Integrand size = 29, antiderivative size = 172 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {\sec ^8(c+d x)}{8 a d}-\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {7 \sec ^3(c+d x) \tan (c+d x)}{384 a d}-\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}-\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d} \]
-7/256*arctanh(sin(d*x+c))/a/d+1/8*sec(d*x+c)^8/a/d-1/10*sec(d*x+c)^10/a/d -7/256*sec(d*x+c)*tan(d*x+c)/a/d-7/384*sec(d*x+c)^3*tan(d*x+c)/a/d-7/480*s ec(d*x+c)^5*tan(d*x+c)/a/d-1/80*sec(d*x+c)^7*tan(d*x+c)/a/d+1/10*sec(d*x+c )^9*tan(d*x+c)/a/d
Time = 1.71 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.71 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {210 \text {arctanh}(\sin (c+d x))-\frac {2 \left (96+201 \sin (c+d x)-279 \sin ^2(c+d x)+511 \sin ^3(c+d x)+511 \sin ^4(c+d x)-385 \sin ^5(c+d x)-385 \sin ^6(c+d x)+105 \sin ^7(c+d x)+105 \sin ^8(c+d x)\right )}{(-1+\sin (c+d x))^4 (1+\sin (c+d x))^5}}{7680 a d} \]
-1/7680*(210*ArcTanh[Sin[c + d*x]] - (2*(96 + 201*Sin[c + d*x] - 279*Sin[c + d*x]^2 + 511*Sin[c + d*x]^3 + 511*Sin[c + d*x]^4 - 385*Sin[c + d*x]^5 - 385*Sin[c + d*x]^6 + 105*Sin[c + d*x]^7 + 105*Sin[c + d*x]^8))/((-1 + Sin [c + d*x])^4*(1 + Sin[c + d*x])^5))/(a*d)
Time = 0.98 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.02, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3314, 3042, 3086, 25, 244, 2009, 3091, 3042, 4255, 3042, 4255, 3042, 4255, 3042, 4255, 3042, 4257}
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.
| \(\displaystyle \int \frac {\tan ^2(c+d x) \sec ^7(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2}{\cos (c+d x)^9 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3314 |
| \(\displaystyle \frac {\int \sec ^9(c+d x) \tan ^2(c+d x)dx}{a}-\frac {\int \sec ^8(c+d x) \tan ^3(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec (c+d x)^9 \tan (c+d x)^2dx}{a}-\frac {\int \sec (c+d x)^8 \tan (c+d x)^3dx}{a}\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle \frac {\int \sec (c+d x)^9 \tan (c+d x)^2dx}{a}-\frac {\int -\sec ^7(c+d x) \left (1-\sec ^2(c+d x)\right )d\sec (c+d x)}{a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \sec ^7(c+d x) \left (1-\sec ^2(c+d x)\right )d\sec (c+d x)}{a d}+\frac {\int \sec (c+d x)^9 \tan (c+d x)^2dx}{a}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \left (\sec ^7(c+d x)-\sec ^9(c+d x)\right )d\sec (c+d x)}{a d}+\frac {\int \sec (c+d x)^9 \tan (c+d x)^2dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \sec (c+d x)^9 \tan (c+d x)^2dx}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}-\frac {1}{10} \int \sec ^9(c+d x)dx}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}-\frac {1}{10} \int \csc \left (c+d x+\frac {\pi }{2}\right )^9dx}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{10} \left (-\frac {7}{8} \int \sec ^7(c+d x)dx-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}\right )+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{10} \left (-\frac {7}{8} \int \csc \left (c+d x+\frac {\pi }{2}\right )^7dx-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}\right )+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{10} \left (-\frac {7}{8} \left (\frac {5}{6} \int \sec ^5(c+d x)dx+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}\right )+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{10} \left (-\frac {7}{8} \left (\frac {5}{6} \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}\right )+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{10} \left (-\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sec ^3(c+d x)dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}\right )+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{10} \left (-\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}\right )+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{10} \left (-\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}\right )+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{10} \left (-\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}\right )+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{10} \left (-\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}\right )+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 d}}{a}-\frac {\frac {1}{10} \sec ^{10}(c+d x)-\frac {1}{8} \sec ^8(c+d x)}{a d}\) |
-((-1/8*Sec[c + d*x]^8 + Sec[c + d*x]^10/10)/(a*d)) + ((Sec[c + d*x]^9*Tan [c + d*x])/(10*d) + (-1/8*(Sec[c + d*x]^7*Tan[c + d*x])/d - (7*((Sec[c + d *x]^5*Tan[c + d*x])/(6*d) + (5*((Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*( ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/4))/6))/ 8)/10)/a
3.10.5.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/(( a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/a Int[Cos[e + f *x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[1/(b*d) Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] & & IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n, -p]))
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.69 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {1}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {1}{128 \sin \left (d x +c \right )-128}+\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(139\) |
| default | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {1}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {1}{128 \sin \left (d x +c \right )-128}+\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(139\) |
| risch | \(\frac {i \left (1610 i {\mathrm e}^{14 i \left (d x +c \right )}+105 \,{\mathrm e}^{17 i \left (d x +c \right )}+210 i {\mathrm e}^{16 i \left (d x +c \right )}+700 \,{\mathrm e}^{15 i \left (d x +c \right )}+51334 i {\mathrm e}^{8 i \left (d x +c \right )}+1876 \,{\mathrm e}^{13 i \left (d x +c \right )}-5362 i {\mathrm e}^{6 i \left (d x +c \right )}+2372 \,{\mathrm e}^{11 i \left (d x +c \right )}+5362 i {\mathrm e}^{12 i \left (d x +c \right )}+14470 \,{\mathrm e}^{9 i \left (d x +c \right )}-51334 i {\mathrm e}^{10 i \left (d x +c \right )}+2372 \,{\mathrm e}^{7 i \left (d x +c \right )}-1610 i {\mathrm e}^{4 i \left (d x +c \right )}+1876 \,{\mathrm e}^{5 i \left (d x +c \right )}-210 i {\mathrm e}^{2 i \left (d x +c \right )}+700 \,{\mathrm e}^{3 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{1920 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} d a}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}\) | \(277\) |
| parallelrisch | \(\frac {\left (105 \cos \left (10 d x +10 c \right )+22050 \cos \left (2 d x +2 c \right )+12600 \cos \left (4 d x +4 c \right )+4725 \cos \left (6 d x +6 c \right )+1050 \cos \left (8 d x +8 c \right )+13230\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-105 \cos \left (10 d x +10 c \right )-22050 \cos \left (2 d x +2 c \right )-12600 \cos \left (4 d x +4 c \right )-4725 \cos \left (6 d x +6 c \right )-1050 \cos \left (8 d x +8 c \right )-13230\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-22440 \sin \left (3 d x +3 c \right )-8792 \sin \left (5 d x +5 c \right )-2030 \sin \left (7 d x +7 c \right )-210 \sin \left (9 d x +9 c \right )-96 \cos \left (10 d x +10 c \right )+102720 \cos \left (2 d x +2 c \right )-11520 \cos \left (4 d x +4 c \right )-4320 \cos \left (6 d x +6 c \right )-960 \cos \left (8 d x +8 c \right )+181140 \sin \left (d x +c \right )-85824}{3840 a d \left (\cos \left (10 d x +10 c \right )+10 \cos \left (8 d x +8 c \right )+45 \cos \left (6 d x +6 c \right )+120 \cos \left (4 d x +4 c \right )+210 \cos \left (2 d x +2 c \right )+126\right )}\) | \(315\) |
1/d/a*(1/256/(sin(d*x+c)-1)^4-1/192/(sin(d*x+c)-1)^3+1/512/(sin(d*x+c)-1)^ 2+1/128/(sin(d*x+c)-1)+7/512*ln(sin(d*x+c)-1)-1/160/(1+sin(d*x+c))^5-1/256 /(1+sin(d*x+c))^4+1/384/(1+sin(d*x+c))^3+5/512/(1+sin(d*x+c))^2+5/256/(1+s in(d*x+c))-7/512*ln(1+sin(d*x+c)))
Time = 0.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {210 \, \cos \left (d x + c\right )^{8} - 70 \, \cos \left (d x + c\right )^{6} - 28 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (105 \, \cos \left (d x + c\right )^{6} + 70 \, \cos \left (d x + c\right )^{4} + 56 \, \cos \left (d x + c\right )^{2} - 432\right )} \sin \left (d x + c\right ) + 96}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
1/7680*(210*cos(d*x + c)^8 - 70*cos(d*x + c)^6 - 28*cos(d*x + c)^4 - 16*co s(d*x + c)^2 - 105*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)^8)*log(sin( d*x + c) + 1) + 105*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)^8)*log(-si n(d*x + c) + 1) - 2*(105*cos(d*x + c)^6 + 70*cos(d*x + c)^4 + 56*cos(d*x + c)^2 - 432)*sin(d*x + c) + 96)/(a*d*cos(d*x + c)^8*sin(d*x + c) + a*d*cos (d*x + c)^8)
Timed out. \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.24 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{7} - 385 \, \sin \left (d x + c\right )^{6} - 385 \, \sin \left (d x + c\right )^{5} + 511 \, \sin \left (d x + c\right )^{4} + 511 \, \sin \left (d x + c\right )^{3} - 279 \, \sin \left (d x + c\right )^{2} + 201 \, \sin \left (d x + c\right ) + 96\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \]
1/7680*(2*(105*sin(d*x + c)^8 + 105*sin(d*x + c)^7 - 385*sin(d*x + c)^6 - 385*sin(d*x + c)^5 + 511*sin(d*x + c)^4 + 511*sin(d*x + c)^3 - 279*sin(d*x + c)^2 + 201*sin(d*x + c) + 96)/(a*sin(d*x + c)^9 + a*sin(d*x + c)^8 - 4* a*sin(d*x + c)^7 - 4*a*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 + 6*a*sin(d*x + c)^4 - 4*a*sin(d*x + c)^3 - 4*a*sin(d*x + c)^2 + a*sin(d*x + c) + a) - 10 5*log(sin(d*x + c) + 1)/a + 105*log(sin(d*x + c) - 1)/a)/d
Time = 0.57 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.91 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (175 \, \sin \left (d x + c\right )^{4} - 748 \, \sin \left (d x + c\right )^{3} + 1182 \, \sin \left (d x + c\right )^{2} - 788 \, \sin \left (d x + c\right ) + 155\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {959 \, \sin \left (d x + c\right )^{5} + 5395 \, \sin \left (d x + c\right )^{4} + 12290 \, \sin \left (d x + c\right )^{3} + 14170 \, \sin \left (d x + c\right )^{2} + 8135 \, \sin \left (d x + c\right ) + 1627}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
-1/30720*(420*log(abs(sin(d*x + c) + 1))/a - 420*log(abs(sin(d*x + c) - 1) )/a + 5*(175*sin(d*x + c)^4 - 748*sin(d*x + c)^3 + 1182*sin(d*x + c)^2 - 7 88*sin(d*x + c) + 155)/(a*(sin(d*x + c) - 1)^4) - (959*sin(d*x + c)^5 + 53 95*sin(d*x + c)^4 + 12290*sin(d*x + c)^3 + 14170*sin(d*x + c)^2 + 8135*sin (d*x + c) + 1627)/(a*(sin(d*x + c) + 1)^5))/d
Time = 17.96 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.88 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}+\frac {221\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+\frac {95\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{192}+\frac {2261\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{480}+\frac {889\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{960}+\frac {7343\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{480}+\frac {1603\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{960}+\frac {2471\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {1603\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{960}+\frac {7343\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{480}+\frac {889\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{960}+\frac {2261\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{480}+\frac {95\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{192}+\frac {221\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}+\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+140\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}-\frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d} \]
((7*tan(c/2 + (d*x)/2))/128 + (7*tan(c/2 + (d*x)/2)^2)/64 + (221*tan(c/2 + (d*x)/2)^3)/96 + (95*tan(c/2 + (d*x)/2)^4)/192 + (2261*tan(c/2 + (d*x)/2) ^5)/480 + (889*tan(c/2 + (d*x)/2)^6)/960 + (7343*tan(c/2 + (d*x)/2)^7)/480 + (1603*tan(c/2 + (d*x)/2)^8)/960 + (2471*tan(c/2 + (d*x)/2)^9)/192 + (16 03*tan(c/2 + (d*x)/2)^10)/960 + (7343*tan(c/2 + (d*x)/2)^11)/480 + (889*ta n(c/2 + (d*x)/2)^12)/960 + (2261*tan(c/2 + (d*x)/2)^13)/480 + (95*tan(c/2 + (d*x)/2)^14)/192 + (221*tan(c/2 + (d*x)/2)^15)/96 + (7*tan(c/2 + (d*x)/2 )^16)/64 + (7*tan(c/2 + (d*x)/2)^17)/128)/(d*(a + 2*a*tan(c/2 + (d*x)/2) - 7*a*tan(c/2 + (d*x)/2)^2 - 16*a*tan(c/2 + (d*x)/2)^3 + 20*a*tan(c/2 + (d* x)/2)^4 + 56*a*tan(c/2 + (d*x)/2)^5 - 28*a*tan(c/2 + (d*x)/2)^6 - 112*a*ta n(c/2 + (d*x)/2)^7 + 14*a*tan(c/2 + (d*x)/2)^8 + 140*a*tan(c/2 + (d*x)/2)^ 9 + 14*a*tan(c/2 + (d*x)/2)^10 - 112*a*tan(c/2 + (d*x)/2)^11 - 28*a*tan(c/ 2 + (d*x)/2)^12 + 56*a*tan(c/2 + (d*x)/2)^13 + 20*a*tan(c/2 + (d*x)/2)^14 - 16*a*tan(c/2 + (d*x)/2)^15 - 7*a*tan(c/2 + (d*x)/2)^16 + 2*a*tan(c/2 + ( d*x)/2)^17 + a*tan(c/2 + (d*x)/2)^18)) - (7*atanh(tan(c/2 + (d*x)/2)))/(12 8*a*d)