6))/d
Time = 5.64 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.13 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {8\,b\,\mathrm {atanh}\left (\frac {4\,b\,\left (a^2+5\,b^2\right )\,\left (a+2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{a\,\left (4\,a^2\,b+20\,b^3\right )}\right )\,\left (a^2+5\,b^2\right )}{a^7\,d}-\frac {\frac {1}{3\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^2+5\,b^2\right )}{a^3}-\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{a^2}+\frac {22\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a^2+5\,b^2\right )}{3\,a^4}+\frac {10\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^2+5\,b^2\right )}{a^5}+\frac {4\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (a^2+5\,b^2\right )}{a^6}}{d\,\left (a^3\,{\mathrm {tan}\left (c+d\,x\right )}^3+3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^4+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^5+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^6\right )} \]
(8*b*atanh((4*b*(a^2 + 5*b^2)*(a + 2*b*tan(c + d*x)))/(a*(4*a^2*b + 20*b^3 )))*(a^2 + 5*b^2))/(a^7*d) - (1/(3*a) + (tan(c + d*x)^2*(a^2 + 5*b^2))/a^3 - (b*tan(c + d*x))/a^2 + (22*b*tan(c + d*x)^3*(a^2 + 5*b^2))/(3*a^4) + (1 0*b^2*tan(c + d*x)^4*(a^2 + 5*b^2))/a^5 + (4*b^3*tan(c + d*x)^5*(a^2 + 5*b ^2))/a^6)/(d*(a^3*tan(c + d*x)^3 + b^3*tan(c + d*x)^6 + 3*a^2*b*tan(c + d* x)^4 + 3*a*b^2*tan(c + d*x)^5))