Integrand size = 29, antiderivative size = 470 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {2 b^5 \left (5 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}+\frac {b^5 \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}+\frac {2 b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{7/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac {b^6 \cos (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {3 b^6 \cos (c+d x)}{2 a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))} \]
2*b^5*(5*a^2-3*b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^3/( a^2-b^2)^(7/2)/d+b^5*(2*a^2+b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2) ^(1/2))/a^3/(a^2-b^2)^(7/2)/d+2*b^5*(15*a^4-17*a^2*b^2+6*b^4)*arctan((b+a* tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^5/(a^2-b^2)^(7/2)/d-1/2*arctanh(cos (d*x+c))/a^3/d-(a^2+6*b^2)*arctanh(cos(d*x+c))/a^5/d+3*b*cot(d*x+c)/a^4/d- 1/2*cot(d*x+c)*csc(d*x+c)/a^3/d+1/2*cos(d*x+c)/(a+b)^3/d/(1-sin(d*x+c))+1/ 2*cos(d*x+c)/(a-b)^3/d/(1+sin(d*x+c))+1/2*b^6*cos(d*x+c)/a^3/(a^2-b^2)^2/d /(a+b*sin(d*x+c))^2+3/2*b^6*cos(d*x+c)/a^2/(a^2-b^2)^3/d/(a+b*sin(d*x+c))+ b^6*(5*a^2-3*b^2)*cos(d*x+c)/a^4/(a^2-b^2)^3/d/(a+b*sin(d*x+c))
Time = 7.03 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.92 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3 b^5 \left (14 a^4-13 a^2 b^2+4 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{7/2} d}+\frac {3 b \cot \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}-\frac {3 \left (a^2+4 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {3 \left (a^2+4 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{(a+b)^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{(a-b)^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {b^6 \cos (c+d x)}{2 a^3 (a-b)^2 (a+b)^2 d (a+b \sin (c+d x))^2}+\frac {13 a^2 b^6 \cos (c+d x)-6 b^8 \cos (c+d x)}{2 a^4 (a-b)^3 (a+b)^3 d (a+b \sin (c+d x))}-\frac {3 b \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d} \]
(3*b^5*(14*a^4 - 13*a^2*b^2 + 4*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(7/2)*d) + (3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(c + d*x)/2]^2/(8*a^3*d) - (3*(a ^2 + 4*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) + (3*(a^2 + 4*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^5*d) + Sec[(c + d*x)/2]^2/(8*a^3*d) + Sin[(c + d*x)/2]/( (a + b)^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - Sin[(c + d*x)/2]/((a - b)^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^6*Cos[c + d*x])/(2*a^ 3*(a - b)^2*(a + b)^2*d*(a + b*Sin[c + d*x])^2) + (13*a^2*b^6*Cos[c + d*x] - 6*b^8*Cos[c + d*x])/(2*a^4*(a - b)^3*(a + b)^3*d*(a + b*Sin[c + d*x])) - (3*b*Tan[(c + d*x)/2])/(2*a^4*d)
Time = 0.96 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3376, 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.
| \(\displaystyle \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^3 \cos (c+d x)^2 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3376 |
| \(\displaystyle \int \left (-\frac {3 b \csc ^2(c+d x)}{a^4}+\frac {\csc ^3(c+d x)}{a^3}+\frac {\left (a^2+6 b^2\right ) \csc (c+d x)}{a^5}+\frac {b^5 \left (5 a^2-3 b^2\right )}{a^4 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {b^5}{a^3 \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}+\frac {b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right )}{a^5 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {1}{2 (a+b)^3 (\sin (c+d x)-1)}-\frac {1}{2 (a-b)^3 (\sin (c+d x)+1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b \cot (c+d x)}{a^4 d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {3 b^6 \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {b^5 \left (2 a^2+b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{7/2}}+\frac {2 b^5 \left (5 a^2-3 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{7/2}}+\frac {b^6 \cos (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {2 b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{7/2}}+\frac {\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}\) |
(2*b^5*(5*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/( a^3*(a^2 - b^2)^(7/2)*d) + (b^5*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/ 2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) + (2*b^5*(15*a^4 - 17*a^2* b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(7/2)*d) - ArcTanh[Cos[c + d*x]]/(2*a^3*d) - ((a^2 + 6*b^2)*ArcTanh[ Cos[c + d*x]])/(a^5*d) + (3*b*Cot[c + d*x])/(a^4*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[ c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) + (b^6*Cos[c + d*x])/(2*a^3*(a ^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*b^6*Cos[c + d*x])/(2*a^2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (b^6*(5*a^2 - 3*b^2)*Cos[c + d*x])/(a^4* (a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))
3.15.77.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(d*sin[ e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x] /; Fr eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && ( LtQ[m, -1] || (EqQ[m, -1] && GtQ[p, 0]))
Time = 3.23 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}+\frac {1}{\left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 b^{5} \left (\frac {\left (\frac {15}{2} a^{3} b^{2}-4 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {7 b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (41 a^{2}-20 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {7 a^{2} b \left (2 a^{2}-b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 \left (14 a^{4}-13 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} a^{5}}-\frac {1}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(355\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}+\frac {1}{\left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 b^{5} \left (\frac {\left (\frac {15}{2} a^{3} b^{2}-4 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {7 b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (41 a^{2}-20 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {7 a^{2} b \left (2 a^{2}-b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 \left (14 a^{4}-13 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} a^{5}}-\frac {1}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(355\) |
| risch | \(\text {Expression too large to display}\) | \(1405\) |
1/d*(1/4/a^4*(1/2*tan(1/2*d*x+1/2*c)^2*a-6*b*tan(1/2*d*x+1/2*c))+1/(a-b)^3 /(tan(1/2*d*x+1/2*c)+1)+2*b^5/(a-b)^3/(a+b)^3/a^5*(((15/2*a^3*b^2-4*a*b^4) *tan(1/2*d*x+1/2*c)^3+7/2*b*(2*a^4+3*a^2*b^2-2*b^4)*tan(1/2*d*x+1/2*c)^2+1 /2*a*b^2*(41*a^2-20*b^2)*tan(1/2*d*x+1/2*c)+7/2*a^2*b*(2*a^2-b^2))/(tan(1/ 2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+3/2*(14*a^4-13*a^2*b^2+4*b^4) /(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))) -1/(a+b)^3/(tan(1/2*d*x+1/2*c)-1)-1/8/a^3/tan(1/2*d*x+1/2*c)^2+1/4/a^5*(6* a^2+24*b^2)*ln(tan(1/2*d*x+1/2*c))+3/2*b/a^4/tan(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 1294 vs. \(2 (441) = 882\).
Time = 2.74 (sec) , antiderivative size = 2672, normalized size of antiderivative = 5.69 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
[1/4*(4*a^12 - 12*a^10*b^2 + 12*a^8*b^4 - 4*a^6*b^6 - 2*(21*a^10*b^2 - 56* a^8*b^4 + 82*a^6*b^6 - 65*a^4*b^8 + 18*a^2*b^10)*cos(d*x + c)^4 - 2*(3*a^1 2 - 31*a^10*b^2 + 68*a^8*b^4 - 88*a^6*b^6 + 66*a^4*b^8 - 18*a^2*b^10)*cos( d*x + c)^2 + 3*((14*a^4*b^7 - 13*a^2*b^9 + 4*b^11)*cos(d*x + c)^5 - (14*a^ 6*b^5 + 15*a^4*b^7 - 22*a^2*b^9 + 8*b^11)*cos(d*x + c)^3 + (14*a^6*b^5 + a ^4*b^7 - 9*a^2*b^9 + 4*b^11)*cos(d*x + c) - 2*((14*a^5*b^6 - 13*a^3*b^8 + 4*a*b^10)*cos(d*x + c)^3 - (14*a^5*b^6 - 13*a^3*b^8 + 4*a*b^10)*cos(d*x + c))*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2* a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b ^2)) - 3*((a^10*b^2 - 10*a^6*b^6 + 20*a^4*b^8 - 15*a^2*b^10 + 4*b^12)*cos( d*x + c)^5 - (a^12 + 2*a^10*b^2 - 10*a^8*b^4 + 25*a^4*b^8 - 26*a^2*b^10 + 8*b^12)*cos(d*x + c)^3 + (a^12 + a^10*b^2 - 10*a^8*b^4 + 10*a^6*b^6 + 5*a^ 4*b^8 - 11*a^2*b^10 + 4*b^12)*cos(d*x + c) - 2*((a^11*b - 10*a^7*b^5 + 20* a^5*b^7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c)^3 - (a^11*b - 10*a^7*b^5 + 2 0*a^5*b^7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c))*sin(d*x + c))*log(1/2*cos (d*x + c) + 1/2) + 3*((a^10*b^2 - 10*a^6*b^6 + 20*a^4*b^8 - 15*a^2*b^10 + 4*b^12)*cos(d*x + c)^5 - (a^12 + 2*a^10*b^2 - 10*a^8*b^4 + 25*a^4*b^8 - 26 *a^2*b^10 + 8*b^12)*cos(d*x + c)^3 + (a^12 + a^10*b^2 - 10*a^8*b^4 + 10*a^ 6*b^6 + 5*a^4*b^8 - 11*a^2*b^10 + 4*b^12)*cos(d*x + c) - 2*((a^11*b - 1...
Timed out. \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (441) = 882\).
Time = 0.44 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.91 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
1/8*(24*(14*a^4*b^5 - 13*a^2*b^7 + 4*b^9)*(pi*floor(1/2*(d*x + c)/pi + 1/2 )*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(a^2 - b^2)) + 16*(3*a^2*b*tan(1/2*d* x + 1/2*c) + b^3*tan(1/2*d*x + 1/2*c) - a^3 - 3*a*b^2)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(tan(1/2*d*x + 1/2*c)^2 - 1)) - (6*a^10*tan(1/2*d*x + 1/ 2*c)^6 + 6*a^8*b^2*tan(1/2*d*x + 1/2*c)^6 - 54*a^6*b^4*tan(1/2*d*x + 1/2*c )^6 + 66*a^4*b^6*tan(1/2*d*x + 1/2*c)^6 - 24*a^2*b^8*tan(1/2*d*x + 1/2*c)^ 6 + 12*a^9*b*tan(1/2*d*x + 1/2*c)^5 + 60*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 - 252*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 156*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 32*a*b^9*tan(1/2*d*x + 1/2*c)^5 + 13*a^10*tan(1/2*d*x + 1/2*c)^4 - 15*a^8* b^2*tan(1/2*d*x + 1/2*c)^4 + 63*a^6*b^4*tan(1/2*d*x + 1/2*c)^4 - 341*a^4*b ^6*tan(1/2*d*x + 1/2*c)^4 + 96*a^2*b^8*tan(1/2*d*x + 1/2*c)^4 + 16*b^10*ta n(1/2*d*x + 1/2*c)^4 + 4*a^9*b*tan(1/2*d*x + 1/2*c)^3 + 36*a^7*b^3*tan(1/2 *d*x + 1/2*c)^3 - 132*a^5*b^5*tan(1/2*d*x + 1/2*c)^3 - 188*a^3*b^7*tan(1/2 *d*x + 1/2*c)^3 + 112*a*b^9*tan(1/2*d*x + 1/2*c)^3 + 8*a^10*tan(1/2*d*x + 1/2*c)^2 - 44*a^8*b^2*tan(1/2*d*x + 1/2*c)^2 + 84*a^6*b^4*tan(1/2*d*x + 1/ 2*c)^2 - 180*a^4*b^6*tan(1/2*d*x + 1/2*c)^2 + 76*a^2*b^8*tan(1/2*d*x + 1/2 *c)^2 - 8*a^9*b*tan(1/2*d*x + 1/2*c) + 24*a^7*b^3*tan(1/2*d*x + 1/2*c) - 2 4*a^5*b^5*tan(1/2*d*x + 1/2*c) + 8*a^3*b^7*tan(1/2*d*x + 1/2*c) + a^10 - 3 *a^8*b^2 + 3*a^6*b^4 - a^4*b^6)/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^...
Time = 16.72 (sec) , antiderivative size = 3266, normalized size of antiderivative = 6.95 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
tan(c/2 + (d*x)/2)^2/(8*a^3*d) + (4*a^2*b*tan(c/2 + (d*x)/2) - a^3/2 + (5* tan(c/2 + (d*x)/2)^2*(3*a^9 - 20*a*b^8 + 49*a^3*b^6 - 27*a^5*b^4 + 19*a^7* b^2))/(2*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (2*tan(c/2 + (d*x)/2)^7*(1 5*a^8*b - 16*b^9 + 27*a^2*b^7 + 9*a^4*b^5 - 5*a^6*b^3))/(a^6 - b^6 + 3*a^2 *b^4 - 3*a^4*b^2) - (4*tan(c/2 + (d*x)/2)^5*(5*a^8*b - 18*b^9 + 43*a^2*b^7 - 7*a^4*b^5 + 7*a^6*b^3))/(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2) + (2*tan(c/ 2 + (d*x)/2)^3*(7*a^8*b - 52*b^9 + 115*a^2*b^7 - 27*a^4*b^5 + 47*a^6*b^3)) /(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2) + (tan(c/2 + (d*x)/2)^4*(33*a^10 - 11 2*b^10 + 220*a^2*b^8 + 11*a^4*b^6 + 119*a^6*b^4 - 31*a^8*b^2))/(2*a*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (tan(c/2 + (d*x)/2)^6*(17*a^10 + 112*b^10 - 120*a^2*b^8 - 257*a^4*b^6 + 83*a^6*b^4 - 195*a^8*b^2))/(2*a*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)))/(d*(4*a^6*tan(c/2 + (d*x)/2)^2 - 4*a^6*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^4*(4*a^6 + 16*a^4*b^2) - tan(c/2 + (d*x) /2)^6*(4*a^6 + 16*a^4*b^2) + 16*a^5*b*tan(c/2 + (d*x)/2)^3 - 16*a^5*b*tan( c/2 + (d*x)/2)^7)) - (3*b*tan(c/2 + (d*x)/2))/(2*a^4*d) + (log(tan(c/2 + ( d*x)/2))*(3*a^2 + 12*b^2))/(2*a^5*d) + (b^5*atan(((b^5*(-(a + b)^7*(a - b) ^7)^(1/2)*(14*a^4 + 4*b^4 - 13*a^2*b^2)*(tan(c/2 + (d*x)/2)*(24*a^30 + 384 *a^8*b^22 - 3552*a^10*b^20 + 14616*a^12*b^18 - 34872*a^14*b^16 + 52800*a^1 6*b^14 - 52176*a^18*b^12 + 33168*a^20*b^10 - 12576*a^22*b^8 + 2112*a^24*b^ 6 + 240*a^26*b^4 - 168*a^28*b^2) - 24*a^29*b + 192*a^9*b^21 - 1728*a^11...