Integrand size = 13, antiderivative size = 90 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {13 x}{2 a^3}+\frac {152 \cos (x)}{15 a^3}-\frac {13 \cos (x) \sin (x)}{2 a^3}+\frac {\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}+\frac {11 \cos (x) \sin ^3(x)}{15 a (a+a \sin (x))^2}+\frac {76 \cos (x) \sin ^2(x)}{15 \left (a^3+a^3 \sin (x)\right )} \] Output:
13/2*x/a^3+152/15*cos(x)/a^3-13/2*cos(x)*sin(x)/a^3+1/5*cos(x)*sin(x)^4/(a +a*sin(x))^3+11/15*cos(x)*sin(x)^3/a/(a+a*sin(x))^2+76*cos(x)*sin(x)^2/(15 *a^3+15*a^3*sin(x))
Time = 0.15 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.89 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-24 \sin \left (\frac {x}{2}\right )+12 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+184 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2-92 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-1016 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4+390 x \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+180 \cos (x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-15 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5 \sin (2 x)\right )}{60 (a+a \sin (x))^3} \] Input:
Integrate[Sin[x]^5/(a + a*Sin[x])^3,x]
Output:
((Cos[x/2] + Sin[x/2])*(-24*Sin[x/2] + 12*(Cos[x/2] + Sin[x/2]) + 184*Sin[ x/2]*(Cos[x/2] + Sin[x/2])^2 - 92*(Cos[x/2] + Sin[x/2])^3 - 1016*Sin[x/2]* (Cos[x/2] + Sin[x/2])^4 + 390*x*(Cos[x/2] + Sin[x/2])^5 + 180*Cos[x]*(Cos[ x/2] + Sin[x/2])^5 - 15*(Cos[x/2] + Sin[x/2])^5*Sin[2*x]))/(60*(a + a*Sin[ x])^3)
Time = 0.60 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 3244, 3042, 3456, 3042, 3456, 3042, 3213}
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.
| \(\displaystyle \int \frac {\sin ^5(x)}{(a \sin (x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^5}{(a \sin (x)+a)^3}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle \frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\int \frac {\sin ^3(x) (4 a-7 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\int \frac {\sin (x)^3 (4 a-7 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\int \frac {\sin ^2(x) \left (33 a^2-43 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{3 a^2}-\frac {11 a \sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\int \frac {\sin (x)^2 \left (33 a^2-43 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{3 a^2}-\frac {11 a \sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {\int \sin (x) \left (152 a^3-195 a^3 \sin (x)\right )dx}{a^2}-\frac {76 a^2 \sin ^2(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {11 a \sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {\int \sin (x) \left (152 a^3-195 a^3 \sin (x)\right )dx}{a^2}-\frac {76 a^2 \sin ^2(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {11 a \sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\frac {-\frac {195 a^3 x}{2}-152 a^3 \cos (x)+\frac {195}{2} a^3 \sin (x) \cos (x)}{a^2}-\frac {76 a^2 \sin ^2(x) \cos (x)}{a \sin (x)+a}}{3 a^2}-\frac {11 a \sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
Input:
Int[Sin[x]^5/(a + a*Sin[x])^3,x]
Output:
(Cos[x]*Sin[x]^4)/(5*(a + a*Sin[x])^3) - ((-11*a*Cos[x]*Sin[x]^3)/(3*(a + a*Sin[x])^2) + ((-76*a^2*Cos[x]*Sin[x]^2)/(a + a*Sin[x]) + ((-195*a^3*x)/2 - 152*a^3*Cos[x] + (195*a^3*Cos[x]*Sin[x])/2)/a^2)/(3*a^2))/(5*a^2)
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{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_)])^(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])
Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08
| method | result | size |
| risch | \(\frac {13 x}{2 a^{3}}+\frac {i {\mathrm e}^{2 i x}}{8 a^{3}}+\frac {3 \,{\mathrm e}^{i x}}{2 a^{3}}+\frac {3 \,{\mathrm e}^{-i x}}{2 a^{3}}-\frac {i {\mathrm e}^{-2 i x}}{8 a^{3}}+\frac {70 i {\mathrm e}^{3 i x}+20 \,{\mathrm e}^{4 i x}-\frac {194 i {\mathrm e}^{i x}}{3}-\frac {298 \,{\mathrm e}^{2 i x}}{3}+\frac {254}{15}}{\left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\) | \(97\) |
| default | \(\frac {\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {6}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {12}{\tan \left (\frac {x}{2}\right )+1}+\frac {4 \left (\frac {\tan \left (\frac {x}{2}\right )^{3}}{4}+\frac {3 \tan \left (\frac {x}{2}\right )^{2}}{2}-\frac {\tan \left (\frac {x}{2}\right )}{4}+\frac {3}{2}\right )}{\left (\tan \left (\frac {x}{2}\right )^{2}+1\right )^{2}}+13 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(100\) |
| parallelrisch | \(\frac {\left (-7800 x -14760\right ) \cos \left (\frac {x}{2}\right )+\left (3900 x +1930\right ) \cos \left (\frac {3 x}{2}\right )+\left (780 x +2982\right ) \cos \left (\frac {5 x}{2}\right )+\left (-7800 x -9560\right ) \sin \left (\frac {x}{2}\right )+\left (-3900 x -10230\right ) \sin \left (\frac {3 x}{2}\right )+\left (780 x -550\right ) \sin \left (\frac {5 x}{2}\right )+105 \sin \left (\frac {7 x}{2}\right )-15 \sin \left (\frac {9 x}{2}\right )+105 \cos \left (\frac {7 x}{2}\right )+15 \cos \left (\frac {9 x}{2}\right )}{120 a^{3} \left (-10 \cos \left (\frac {x}{2}\right )+5 \cos \left (\frac {3 x}{2}\right )-5 \sin \left (\frac {3 x}{2}\right )-10 \sin \left (\frac {x}{2}\right )+\sin \left (\frac {5 x}{2}\right )+\cos \left (\frac {5 x}{2}\right )\right )}\) | \(126\) |
| norman | \(\frac {\frac {13 x}{2 a}+\frac {3491 \tan \left (\frac {x}{2}\right )^{9}}{3 a}+\frac {1313 x \tan \left (\frac {x}{2}\right )^{10}}{2 a}+\frac {845 x \tan \left (\frac {x}{2}\right )^{11}}{2 a}+\frac {455 x \tan \left (\frac {x}{2}\right )^{12}}{2 a}+\frac {195 x \tan \left (\frac {x}{2}\right )^{13}}{2 a}+\frac {65 x \tan \left (\frac {x}{2}\right )^{14}}{2 a}+\frac {13 x \tan \left (\frac {x}{2}\right )^{15}}{2 a}+\frac {304}{15 a}+\frac {1313 x \tan \left (\frac {x}{2}\right )^{5}}{2 a}+\frac {65 x \tan \left (\frac {x}{2}\right )}{2 a}+\frac {195 x \tan \left (\frac {x}{2}\right )^{2}}{2 a}+\frac {455 x \tan \left (\frac {x}{2}\right )^{3}}{2 a}+\frac {845 x \tan \left (\frac {x}{2}\right )^{4}}{2 a}+\frac {1755 x \tan \left (\frac {x}{2}\right )^{6}}{2 a}+\frac {2015 x \tan \left (\frac {x}{2}\right )^{7}}{2 a}+\frac {1755 x \tan \left (\frac {x}{2}\right )^{9}}{2 a}+\frac {3913 \tan \left (\frac {x}{2}\right )^{10}}{5 a}+\frac {2015 x \tan \left (\frac {x}{2}\right )^{8}}{2 a}+\frac {1300 \tan \left (\frac {x}{2}\right )^{11}}{3 a}+\frac {572 \tan \left (\frac {x}{2}\right )^{12}}{3 a}+\frac {65 \tan \left (\frac {x}{2}\right )^{13}}{a}+\frac {13 \tan \left (\frac {x}{2}\right )^{14}}{a}+\frac {4717 \tan \left (\frac {x}{2}\right )^{6}}{3 a}+\frac {4888 \tan \left (\frac {x}{2}\right )^{7}}{3 a}+\frac {3793 \tan \left (\frac {x}{2}\right )^{5}}{3 a}+\frac {884 \tan \left (\frac {x}{2}\right )^{4}}{a}+\frac {265 \tan \left (\frac {x}{2}\right )}{3 a}+\frac {239 \tan \left (\frac {x}{2}\right )^{2}}{a}+\frac {1556 \tan \left (\frac {x}{2}\right )^{3}}{3 a}+\frac {1512 \tan \left (\frac {x}{2}\right )^{8}}{a}}{\left (\tan \left (\frac {x}{2}\right )^{2}+1\right )^{5} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(365\) |
Input:
int(sin(x)^5/(a+a*sin(x))^3,x,method=_RETURNVERBOSE)
Output:
13/2*x/a^3+1/8*I/a^3*exp(2*I*x)+3/2/a^3*exp(I*x)+3/2/a^3*exp(-I*x)-1/8*I/a ^3*exp(-2*I*x)+2/15*(525*I*exp(3*I*x)+150*exp(4*I*x)-485*I*exp(I*x)-745*ex p(2*I*x)+127)/(exp(I*x)+I)^5/a^3
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.61 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {15 \, \cos \left (x\right )^{5} + {\left (195 \, x + 449\right )} \cos \left (x\right )^{3} + 60 \, \cos \left (x\right )^{4} + {\left (585 \, x - 358\right )} \cos \left (x\right )^{2} - 6 \, {\left (65 \, x + 128\right )} \cos \left (x\right ) - {\left (15 \, \cos \left (x\right )^{4} - {\left (195 \, x - 404\right )} \cos \left (x\right )^{2} - 45 \, \cos \left (x\right )^{3} + 6 \, {\left (65 \, x + 127\right )} \cos \left (x\right ) + 780 \, x - 6\right )} \sin \left (x\right ) - 780 \, x - 6}{30 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(sin(x)^5/(a+a*sin(x))^3,x, algorithm="fricas")
Output:
1/30*(15*cos(x)^5 + (195*x + 449)*cos(x)^3 + 60*cos(x)^4 + (585*x - 358)*c os(x)^2 - 6*(65*x + 128)*cos(x) - (15*cos(x)^4 - (195*x - 404)*cos(x)^2 - 45*cos(x)^3 + 6*(65*x + 127)*cos(x) + 780*x - 6)*sin(x) - 780*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*c os(x) - 4*a^3)*sin(x))
Leaf count of result is larger than twice the leaf count of optimal. 2259 vs. \(2 (95) = 190\).
Time = 11.68 (sec) , antiderivative size = 2259, normalized size of antiderivative = 25.10 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\text {Too large to display} \] Input:
integrate(sin(x)**5/(a+a*sin(x))**3,x)
Output:
195*x*tan(x/2)**9/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*t an(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/ 2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 975*x*tan(x/2)**8/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780* a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3* tan(x/2) + 30*a**3) + 2340*x*tan(x/2)**7/(30*a**3*tan(x/2)**9 + 150*a**3*t an(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/ 2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 3900*x*tan(x/2)**6/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780 *a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3 *tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 5070*x*tan(x/2)**5/(30*a**3* tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x /2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)** 3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 5070*x*tan(x/2)* *4/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 60 0*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a** 3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 3900 *x*tan(x/2)**3/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*t...
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (78) = 156\).
Time = 0.11 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.80 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {\frac {1325 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2673 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3805 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {4329 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {3575 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {2275 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {975 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {195 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + 304}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {12 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {20 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {26 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {26 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {20 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {12 \, a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {5 \, a^{3} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (x\right )^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}}\right )}} + \frac {13 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \] Input:
integrate(sin(x)^5/(a+a*sin(x))^3,x, algorithm="maxima")
Output:
1/15*(1325*sin(x)/(cos(x) + 1) + 2673*sin(x)^2/(cos(x) + 1)^2 + 3805*sin(x )^3/(cos(x) + 1)^3 + 4329*sin(x)^4/(cos(x) + 1)^4 + 3575*sin(x)^5/(cos(x) + 1)^5 + 2275*sin(x)^6/(cos(x) + 1)^6 + 975*sin(x)^7/(cos(x) + 1)^7 + 195* sin(x)^8/(cos(x) + 1)^8 + 304)/(a^3 + 5*a^3*sin(x)/(cos(x) + 1) + 12*a^3*s in(x)^2/(cos(x) + 1)^2 + 20*a^3*sin(x)^3/(cos(x) + 1)^3 + 26*a^3*sin(x)^4/ (cos(x) + 1)^4 + 26*a^3*sin(x)^5/(cos(x) + 1)^5 + 20*a^3*sin(x)^6/(cos(x) + 1)^6 + 12*a^3*sin(x)^7/(cos(x) + 1)^7 + 5*a^3*sin(x)^8/(cos(x) + 1)^8 + a^3*sin(x)^9/(cos(x) + 1)^9) + 13*arctan(sin(x)/(cos(x) + 1))/a^3
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {13 \, x}{2 \, a^{3}} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 6}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a^{3}} + \frac {2 \, {\left (90 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 405 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 665 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 445 \, \tan \left (\frac {1}{2} \, x\right ) + 107\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \] Input:
integrate(sin(x)^5/(a+a*sin(x))^3,x, algorithm="giac")
Output:
13/2*x/a^3 + (tan(1/2*x)^3 + 6*tan(1/2*x)^2 - tan(1/2*x) + 6)/((tan(1/2*x) ^2 + 1)^2*a^3) + 2/15*(90*tan(1/2*x)^4 + 405*tan(1/2*x)^3 + 665*tan(1/2*x) ^2 + 445*tan(1/2*x) + 107)/(a^3*(tan(1/2*x) + 1)^5)
Time = 17.62 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {13\,x}{2\,a^3}+\frac {13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+65\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7+\frac {455\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{3}+\frac {715\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{3}+\frac {1443\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{5}+\frac {761\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{3}+\frac {891\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+\frac {265\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {304}{15}}{a^3\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \] Input:
int(sin(x)^5/(a + a*sin(x))^3,x)
Output:
(13*x)/(2*a^3) + ((265*tan(x/2))/3 + (891*tan(x/2)^2)/5 + (761*tan(x/2)^3) /3 + (1443*tan(x/2)^4)/5 + (715*tan(x/2)^5)/3 + (455*tan(x/2)^6)/3 + 65*ta n(x/2)^7 + 13*tan(x/2)^8 + 304/15)/(a^3*(tan(x/2)^2 + 1)^2*(tan(x/2) + 1)^ 5)
Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.63 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {15 \cos \left (x \right ) \sin \left (x \right )^{4}-45 \cos \left (x \right ) \sin \left (x \right )^{3}+195 \cos \left (x \right ) \sin \left (x \right )^{2} x -253 \cos \left (x \right ) \sin \left (x \right )^{2}+390 \cos \left (x \right ) \sin \left (x \right ) x -265 \cos \left (x \right ) \sin \left (x \right )+195 \cos \left (x \right ) x -78 \cos \left (x \right )+15 \sin \left (x \right )^{5}-60 \sin \left (x \right )^{4}-195 \sin \left (x \right )^{3} x -660 \sin \left (x \right )^{3}-585 \sin \left (x \right )^{2} x -916 \sin \left (x \right )^{2}-585 \sin \left (x \right ) x -265 \sin \left (x \right )-195 x +78}{30 a^{3} \left (\cos \left (x \right ) \sin \left (x \right )^{2}+2 \cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right )-\sin \left (x \right )^{3}-3 \sin \left (x \right )^{2}-3 \sin \left (x \right )-1\right )} \] Input:
int(sin(x)^5/(a+a*sin(x))^3,x)
Output:
(15*cos(x)*sin(x)**4 - 45*cos(x)*sin(x)**3 + 195*cos(x)*sin(x)**2*x - 253* cos(x)*sin(x)**2 + 390*cos(x)*sin(x)*x - 265*cos(x)*sin(x) + 195*cos(x)*x - 78*cos(x) + 15*sin(x)**5 - 60*sin(x)**4 - 195*sin(x)**3*x - 660*sin(x)** 3 - 585*sin(x)**2*x - 916*sin(x)**2 - 585*sin(x)*x - 265*sin(x) - 195*x + 78)/(30*a**3*(cos(x)*sin(x)**2 + 2*cos(x)*sin(x) + cos(x) - sin(x)**3 - 3* sin(x)**2 - 3*sin(x) - 1))