Integrand size = 29, antiderivative size = 200 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {3 \sec ^7(c+d x)}{7 a^3 d}-\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {8 \tan (c+d x)}{a^3 d}+\frac {22 \tan ^3(c+d x)}{3 a^3 d}+\frac {28 \tan ^5(c+d x)}{5 a^3 d}+\frac {17 \tan ^7(c+d x)}{7 a^3 d}+\frac {4 \tan ^9(c+d x)}{9 a^3 d} \]
3*arctanh(cos(d*x+c))/a^3/d-cot(d*x+c)/a^3/d-3*sec(d*x+c)/a^3/d-sec(d*x+c) ^3/a^3/d-3/5*sec(d*x+c)^5/a^3/d-3/7*sec(d*x+c)^7/a^3/d-4/9*sec(d*x+c)^9/a^ 3/d+8*tan(d*x+c)/a^3/d+22/3*tan(d*x+c)^3/a^3/d+28/5*tan(d*x+c)^5/a^3/d+17/ 7*tan(d*x+c)^7/a^3/d+4/9*tan(d*x+c)^9/a^3/d
Time = 1.14 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {1935360 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-1935360 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {\csc (c+d x) (-590976+1083321 \cos (c+d x)-653248 \cos (2 (c+d x))-601845 \cos (3 (c+d x))+340096 \cos (4 (c+d x))-521599 \cos (5 (c+d x))+259008 \cos (6 (c+d x))+40123 \cos (7 (c+d x))-707328 \sin (c+d x)+1364182 \sin (2 (c+d x))-1161600 \sin (3 (c+d x))+320984 \sin (4 (c+d x))-329344 \sin (5 (c+d x))-240738 \sin (6 (c+d x))+53248 \sin (7 (c+d x)))}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^9}}{645120 a^3 d} \]
(1935360*Log[Cos[(c + d*x)/2]] - 1935360*Log[Sin[(c + d*x)/2]] + (Csc[c + d*x]*(-590976 + 1083321*Cos[c + d*x] - 653248*Cos[2*(c + d*x)] - 601845*Co s[3*(c + d*x)] + 340096*Cos[4*(c + d*x)] - 521599*Cos[5*(c + d*x)] + 25900 8*Cos[6*(c + d*x)] + 40123*Cos[7*(c + d*x)] - 707328*Sin[c + d*x] + 136418 2*Sin[2*(c + d*x)] - 1161600*Sin[3*(c + d*x)] + 320984*Sin[4*(c + d*x)] - 329344*Sin[5*(c + d*x)] - 240738*Sin[6*(c + d*x)] + 53248*Sin[7*(c + d*x)] ))/((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d *x)/2])^9))/(645120*a^3*d)
Time = 0.62 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3354, 3042, 3352, 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.
| \(\displaystyle \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^2 \cos (c+d x)^4 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \csc ^2(c+d x) \sec ^{10}(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3}{\cos (c+d x)^{10} \sin (c+d x)^2}dx}{a^6}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (3 a^3 \sec ^{10}(c+d x)+a^3 \csc ^2(c+d x) \sec ^{10}(c+d x)-3 a^3 \csc (c+d x) \sec ^{10}(c+d x)-a^3 \tan (c+d x) \sec ^9(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 \tan ^9(c+d x)}{9 d}+\frac {17 a^3 \tan ^7(c+d x)}{7 d}+\frac {28 a^3 \tan ^5(c+d x)}{5 d}+\frac {22 a^3 \tan ^3(c+d x)}{3 d}+\frac {8 a^3 \tan (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}-\frac {4 a^3 \sec ^9(c+d x)}{9 d}-\frac {3 a^3 \sec ^7(c+d x)}{7 d}-\frac {3 a^3 \sec ^5(c+d x)}{5 d}-\frac {a^3 \sec ^3(c+d x)}{d}-\frac {3 a^3 \sec (c+d x)}{d}}{a^6}\) |
((3*a^3*ArcTanh[Cos[c + d*x]])/d - (a^3*Cot[c + d*x])/d - (3*a^3*Sec[c + d *x])/d - (a^3*Sec[c + d*x]^3)/d - (3*a^3*Sec[c + d*x]^5)/(5*d) - (3*a^3*Se c[c + d*x]^7)/(7*d) - (4*a^3*Sec[c + d*x]^9)/(9*d) + (8*a^3*Tan[c + d*x])/ d + (22*a^3*Tan[c + d*x]^3)/(3*d) + (28*a^3*Tan[c + d*x]^5)/(5*d) + (17*a^ 3*Tan[c + d*x]^7)/(7*d) + (4*a^3*Tan[c + d*x]^9)/(9*d))/a^6
3.9.47.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Time = 1.84 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {11}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {152}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {116}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {267}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {111}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {50}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {67}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {501}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{2 d \,a^{3}}\) | \(224\) |
| default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {11}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {152}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {116}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {267}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {111}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {50}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {67}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {501}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{2 d \,a^{3}}\) | \(224\) |
| parallelrisch | \(\frac {-1890 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-17955 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-64260 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-63105 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+84420 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+225729 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+104664 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-158607 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-203832 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-41753 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+63852 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+46101 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+10676}{630 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}\) | \(240\) |
| risch | \(-\frac {2 \left (-18081 \,{\mathrm e}^{9 i \left (d x +c \right )}+38495 \,{\mathrm e}^{5 i \left (d x +c \right )}+945 \,{\mathrm e}^{13 i \left (d x +c \right )}-1664 i-9039 \,{\mathrm e}^{i \left (d x +c \right )}-11970 \,{\mathrm e}^{11 i \left (d x +c \right )}+1342 \,{\mathrm e}^{3 i \left (d x +c \right )}+18468 \,{\mathrm e}^{7 i \left (d x +c \right )}-5670 i {\mathrm e}^{10 i \left (d x +c \right )}+5670 i {\mathrm e}^{12 i \left (d x +c \right )}-11412 i {\mathrm e}^{6 i \left (d x +c \right )}+30630 i {\mathrm e}^{4 i \left (d x +c \right )}-33516 i {\mathrm e}^{8 i \left (d x +c \right )}+15962 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{315 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d \,a^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}\) | \(243\) |
| norman | \(\frac {\frac {1}{2 a d}+\frac {2492 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}+\frac {\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {102 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {57 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {601 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {134 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {5338 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{315 d a}+\frac {3583 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 d a}+\frac {10642 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d a}-\frac {11324 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}+\frac {15367 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{210 d a}-\frac {41753 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{630 d a}-\frac {17623 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(315\) |
1/2/d/a^3*(tan(1/2*d*x+1/2*c)-1/12/(tan(1/2*d*x+1/2*c)-1)^3-1/8/(tan(1/2*d *x+1/2*c)-1)^2-11/16/(tan(1/2*d*x+1/2*c)-1)-1/tan(1/2*d*x+1/2*c)-6*ln(tan( 1/2*d*x+1/2*c))-16/9/(tan(1/2*d*x+1/2*c)+1)^9+8/(tan(1/2*d*x+1/2*c)+1)^8-1 52/7/(tan(1/2*d*x+1/2*c)+1)^7+116/3/(tan(1/2*d*x+1/2*c)+1)^6-267/5/(tan(1/ 2*d*x+1/2*c)+1)^5+111/2/(tan(1/2*d*x+1/2*c)+1)^4-50/(tan(1/2*d*x+1/2*c)+1) ^3+67/2/(tan(1/2*d*x+1/2*c)+1)^2-501/16/(tan(1/2*d*x+1/2*c)+1))
Time = 0.30 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.48 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {8094 \, \cos \left (d x + c\right )^{6} - 9484 \, \cos \left (d x + c\right )^{4} + 620 \, \cos \left (d x + c\right )^{2} + 945 \, {\left (\cos \left (d x + c\right )^{7} - 5 \, \cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{3} - {\left (3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 945 \, {\left (\cos \left (d x + c\right )^{7} - 5 \, \cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{3} - {\left (3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (1664 \, \cos \left (d x + c\right )^{6} - 4653 \, \cos \left (d x + c\right )^{4} + 285 \, \cos \left (d x + c\right )^{2} + 35\right )} \sin \left (d x + c\right ) + 140}{630 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} - 5 \, a^{3} d \cos \left (d x + c\right )^{5} + 4 \, a^{3} d \cos \left (d x + c\right )^{3} - {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
1/630*(8094*cos(d*x + c)^6 - 9484*cos(d*x + c)^4 + 620*cos(d*x + c)^2 + 94 5*(cos(d*x + c)^7 - 5*cos(d*x + c)^5 + 4*cos(d*x + c)^3 - (3*cos(d*x + c)^ 5 - 4*cos(d*x + c)^3)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 945*(cos (d*x + c)^7 - 5*cos(d*x + c)^5 + 4*cos(d*x + c)^3 - (3*cos(d*x + c)^5 - 4* cos(d*x + c)^3)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*(1664*cos(d *x + c)^6 - 4653*cos(d*x + c)^4 + 285*cos(d*x + c)^2 + 35)*sin(d*x + c) + 140)/(a^3*d*cos(d*x + c)^7 - 5*a^3*d*cos(d*x + c)^5 + 4*a^3*d*cos(d*x + c) ^3 - (3*a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x + c)^3)*sin(d*x + c))
Timed out. \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (186) = 372\).
Time = 0.22 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.84 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {8786 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35076 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {43062 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {41753 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {152172 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {99072 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {93324 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {157689 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {44730 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {50820 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {42210 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {10395 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + 315}{\frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {36 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {a^{3} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}}} + \frac {1890 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {315 \, \sin \left (d x + c\right )}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}}{630 \, d} \]
-1/630*((8786*sin(d*x + c)/(cos(d*x + c) + 1) + 35076*sin(d*x + c)^2/(cos( d*x + c) + 1)^2 + 43062*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 41753*sin(d* x + c)^4/(cos(d*x + c) + 1)^4 - 152172*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 99072*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 93324*sin(d*x + c)^7/(cos(d *x + c) + 1)^7 + 157689*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 44730*sin(d* x + c)^9/(cos(d*x + c) + 1)^9 - 50820*sin(d*x + c)^10/(cos(d*x + c) + 1)^1 0 - 42210*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 10395*sin(d*x + c)^12/(c os(d*x + c) + 1)^12 + 315)/(a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 6*a^3*si n(d*x + c)^2/(cos(d*x + c) + 1)^2 + 12*a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 27*a^3*sin(d*x + c)^5/( cos(d*x + c) + 1)^5 - 36*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 36*a^3* sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 27*a^3*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 2*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 12*a^3*sin(d*x + c) ^11/(cos(d*x + c) + 1)^11 - 6*a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - a^3*sin(d*x + c)^13/(cos(d*x + c) + 1)^13) + 1890*log(sin(d*x + c)/(cos(d* x + c) + 1))/a^3 - 315*sin(d*x + c)/(a^3*(cos(d*x + c) + 1)))/d
Time = 0.46 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {30240 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {5040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {5040 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {105 \, {\left (33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 31\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {157815 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1093680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3488940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6524280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7788186 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6052704 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2995596 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 864504 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 113591}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{10080 \, d} \]
-1/10080*(30240*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - 5040*tan(1/2*d*x + 1/ 2*c)/a^3 - 5040*(6*tan(1/2*d*x + 1/2*c) - 1)/(a^3*tan(1/2*d*x + 1/2*c)) + 105*(33*tan(1/2*d*x + 1/2*c)^2 - 60*tan(1/2*d*x + 1/2*c) + 31)/(a^3*(tan(1 /2*d*x + 1/2*c) - 1)^3) + (157815*tan(1/2*d*x + 1/2*c)^8 + 1093680*tan(1/2 *d*x + 1/2*c)^7 + 3488940*tan(1/2*d*x + 1/2*c)^6 + 6524280*tan(1/2*d*x + 1 /2*c)^5 + 7788186*tan(1/2*d*x + 1/2*c)^4 + 6052704*tan(1/2*d*x + 1/2*c)^3 + 2995596*tan(1/2*d*x + 1/2*c)^2 + 864504*tan(1/2*d*x + 1/2*c) + 113591)/( a^3*(tan(1/2*d*x + 1/2*c) + 1)^9))/d
Time = 11.73 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.95 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {-33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-134\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-\frac {484\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+142\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {2503\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}+\frac {4444\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15}-\frac {11008\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{35}-\frac {16908\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{35}-\frac {41753\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{315}+\frac {14354\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{105}+\frac {11692\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}+\frac {8786\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{315}+1}{d\,\left (-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-24\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+72\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-72\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+24\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \]
tan(c/2 + (d*x)/2)/(2*a^3*d) - (3*log(tan(c/2 + (d*x)/2)))/(a^3*d) - ((878 6*tan(c/2 + (d*x)/2))/315 + (11692*tan(c/2 + (d*x)/2)^2)/105 + (14354*tan( c/2 + (d*x)/2)^3)/105 - (41753*tan(c/2 + (d*x)/2)^4)/315 - (16908*tan(c/2 + (d*x)/2)^5)/35 - (11008*tan(c/2 + (d*x)/2)^6)/35 + (4444*tan(c/2 + (d*x) /2)^7)/15 + (2503*tan(c/2 + (d*x)/2)^8)/5 + 142*tan(c/2 + (d*x)/2)^9 - (48 4*tan(c/2 + (d*x)/2)^10)/3 - 134*tan(c/2 + (d*x)/2)^11 - 33*tan(c/2 + (d*x )/2)^12 + 1)/(d*(12*a^3*tan(c/2 + (d*x)/2)^2 + 24*a^3*tan(c/2 + (d*x)/2)^3 + 4*a^3*tan(c/2 + (d*x)/2)^4 - 54*a^3*tan(c/2 + (d*x)/2)^5 - 72*a^3*tan(c /2 + (d*x)/2)^6 + 72*a^3*tan(c/2 + (d*x)/2)^8 + 54*a^3*tan(c/2 + (d*x)/2)^ 9 - 4*a^3*tan(c/2 + (d*x)/2)^10 - 24*a^3*tan(c/2 + (d*x)/2)^11 - 12*a^3*ta n(c/2 + (d*x)/2)^12 - 2*a^3*tan(c/2 + (d*x)/2)^13 + 2*a^3*tan(c/2 + (d*x)/ 2)))