Integrand size = 29, antiderivative size = 178 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {x}{a^3}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {\tan (c+d x)}{a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d} \]
x/a^3+3*sec(d*x+c)/a^3/d-13/3*sec(d*x+c)^3/a^3/d+21/5*sec(d*x+c)^5/a^3/d-1 5/7*sec(d*x+c)^7/a^3/d+4/9*sec(d*x+c)^9/a^3/d-tan(d*x+c)/a^3/d+1/3*tan(d*x +c)^3/a^3/d-1/5*tan(d*x+c)^5/a^3/d+1/7*tan(d*x+c)^7/a^3/d-4/9*tan(d*x+c)^9 /a^3/d
Time = 1.21 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.53 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {169344-675036 \cos (c+d x)+362880 (c+d x) \cos (c+d x)+173952 \cos (2 (c+d x))-37502 \cos (3 (c+d x))+20160 (c+d x) \cos (3 (c+d x))+54912 \cos (4 (c+d x))+112506 \cos (5 (c+d x))-60480 (c+d x) \cos (5 (c+d x))-21376 \cos (6 (c+d x))+93312 \sin (c+d x)-506277 \sin (2 (c+d x))+272160 (c+d x) \sin (2 (c+d x))+125248 \sin (3 (c+d x))-225012 \sin (4 (c+d x))+120960 (c+d x) \sin (4 (c+d x))+67776 \sin (5 (c+d x))+18751 \sin (6 (c+d x))-10080 (c+d x) \sin (6 (c+d x))}{322560 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a+a \sin (c+d x))^3} \]
(169344 - 675036*Cos[c + d*x] + 362880*(c + d*x)*Cos[c + d*x] + 173952*Cos [2*(c + d*x)] - 37502*Cos[3*(c + d*x)] + 20160*(c + d*x)*Cos[3*(c + d*x)] + 54912*Cos[4*(c + d*x)] + 112506*Cos[5*(c + d*x)] - 60480*(c + d*x)*Cos[5 *(c + d*x)] - 21376*Cos[6*(c + d*x)] + 93312*Sin[c + d*x] - 506277*Sin[2*( c + d*x)] + 272160*(c + d*x)*Sin[2*(c + d*x)] + 125248*Sin[3*(c + d*x)] - 225012*Sin[4*(c + d*x)] + 120960*(c + d*x)*Sin[4*(c + d*x)] + 67776*Sin[5* (c + d*x)] + 18751*Sin[6*(c + d*x)] - 10080*(c + d*x)*Sin[6*(c + d*x)])/(3 22560*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)
Time = 0.61 (sec) , antiderivative size = 182, 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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^7}{\cos (c+d x)^4 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \sec ^3(c+d x) (a-a \sin (c+d x))^3 \tan ^7(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^7 (a-a \sin (c+d x))^3}{\cos (c+d x)^{10}}dx}{a^6}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (-a^3 \tan ^{10}(c+d x)+3 a^3 \sec (c+d x) \tan ^9(c+d x)-3 a^3 \sec ^2(c+d x) \tan ^8(c+d x)+a^3 \sec ^3(c+d x) \tan ^7(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {4 a^3 \tan ^9(c+d x)}{9 d}+\frac {a^3 \tan ^7(c+d x)}{7 d}-\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {a^3 \tan (c+d x)}{d}+\frac {4 a^3 \sec ^9(c+d x)}{9 d}-\frac {15 a^3 \sec ^7(c+d x)}{7 d}+\frac {21 a^3 \sec ^5(c+d x)}{5 d}-\frac {13 a^3 \sec ^3(c+d x)}{3 d}+\frac {3 a^3 \sec (c+d x)}{d}+a^3 x}{a^6}\) |
(a^3*x + (3*a^3*Sec[c + d*x])/d - (13*a^3*Sec[c + d*x]^3)/(3*d) + (21*a^3* Sec[c + d*x]^5)/(5*d) - (15*a^3*Sec[c + d*x]^7)/(7*d) + (4*a^3*Sec[c + d*x ]^9)/(9*d) - (a^3*Tan[c + d*x])/d + (a^3*Tan[c + d*x]^3)/(3*d) - (a^3*Tan[ c + d*x]^5)/(5*d) + (a^3*Tan[c + d*x]^7)/(7*d) - (4*a^3*Tan[c + d*x]^9)/(9 *d))/a^6
3.9.39.3.1 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]
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {x}{a^{3}}+\frac {20 i {\mathrm e}^{10 i \left (d x +c \right )}+6 \,{\mathrm e}^{11 i \left (d x +c \right )}+40 i {\mathrm e}^{8 i \left (d x +c \right )}-\frac {50 \,{\mathrm e}^{9 i \left (d x +c \right )}}{3}-\frac {168 i {\mathrm e}^{6 i \left (d x +c \right )}}{5}-\frac {428 \,{\mathrm e}^{7 i \left (d x +c \right )}}{5}-\frac {2608 i {\mathrm e}^{4 i \left (d x +c \right )}}{35}-\frac {2348 \,{\mathrm e}^{5 i \left (d x +c \right )}}{35}-\frac {3244 i {\mathrm e}^{2 i \left (d x +c \right )}}{105}+\frac {2578 \,{\mathrm e}^{3 i \left (d x +c \right )}}{315}+\frac {1336 i}{315}+\frac {2042 \,{\mathrm e}^{i \left (d x +c \right )}}{105}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d \,a^{3}}\) | \(172\) |
| derivativedivides | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {40}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {21}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {13}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {57}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{3}}\) | \(202\) |
| default | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {40}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {21}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {13}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {57}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{3}}\) | \(202\) |
| parallelrisch | \(\frac {315 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +\left (1890 d x +630\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3780 d x +3780\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (630 d x +7350\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8505 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +\left (-11340 d x -19404\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-22344 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (11340 d x +7092\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (8505 d x +19872\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-630 d x +5878\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3780 d x -5052\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1890 d x -3786\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-315 d x -736}{315 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}\) | \(244\) |
x/a^3+2/315*(3150*I*exp(10*I*(d*x+c))+945*exp(11*I*(d*x+c))+6300*I*exp(8*I *(d*x+c))-2625*exp(9*I*(d*x+c))-5292*I*exp(6*I*(d*x+c))-13482*exp(7*I*(d*x +c))-11736*I*exp(4*I*(d*x+c))-10566*exp(5*I*(d*x+c))-4866*I*exp(2*I*(d*x+c ))+1289*exp(3*I*(d*x+c))+668*I+3063*exp(I*(d*x+c)))/(exp(I*(d*x+c))+I)^9/( exp(I*(d*x+c))-I)^3/d/a^3
Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.99 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {945 \, d x \cos \left (d x + c\right )^{5} + 668 \, \cos \left (d x + c\right )^{6} - 1260 \, d x \cos \left (d x + c\right )^{3} - 1431 \, \cos \left (d x + c\right )^{4} + 465 \, \cos \left (d x + c\right )^{2} + {\left (315 \, d x \cos \left (d x + c\right )^{5} - 1260 \, d x \cos \left (d x + c\right )^{3} - 1059 \, \cos \left (d x + c\right )^{4} + 305 \, \cos \left (d x + c\right )^{2} - 35\right )} \sin \left (d x + c\right ) - 70}{315 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
1/315*(945*d*x*cos(d*x + c)^5 + 668*cos(d*x + c)^6 - 1260*d*x*cos(d*x + c) ^3 - 1431*cos(d*x + c)^4 + 465*cos(d*x + c)^2 + (315*d*x*cos(d*x + c)^5 - 1260*d*x*cos(d*x + c)^3 - 1059*cos(d*x + c)^4 + 305*cos(d*x + c)^2 - 35)*s in(d*x + c) - 70)/(3*a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x + c)^3 + (a^3* d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x + c)^3)*sin(d*x + c))
Timed out. \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (162) = 324\).
Time = 0.32 (sec) , antiderivative size = 487, normalized size of antiderivative = 2.74 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2 \, {\left (\frac {\frac {1893 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2526 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2939 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {9936 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3546 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {11172 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {9702 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3675 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {1890 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 368}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, 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 )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {315 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{315 \, d} \]
2/315*((1893*sin(d*x + c)/(cos(d*x + c) + 1) + 2526*sin(d*x + c)^2/(cos(d* x + c) + 1)^2 - 2939*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 9936*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3546*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 111 72*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 9702*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 3675*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 1890*sin(d*x + c)^10/ (cos(d*x + c) + 1)^10 - 315*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 368)/( a^3 + 6*a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 12*a^3*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 + 2*a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 27*a^3*sin(d* x + c)^4/(cos(d*x + c) + 1)^4 - 36*a^3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 36*a^3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 27*a^3*sin(d*x + c)^8/(cos (d*x + c) + 1)^8 - 2*a^3*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 12*a^3*sin( d*x + c)^10/(cos(d*x + c) + 1)^10 - 6*a^3*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) + 315*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.44 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {10080 \, {\left (d x + c\right )}}{a^{3}} + \frac {105 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 23\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {17955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 160020 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 624960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1387260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1884582 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1556268 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 774792 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 215748 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25967}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{10080 \, d} \]
1/10080*(10080*(d*x + c)/a^3 + 105*(21*tan(1/2*d*x + 1/2*c)^2 - 48*tan(1/2 *d*x + 1/2*c) + 23)/(a^3*(tan(1/2*d*x + 1/2*c) - 1)^3) + (17955*tan(1/2*d* x + 1/2*c)^8 + 160020*tan(1/2*d*x + 1/2*c)^7 + 624960*tan(1/2*d*x + 1/2*c) ^6 + 1387260*tan(1/2*d*x + 1/2*c)^5 + 1884582*tan(1/2*d*x + 1/2*c)^4 + 155 6268*tan(1/2*d*x + 1/2*c)^3 + 774792*tan(1/2*d*x + 1/2*c)^2 + 215748*tan(1 /2*d*x + 1/2*c) + 25967)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^9))/d
Time = 20.07 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {x}{a^3}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}-\frac {308\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {1064\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {788\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{35}+\frac {2208\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}+\frac {5878\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{315}-\frac {1684\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}-\frac {1262\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-\frac {736}{315}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^9} \]
x/a^3 + ((5878*tan(c/2 + (d*x)/2)^3)/315 - (1684*tan(c/2 + (d*x)/2)^2)/105 - (1262*tan(c/2 + (d*x)/2))/105 + (2208*tan(c/2 + (d*x)/2)^4)/35 + (788*t an(c/2 + (d*x)/2)^5)/35 - (1064*tan(c/2 + (d*x)/2)^6)/15 - (308*tan(c/2 + (d*x)/2)^7)/5 + (70*tan(c/2 + (d*x)/2)^9)/3 + 12*tan(c/2 + (d*x)/2)^10 + 2 *tan(c/2 + (d*x)/2)^11 - 736/315)/(a^3*d*(tan(c/2 + (d*x)/2) - 1)^3*(tan(c /2 + (d*x)/2) + 1)^9)