Integrand size = 29, antiderivative size = 162 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {7 \tan (c+d x)}{a^3 d}+\frac {5 \tan ^3(c+d x)}{a^3 d}+\frac {13 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d} \]
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-4/7*sec(d*x+c)^7/a^3/d+7*tan(d*x+c)/a^3/d+ 5*tan(d*x+c)^3/a^3/d+13/5*tan(d*x+c)^5/a^3/d+4/7*tan(d*x+c)^7/a^3/d
Leaf count is larger than twice the leaf count of optimal. \(351\) vs. \(2(162)=324\).
Time = 1.17 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.17 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\csc ^3(c+d x) \left (-966-440 \cos (2 (c+d x))-2640 \cos (3 (c+d x))+846 \cos (4 (c+d x))+176 \cos (5 (c+d x))-1575 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+14 \cos (c+d x) \left (176+105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+1575 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1316 \sin (c+d x)+3520 \sin (2 (c+d x))+2100 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (2 (c+d x))-2100 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (2 (c+d x))-1380 \sin (3 (c+d x))-1056 \sin (4 (c+d x))-630 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))+630 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))+176 \sin (5 (c+d x))\right )}{140 a^3 d \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\sin (c+d x))^3} \]
(Csc[c + d*x]^3*(-966 - 440*Cos[2*(c + d*x)] - 2640*Cos[3*(c + d*x)] + 846 *Cos[4*(c + d*x)] + 176*Cos[5*(c + d*x)] - 1575*Cos[3*(c + d*x)]*Log[Cos[( c + d*x)/2]] + 105*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 14*Cos[c + d*x ]*(176 + 105*Log[Cos[(c + d*x)/2]] - 105*Log[Sin[(c + d*x)/2]]) + 1575*Cos [3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 105*Cos[5*(c + d*x)]*Log[Sin[(c + d* x)/2]] - 1316*Sin[c + d*x] + 3520*Sin[2*(c + d*x)] + 2100*Log[Cos[(c + d*x )/2]]*Sin[2*(c + d*x)] - 2100*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] - 138 0*Sin[3*(c + d*x)] - 1056*Sin[4*(c + d*x)] - 630*Log[Cos[(c + d*x)/2]]*Sin [4*(c + d*x)] + 630*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] + 176*Sin[5*(c + d*x)]))/(140*a^3*d*(Csc[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^2)*(1 + Sin[c + d*x])^3)
Time = 0.58 (sec) , antiderivative size = 166, 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.
| \(\displaystyle \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^2 \cos (c+d x)^2 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \csc ^2(c+d x) \sec ^8(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3}{\cos (c+d x)^8 \sin (c+d x)^2}dx}{a^6}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (3 a^3 \sec ^8(c+d x)+a^3 \csc ^2(c+d x) \sec ^8(c+d x)-3 a^3 \csc (c+d x) \sec ^8(c+d x)-a^3 \tan (c+d x) \sec ^7(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 \tan ^7(c+d x)}{7 d}+\frac {13 a^3 \tan ^5(c+d x)}{5 d}+\frac {5 a^3 \tan ^3(c+d x)}{d}+\frac {7 a^3 \tan (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}-\frac {4 a^3 \sec ^7(c+d x)}{7 d}-\frac {3 a^3 \sec ^5(c+d x)}{5 d}-\frac {a^3 \sec ^3(c+d x)}{d}-\frac {3 a^3 \sec (c+d x)}{d}}{a^6}\) |
((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) - (4*a^3*Se c[c + d*x]^7)/(7*d) + (7*a^3*Tan[c + d*x])/d + (5*a^3*Tan[c + d*x]^3)/d + (13*a^3*Tan[c + d*x]^5)/(5*d) + (4*a^3*Tan[c + d*x]^7)/(7*d))/a^6
3.8.94.3.1 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 = 0.95 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {92}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {26}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {31}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {49}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {111}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{2 d \,a^{3}}\) | \(164\) |
| default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {92}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {26}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {31}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {49}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {111}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{2 d \,a^{3}}\) | \(164\) |
| parallelrisch | \(\frac {-210 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1785 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6720 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10010 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4200 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6006 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+9296 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+5079 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1144}{70 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}\) | \(186\) |
| risch | \(-\frac {2 \left (-1540 \,{\mathrm e}^{7 i \left (d x +c \right )}+630 i {\mathrm e}^{8 i \left (d x +c \right )}+966 \,{\mathrm e}^{5 i \left (d x +c \right )}-1890 i {\mathrm e}^{6 i \left (d x +c \right )}+1980 \,{\mathrm e}^{3 i \left (d x +c \right )}-951 \,{\mathrm e}^{i \left (d x +c \right )}+2010 i {\mathrm e}^{2 i \left (d x +c \right )}-176 i-574 i {\mathrm e}^{4 i \left (d x +c \right )}+105 \,{\mathrm e}^{9 i \left (d x +c \right )}\right )}{35 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) d \,a^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}\) | \(197\) |
| norman | \(\frac {-\frac {143 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {1}{2 a d}-\frac {60 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {51 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {96 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {572 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{35 d a}+\frac {664 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {429 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {5079 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 d a}}{a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(239\) |
1/2/d/a^3*(tan(1/2*d*x+1/2*c)-1/4/(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))-16/7/(tan(1/2*d*x+1/2*c)+1)^7+8/(tan(1/2*d*x+ 1/2*c)+1)^6-92/5/(tan(1/2*d*x+1/2*c)+1)^5+26/(tan(1/2*d*x+1/2*c)+1)^4-31/( tan(1/2*d*x+1/2*c)+1)^3+49/2/(tan(1/2*d*x+1/2*c)+1)^2-111/4/(tan(1/2*d*x+1 /2*c)+1))
Time = 0.27 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.64 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {846 \, \cos \left (d x + c\right )^{4} - 956 \, \cos \left (d x + c\right )^{2} + 105 \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} - {\left (3 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 105 \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} - {\left (3 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (176 \, \cos \left (d x + c\right )^{4} - 477 \, \cos \left (d x + c\right )^{2} + 15\right )} \sin \left (d x + c\right ) + 40}{70 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + 4 \, a^{3} d \cos \left (d x + c\right ) - {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
1/70*(846*cos(d*x + c)^4 - 956*cos(d*x + c)^2 + 105*(cos(d*x + c)^5 - 5*co s(d*x + c)^3 - (3*cos(d*x + c)^3 - 4*cos(d*x + c))*sin(d*x + c) + 4*cos(d* x + c))*log(1/2*cos(d*x + c) + 1/2) - 105*(cos(d*x + c)^5 - 5*cos(d*x + c) ^3 - (3*cos(d*x + c)^3 - 4*cos(d*x + c))*sin(d*x + c) + 4*cos(d*x + c))*lo g(-1/2*cos(d*x + c) + 1/2) + 2*(176*cos(d*x + c)^4 - 477*cos(d*x + c)^2 + 15)*sin(d*x + c) + 40)/(a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^3 + 4* a^3*d*cos(d*x + c) - (3*a^3*d*cos(d*x + c)^3 - 4*a^3*d*cos(d*x + c))*sin(d *x + c))
\[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\csc ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Integral(csc(c + d*x)**2*sec(c + d*x)**2/(sin(c + d*x)**3 + 3*sin(c + d*x) **2 + 3*sin(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (154) = 308\).
Time = 0.22 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.44 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {934 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3854 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6566 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3556 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3710 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {7070 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {4270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1015 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 35}{\frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} + \frac {210 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {35 \, \sin \left (d x + c\right )}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}}{70 \, d} \]
-1/70*((934*sin(d*x + c)/(cos(d*x + c) + 1) + 3854*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6566*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3556*sin(d*x + c )^4/(cos(d*x + c) + 1)^4 - 3710*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 7070 *sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 4270*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 1015*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 35)/(a^3*sin(d*x + c)/( cos(d*x + c) + 1) + 6*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 14*a^3*sin (d*x + c)^3/(cos(d*x + c) + 1)^3 + 14*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1 )^4 - 14*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 14*a^3*sin(d*x + c)^7/( cos(d*x + c) + 1)^7 - 6*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - a^3*sin( d*x + c)^9/(cos(d*x + c) + 1)^9) + 210*log(sin(d*x + c)/(cos(d*x + c) + 1) )/a^3 - 35*sin(d*x + c)/(a^3*(cos(d*x + c) + 1)))/d
Time = 0.33 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {35 \, {\left (12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 17 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} a^{3}} + \frac {3885 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 19880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 57120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 41671 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16632 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2931}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \]
-1/280*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - 140*tan(1/2*d*x + 1/2*c)/ a^3 - 35*(12*tan(1/2*d*x + 1/2*c)^2 - 17*tan(1/2*d*x + 1/2*c) + 4)/((tan(1 /2*d*x + 1/2*c)^2 - tan(1/2*d*x + 1/2*c))*a^3) + (3885*tan(1/2*d*x + 1/2*c )^6 + 19880*tan(1/2*d*x + 1/2*c)^5 + 45465*tan(1/2*d*x + 1/2*c)^4 + 57120* tan(1/2*d*x + 1/2*c)^3 + 41671*tan(1/2*d*x + 1/2*c)^2 + 16632*tan(1/2*d*x + 1/2*c) + 2931)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^7))/d
Time = 11.88 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.69 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {-29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-122\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-202\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-106\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {508\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {938\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+\frac {3854\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}+\frac {934\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}+1}{d\,\left (-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d} \]
tan(c/2 + (d*x)/2)/(2*a^3*d) - ((934*tan(c/2 + (d*x)/2))/35 + (3854*tan(c/ 2 + (d*x)/2)^2)/35 + (938*tan(c/2 + (d*x)/2)^3)/5 + (508*tan(c/2 + (d*x)/2 )^4)/5 - 106*tan(c/2 + (d*x)/2)^5 - 202*tan(c/2 + (d*x)/2)^6 - 122*tan(c/2 + (d*x)/2)^7 - 29*tan(c/2 + (d*x)/2)^8 + 1)/(d*(12*a^3*tan(c/2 + (d*x)/2) ^2 + 28*a^3*tan(c/2 + (d*x)/2)^3 + 28*a^3*tan(c/2 + (d*x)/2)^4 - 28*a^3*ta n(c/2 + (d*x)/2)^6 - 28*a^3*tan(c/2 + (d*x)/2)^7 - 12*a^3*tan(c/2 + (d*x)/ 2)^8 - 2*a^3*tan(c/2 + (d*x)/2)^9 + 2*a^3*tan(c/2 + (d*x)/2))) - (3*log(ta n(c/2 + (d*x)/2)))/(a^3*d)