Integrand size = 13, antiderivative size = 71 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^3} \, dx=-\frac {3 x}{a^3}-\frac {9 \cos (x)}{5 a^3}+\frac {\cos (x) \sin ^3(x)}{5 (a+a \sin (x))^3}+\frac {3 \cos (x) \sin ^2(x)}{5 a (a+a \sin (x))^2}-\frac {3 \cos (x)}{a^3+a^3 \sin (x)} \]
-3*x/a^3-9/5*cos(x)/a^3+1/5*cos(x)*sin(x)^3/(a+a*sin(x))^3+3/5*cos(x)*sin( x)^2/a/(a+a*sin(x))^2-3*cos(x)/(a^3+a^3*sin(x))
Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.97 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )-12 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2+6 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+48 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4-15 x \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-5 \cos (x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5\right )}{5 (a+a \sin (x))^3} \]
((Cos[x/2] + Sin[x/2])*(-Cos[x/2] + Sin[x/2] - 12*Sin[x/2]*(Cos[x/2] + Sin [x/2])^2 + 6*(Cos[x/2] + Sin[x/2])^3 + 48*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 - 15*x*(Cos[x/2] + Sin[x/2])^5 - 5*Cos[x]*(Cos[x/2] + Sin[x/2])^5))/(5*(a + a*Sin[x])^3)
Time = 0.71 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 3244, 27, 3042, 3456, 27, 3042, 3447, 3042, 3502, 27, 3042, 3214, 3042, 3127}
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.
| \(\displaystyle \int \frac {\sin ^4(x)}{(a \sin (x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^4}{(a \sin (x)+a)^3}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\int \frac {3 \sin ^2(x) (a-2 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \int \frac {\sin ^2(x) (a-2 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \int \frac {\sin (x)^2 (a-2 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {\int \frac {3 \sin (x) \left (2 a^2-3 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{3 a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {\int \frac {\sin (x) \left (2 a^2-3 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {\int \frac {\sin (x) \left (2 a^2-3 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {\int \frac {2 a^2 \sin (x)-3 a^2 \sin ^2(x)}{\sin (x) a+a}dx}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {\int \frac {2 a^2 \sin (x)-3 a^2 \sin (x)^2}{\sin (x) a+a}dx}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {\frac {\int \frac {5 a^3 \sin (x)}{\sin (x) a+a}dx}{a}+3 a \cos (x)}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {5 a^2 \int \frac {\sin (x)}{\sin (x) a+a}dx+3 a \cos (x)}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {5 a^2 \int \frac {\sin (x)}{\sin (x) a+a}dx+3 a \cos (x)}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {5 a^2 \left (\frac {x}{a}-\int \frac {1}{\sin (x) a+a}dx\right )+3 a \cos (x)}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {5 a^2 \left (\frac {x}{a}-\int \frac {1}{\sin (x) a+a}dx\right )+3 a \cos (x)}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {3 \left (\frac {5 a^2 \left (\frac {x}{a}+\frac {\cos (x)}{a \sin (x)+a}\right )+3 a \cos (x)}{a^2}-\frac {a \sin ^2(x) \cos (x)}{(a \sin (x)+a)^2}\right )}{5 a^2}\) |
(Cos[x]*Sin[x]^3)/(5*(a + a*Sin[x])^3) - (3*(-((a*Cos[x]*Sin[x]^2)/(a + a* Sin[x])^2) + (3*a*Cos[x] + 5*a^2*(x/a + Cos[x]/(a + a*Sin[x])))/a^2))/(5*a ^2)
3.1.23.3.1 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[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
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_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 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])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 0.51 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72
| method | result | size |
| parallelrisch | \(\frac {4 \tan \left (x \right ) \left (\sec ^{4}\left (x \right )\right )-4 \left (\sec ^{5}\left (x \right )\right )-13 \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )+15 \left (\sec ^{3}\left (x \right )\right )-5 \cos \left (x \right )+24 \tan \left (x \right )-30 \sec \left (x \right )-15 x -24}{5 a^{3}}\) | \(51\) |
| default | \(\frac {-\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {6}{\tan \left (\frac {x}{2}\right )+1}-\frac {2}{1+\tan ^{2}\left (\frac {x}{2}\right )}-6 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(66\) |
| risch | \(-\frac {3 x}{a^{3}}-\frac {{\mathrm e}^{i x}}{2 a^{3}}-\frac {{\mathrm e}^{-i x}}{2 a^{3}}-\frac {4 \left (-70 \,{\mathrm e}^{2 i x}+50 i {\mathrm e}^{3 i x}+15 \,{\mathrm e}^{4 i x}-45 i {\mathrm e}^{i x}+12\right )}{5 \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\) | \(75\) |
| norman | \(\frac {-\frac {90 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {90 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}-\frac {48}{5 a}-\frac {15 x \tan \left (\frac {x}{2}\right )}{a}-\frac {15 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{a}-\frac {170 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}-\frac {522 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{5 a}-\frac {206 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {2172 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{5 a}-\frac {252 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {153 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}-\frac {153 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}-\frac {252 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {30 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{a}-\frac {42 \tan \left (\frac {x}{2}\right )}{a}-\frac {1598 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{5 a}-\frac {213 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {42 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {396 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {42 x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x}{a}-\frac {82 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}-\frac {1428 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{5 a}-\frac {372 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {213 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(319\) |
1/5*(4*tan(x)*sec(x)^4-4*sec(x)^5-13*tan(x)*sec(x)^2+15*sec(x)^3-5*cos(x)+ 24*tan(x)-30*sec(x)-15*x-24)/a^3
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (65) = 130\).
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.86 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^3} \, dx=-\frac {3 \, {\left (5 \, x + 13\right )} \cos \left (x\right )^{3} + 5 \, \cos \left (x\right )^{4} + {\left (45 \, x - 28\right )} \cos \left (x\right )^{2} - 3 \, {\left (10 \, x + 21\right )} \cos \left (x\right ) + {\left ({\left (15 \, x - 34\right )} \cos \left (x\right )^{2} + 5 \, \cos \left (x\right )^{3} - 2 \, {\left (15 \, x + 31\right )} \cos \left (x\right ) - 60 \, x + 1\right )} \sin \left (x\right ) - 60 \, x - 1}{5 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \]
-1/5*(3*(5*x + 13)*cos(x)^3 + 5*cos(x)^4 + (45*x - 28)*cos(x)^2 - 3*(10*x + 21)*cos(x) + ((15*x - 34)*cos(x)^2 + 5*cos(x)^3 - 2*(15*x + 31)*cos(x) - 60*x + 1)*sin(x) - 60*x - 1)/(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. 1425 vs. \(2 (71) = 142\).
Time = 6.70 (sec) , antiderivative size = 1425, normalized size of antiderivative = 20.07 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^3} \, dx=\text {Too large to display} \]
-15*x*tan(x/2)**7/(5*a**3*tan(x/2)**7 + 25*a**3*tan(x/2)**6 + 55*a**3*tan( x/2)**5 + 75*a**3*tan(x/2)**4 + 75*a**3*tan(x/2)**3 + 55*a**3*tan(x/2)**2 + 25*a**3*tan(x/2) + 5*a**3) - 75*x*tan(x/2)**6/(5*a**3*tan(x/2)**7 + 25*a **3*tan(x/2)**6 + 55*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 75*a**3*tan( x/2)**3 + 55*a**3*tan(x/2)**2 + 25*a**3*tan(x/2) + 5*a**3) - 165*x*tan(x/2 )**5/(5*a**3*tan(x/2)**7 + 25*a**3*tan(x/2)**6 + 55*a**3*tan(x/2)**5 + 75* a**3*tan(x/2)**4 + 75*a**3*tan(x/2)**3 + 55*a**3*tan(x/2)**2 + 25*a**3*tan (x/2) + 5*a**3) - 225*x*tan(x/2)**4/(5*a**3*tan(x/2)**7 + 25*a**3*tan(x/2) **6 + 55*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 75*a**3*tan(x/2)**3 + 55 *a**3*tan(x/2)**2 + 25*a**3*tan(x/2) + 5*a**3) - 225*x*tan(x/2)**3/(5*a**3 *tan(x/2)**7 + 25*a**3*tan(x/2)**6 + 55*a**3*tan(x/2)**5 + 75*a**3*tan(x/2 )**4 + 75*a**3*tan(x/2)**3 + 55*a**3*tan(x/2)**2 + 25*a**3*tan(x/2) + 5*a* *3) - 165*x*tan(x/2)**2/(5*a**3*tan(x/2)**7 + 25*a**3*tan(x/2)**6 + 55*a** 3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 75*a**3*tan(x/2)**3 + 55*a**3*tan(x/ 2)**2 + 25*a**3*tan(x/2) + 5*a**3) - 75*x*tan(x/2)/(5*a**3*tan(x/2)**7 + 2 5*a**3*tan(x/2)**6 + 55*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 75*a**3*t an(x/2)**3 + 55*a**3*tan(x/2)**2 + 25*a**3*tan(x/2) + 5*a**3) - 15*x/(5*a* *3*tan(x/2)**7 + 25*a**3*tan(x/2)**6 + 55*a**3*tan(x/2)**5 + 75*a**3*tan(x /2)**4 + 75*a**3*tan(x/2)**3 + 55*a**3*tan(x/2)**2 + 25*a**3*tan(x/2) + 5* a**3) - 30*tan(x/2)**6/(5*a**3*tan(x/2)**7 + 25*a**3*tan(x/2)**6 + 55*a...
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (65) = 130\).
Time = 0.30 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.79 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^3} \, dx=-\frac {2 \, {\left (\frac {105 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {189 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {160 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {75 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 24\right )}}{5 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {11 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {15 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {11 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {5 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac {6 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \]
-2/5*(105*sin(x)/(cos(x) + 1) + 189*sin(x)^2/(cos(x) + 1)^2 + 200*sin(x)^3 /(cos(x) + 1)^3 + 160*sin(x)^4/(cos(x) + 1)^4 + 75*sin(x)^5/(cos(x) + 1)^5 + 15*sin(x)^6/(cos(x) + 1)^6 + 24)/(a^3 + 5*a^3*sin(x)/(cos(x) + 1) + 11* a^3*sin(x)^2/(cos(x) + 1)^2 + 15*a^3*sin(x)^3/(cos(x) + 1)^3 + 15*a^3*sin( x)^4/(cos(x) + 1)^4 + 11*a^3*sin(x)^5/(cos(x) + 1)^5 + 5*a^3*sin(x)^6/(cos (x) + 1)^6 + a^3*sin(x)^7/(cos(x) + 1)^7) - 6*arctan(sin(x)/(cos(x) + 1))/ a^3
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^3} \, dx=-\frac {3 \, x}{a^{3}} - \frac {2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )} a^{3}} - \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 70 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 120 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 80 \, \tan \left (\frac {1}{2} \, x\right ) + 19\right )}}{5 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
-3*x/a^3 - 2/((tan(1/2*x)^2 + 1)*a^3) - 2/5*(15*tan(1/2*x)^4 + 70*tan(1/2* x)^3 + 120*tan(1/2*x)^2 + 80*tan(1/2*x) + 19)/(a^3*(tan(1/2*x) + 1)^5)
Time = 5.96 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.10 \[ \int \frac {\sin ^4(x)}{(a+a \sin (x))^3} \, dx=-\frac {3\,x}{a^3}-\frac {6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+30\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+64\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+80\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {378\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+42\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {48}{5}}{a^3\,\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]