Integrand size = 13, antiderivative size = 101 \[ \int \frac {\sin ^6(x)}{(a+a \sin (x))^3} \, dx=-\frac {23 x}{2 a^3}-\frac {136 \cos (x)}{5 a^3}+\frac {136 \cos ^3(x)}{15 a^3}+\frac {23 \cos (x) \sin (x)}{2 a^3}+\frac {\cos (x) \sin ^5(x)}{5 (a+a \sin (x))^3}+\frac {13 \cos (x) \sin ^4(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cos (x) \sin ^3(x)}{3 \left (a^3+a^3 \sin (x)\right )} \]
-23/2*x/a^3-136/5*cos(x)/a^3+136/15*cos(x)^3/a^3+23/2*cos(x)*sin(x)/a^3+1/ 5*cos(x)*sin(x)^5/(a+a*sin(x))^3+13/15*cos(x)*sin(x)^4/a/(a+a*sin(x))^2+23 /3*cos(x)*sin(x)^3/(a^3+a^3*sin(x))
Time = 0.13 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.89 \[ \int \frac {\sin ^6(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (24 \sin \left (\frac {x}{2}\right )-12 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-224 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2+112 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+1576 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4-690 x \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-405 \cos (x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+5 \cos (3 x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+45 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5 \sin (2 x)\right )}{60 (a+a \sin (x))^3} \]
((Cos[x/2] + Sin[x/2])*(24*Sin[x/2] - 12*(Cos[x/2] + Sin[x/2]) - 224*Sin[x /2]*(Cos[x/2] + Sin[x/2])^2 + 112*(Cos[x/2] + Sin[x/2])^3 + 1576*Sin[x/2]* (Cos[x/2] + Sin[x/2])^4 - 690*x*(Cos[x/2] + Sin[x/2])^5 - 405*Cos[x]*(Cos[ x/2] + Sin[x/2])^5 + 5*Cos[3*x]*(Cos[x/2] + Sin[x/2])^5 + 45*(Cos[x/2] + S in[x/2])^5*Sin[2*x]))/(60*(a + a*Sin[x])^3)
Time = 0.72 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {3042, 3244, 3042, 3456, 3042, 3456, 27, 3042, 3227, 3042, 3113, 2009, 3115, 24}
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.
| \(\displaystyle \int \frac {\sin ^6(x)}{(a \sin (x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^6}{(a \sin (x)+a)^3}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\int \frac {\sin ^4(x) (5 a-8 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\int \frac {\sin (x)^4 (5 a-8 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\int \frac {\sin ^3(x) \left (52 a^2-63 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\int \frac {\sin (x)^3 \left (52 a^2-63 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {\int 3 \sin ^2(x) \left (115 a^3-136 a^3 \sin (x)\right )dx}{a^2}-\frac {115 a^2 \sin ^3(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {3 \int \sin ^2(x) \left (115 a^3-136 a^3 \sin (x)\right )dx}{a^2}-\frac {115 a^2 \sin ^3(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {3 \int \sin (x)^2 \left (115 a^3-136 a^3 \sin (x)\right )dx}{a^2}-\frac {115 a^2 \sin ^3(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {3 \left (115 a^3 \int \sin ^2(x)dx-136 a^3 \int \sin ^3(x)dx\right )}{a^2}-\frac {115 a^2 \sin ^3(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {3 \left (115 a^3 \int \sin (x)^2dx-136 a^3 \int \sin (x)^3dx\right )}{a^2}-\frac {115 a^2 \sin ^3(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {3 \left (115 a^3 \int \sin (x)^2dx+136 a^3 \int \left (1-\cos ^2(x)\right )d\cos (x)\right )}{a^2}-\frac {115 a^2 \sin ^3(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {3 \left (115 a^3 \int \sin (x)^2dx+136 a^3 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )\right )}{a^2}-\frac {115 a^2 \sin ^3(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {3 \left (115 a^3 \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )+136 a^3 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )\right )}{a^2}-\frac {115 a^2 \sin ^3(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {3 \left (136 a^3 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )+115 a^3 \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )\right )}{a^2}-\frac {115 a^2 \sin ^3(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {13 a \sin ^4(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
(Cos[x]*Sin[x]^5)/(5*(a + a*Sin[x])^3) - ((-13*a*Cos[x]*Sin[x]^4)/(3*(a + a*Sin[x])^2) + ((-115*a^2*Cos[x]*Sin[x]^3)/(a + a*Sin[x]) + (3*(136*a^3*(C os[x] - Cos[x]^3/3) + 115*a^3*(x/2 - (Cos[x]*Sin[x])/2)))/a^2)/(3*a^2))/(5 *a^2)
3.1.21.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* (2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & & NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
Time = 0.67 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62
| method | result | size |
| parallelrisch | \(\frac {48 \tan \left (x \right ) \left (\sec ^{4}\left (x \right )\right )-48 \left (\sec ^{5}\left (x \right )\right )-236 \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )+260 \left (\sec ^{3}\left (x \right )\right )-405 \cos \left (x \right )+788 \tan \left (x \right )-900 \sec \left (x \right )+5 \cos \left (3 x \right )+45 \sin \left (2 x \right )-690 x -1088}{60 a^{3}}\) | \(63\) |
| default | \(\frac {-\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {8}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {8}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {20}{\tan \left (\frac {x}{2}\right )+1}-\frac {4 \left (\frac {3 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}+3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+7 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {3 \tan \left (\frac {x}{2}\right )}{4}+\frac {10}{3}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}-23 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(108\) |
| risch | \(-\frac {23 x}{2 a^{3}}+\frac {{\mathrm e}^{3 i x}}{24 a^{3}}-\frac {3 i {\mathrm e}^{2 i x}}{8 a^{3}}-\frac {27 \,{\mathrm e}^{i x}}{8 a^{3}}-\frac {27 \,{\mathrm e}^{-i x}}{8 a^{3}}+\frac {3 i {\mathrm e}^{-2 i x}}{8 a^{3}}+\frac {{\mathrm e}^{-3 i x}}{24 a^{3}}-\frac {2 \left (810 i {\mathrm e}^{3 i x}+225 \,{\mathrm e}^{4 i x}-1160 \,{\mathrm e}^{2 i x}-760 i {\mathrm e}^{i x}+197\right )}{15 \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\) | \(117\) |
| norman | \(\frac {-\frac {184 x \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{a}-\frac {460 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {2944 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}-\frac {544}{15 a}-\frac {115 x \tan \left (\frac {x}{2}\right )}{2 a}-\frac {1564 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{a}-\frac {15025 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {6979 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{15 a}-\frac {3271 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {4409 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {2300 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {920 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {460 x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{a}-\frac {920 x \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}-\frac {3335 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}-\frac {2944 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {8533 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {25829 \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{15 a}-\frac {2645 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {1081 \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {115 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{a}-\frac {23 \left (\tan ^{16}\left (\frac {x}{2}\right )\right )}{a}-\frac {115 x \left (\tan ^{16}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {475 \tan \left (\frac {x}{2}\right )}{3 a}-\frac {6059 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {3335 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {184 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {23 x \left (\tan ^{17}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {9631 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {2300 x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}-\frac {23 x}{2 a}-\frac {61129 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{15 a}-\frac {5509 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {5209 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {1564 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(411\) |
1/60*(48*tan(x)*sec(x)^4-48*sec(x)^5-236*tan(x)*sec(x)^2+260*sec(x)^3-405* cos(x)+788*tan(x)-900*sec(x)+5*cos(3*x)+45*sin(2*x)-690*x-1088)/a^3
Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.56 \[ \int \frac {\sin ^6(x)}{(a+a \sin (x))^3} \, dx=\frac {10 \, \cos \left (x\right )^{6} - 15 \, \cos \left (x\right )^{5} - {\left (345 \, x + 839\right )} \cos \left (x\right )^{3} - 140 \, \cos \left (x\right )^{4} - {\left (1035 \, x - 668\right )} \cos \left (x\right )^{2} + 6 \, {\left (115 \, x + 233\right )} \cos \left (x\right ) + {\left (10 \, \cos \left (x\right )^{5} + 25 \, \cos \left (x\right )^{4} - {\left (345 \, x - 724\right )} \cos \left (x\right )^{2} - 115 \, \cos \left (x\right )^{3} + 6 \, {\left (115 \, x + 232\right )} \cos \left (x\right ) + 1380 \, x - 6\right )} \sin \left (x\right ) + 1380 \, x + 6}{30 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \]
1/30*(10*cos(x)^6 - 15*cos(x)^5 - (345*x + 839)*cos(x)^3 - 140*cos(x)^4 - (1035*x - 668)*cos(x)^2 + 6*(115*x + 233)*cos(x) + (10*cos(x)^5 + 25*cos(x )^4 - (345*x - 724)*cos(x)^2 - 115*cos(x)^3 + 6*(115*x + 232)*cos(x) + 138 0*x - 6)*sin(x) + 1380*x + 6)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x ) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))
Leaf count of result is larger than twice the leaf count of optimal. 3288 vs. \(2 (107) = 214\).
Time = 17.82 (sec) , antiderivative size = 3288, normalized size of antiderivative = 32.55 \[ \int \frac {\sin ^6(x)}{(a+a \sin (x))^3} \, dx=\text {Too large to display} \]
-345*x*tan(x/2)**11/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a* *3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3* tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan (x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 1725*x*ta n(x/2)**10/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/ 2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)* *6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 4485*x*tan(x/2)**9 /(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 75 0*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380* a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3 *tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 8625*x*tan(x/2)**8/(30*a**3* tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan (x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x /2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)* *2 + 150*a**3*tan(x/2) + 30*a**3) - 13110*x*tan(x/2)**7/(30*a**3*tan(x/2)* *11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150* a**3*tan(x/2) + 30*a**3) - 15870*x*tan(x/2)**6/(30*a**3*tan(x/2)**11 + ...
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (87) = 174\).
Time = 0.31 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.03 \[ \int \frac {\sin ^6(x)}{(a+a \sin (x))^3} \, dx=-\frac {\frac {2375 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {5347 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {9230 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {12622 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {13340 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {11684 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {8050 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {4370 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {1725 \, \sin \left (x\right )^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}} + \frac {345 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + 544}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {13 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {25 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {38 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {46 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {46 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {38 \, a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {25 \, a^{3} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {13 \, a^{3} \sin \left (x\right )^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}} + \frac {5 \, a^{3} \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (x\right )^{11}}{{\left (\cos \left (x\right ) + 1\right )}^{11}}\right )}} - \frac {23 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \]
-1/15*(2375*sin(x)/(cos(x) + 1) + 5347*sin(x)^2/(cos(x) + 1)^2 + 9230*sin( x)^3/(cos(x) + 1)^3 + 12622*sin(x)^4/(cos(x) + 1)^4 + 13340*sin(x)^5/(cos( x) + 1)^5 + 11684*sin(x)^6/(cos(x) + 1)^6 + 8050*sin(x)^7/(cos(x) + 1)^7 + 4370*sin(x)^8/(cos(x) + 1)^8 + 1725*sin(x)^9/(cos(x) + 1)^9 + 345*sin(x)^ 10/(cos(x) + 1)^10 + 544)/(a^3 + 5*a^3*sin(x)/(cos(x) + 1) + 13*a^3*sin(x) ^2/(cos(x) + 1)^2 + 25*a^3*sin(x)^3/(cos(x) + 1)^3 + 38*a^3*sin(x)^4/(cos( x) + 1)^4 + 46*a^3*sin(x)^5/(cos(x) + 1)^5 + 46*a^3*sin(x)^6/(cos(x) + 1)^ 6 + 38*a^3*sin(x)^7/(cos(x) + 1)^7 + 25*a^3*sin(x)^8/(cos(x) + 1)^8 + 13*a ^3*sin(x)^9/(cos(x) + 1)^9 + 5*a^3*sin(x)^10/(cos(x) + 1)^10 + a^3*sin(x)^ 11/(cos(x) + 1)^11) - 23*arctan(sin(x)/(cos(x) + 1))/a^3
Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^6(x)}{(a+a \sin (x))^3} \, dx=-\frac {23 \, x}{2 \, a^{3}} - \frac {9 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 36 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 84 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, x\right ) + 40}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3} a^{3}} - \frac {4 \, {\left (75 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 330 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 530 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 355 \, \tan \left (\frac {1}{2} \, x\right ) + 86\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
-23/2*x/a^3 - 1/3*(9*tan(1/2*x)^5 + 36*tan(1/2*x)^4 + 84*tan(1/2*x)^2 - 9* tan(1/2*x) + 40)/((tan(1/2*x)^2 + 1)^3*a^3) - 4/15*(75*tan(1/2*x)^4 + 330* tan(1/2*x)^3 + 530*tan(1/2*x)^2 + 355*tan(1/2*x) + 86)/(a^3*(tan(1/2*x) + 1)^5)
Time = 6.62 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^6(x)}{(a+a \sin (x))^3} \, dx=-\frac {23\,x}{2\,a^3}-\frac {23\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}+115\,{\mathrm {tan}\left (\frac {x}{2}\right )}^9+\frac {874\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8}{3}+\frac {1610\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7}{3}+\frac {11684\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{15}+\frac {2668\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{3}+\frac {12622\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{15}+\frac {1846\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{3}+\frac {5347\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{15}+\frac {475\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {544}{15}}{a^3\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]