Integrand size = 13, antiderivative size = 86 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=-\frac {13 \text {arctanh}(\cos (x))}{2 a^3}+\frac {152 \cot (x)}{15 a^3}-\frac {13 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac {11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac {76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )} \] Output:
-13/2*arctanh(cos(x))/a^3+152/15*cot(x)/a^3-13/2*cot(x)*csc(x)/a^3+1/5*cot (x)*csc(x)/(a+a*sin(x))^3+11/15*cot(x)*csc(x)/a/(a+a*sin(x))^2+76*cot(x)*c sc(x)/(15*a^3+15*a^3*sin(x))
Leaf count is larger than twice the leaf count of optimal. \(247\) vs. \(2(86)=172\).
Time = 1.42 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.87 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-48 \sin \left (\frac {x}{2}\right )-15 \left (1+\cot \left (\frac {x}{2}\right )\right )^5 \sin ^3\left (\frac {x}{2}\right )+24 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-272 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2+136 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-1712 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4+180 \cot \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-780 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+780 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-180 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5 \tan \left (\frac {x}{2}\right )+15 \cos ^3\left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^5\right )}{120 a^3 (1+\sin (x))^3} \] Input:
Integrate[Csc[x]^3/(a + a*Sin[x])^3,x]
Output:
((Cos[x/2] + Sin[x/2])*(-48*Sin[x/2] - 15*(1 + Cot[x/2])^5*Sin[x/2]^3 + 24 *(Cos[x/2] + Sin[x/2]) - 272*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 + 136*(Cos[x /2] + Sin[x/2])^3 - 1712*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 + 180*Cot[x/2]*( Cos[x/2] + Sin[x/2])^5 - 780*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^5 + 780*L og[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^5 - 180*(Cos[x/2] + Sin[x/2])^5*Tan[x/2 ] + 15*Cos[x/2]^3*(1 + Tan[x/2])^5))/(120*a^3*(1 + Sin[x])^3)
Time = 0.86 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.16, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {3042, 3245, 3042, 3457, 3042, 3457, 3042, 3227, 3042, 4254, 24, 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.
| \(\displaystyle \int \frac {\csc ^3(x)}{(a \sin (x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^3 (a \sin (x)+a)^3}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle \frac {\int \frac {\csc ^3(x) (7 a-4 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {7 a-4 a \sin (x)}{\sin (x)^3 (\sin (x) a+a)^2}dx}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \frac {\csc ^3(x) \left (43 a^2-33 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {43 a^2-33 a^2 \sin (x)}{\sin (x)^3 (\sin (x) a+a)}dx}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\int \csc ^3(x) \left (195 a^3-152 a^3 \sin (x)\right )dx}{a^2}+\frac {76 a^2 \cot (x) \csc (x)}{a \sin (x)+a}}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {195 a^3-152 a^3 \sin (x)}{\sin (x)^3}dx}{a^2}+\frac {76 a^2 \cot (x) \csc (x)}{a \sin (x)+a}}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {\frac {195 a^3 \int \csc ^3(x)dx-152 a^3 \int \csc ^2(x)dx}{a^2}+\frac {76 a^2 \cot (x) \csc (x)}{a \sin (x)+a}}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {195 a^3 \int \csc (x)^3dx-152 a^3 \int \csc (x)^2dx}{a^2}+\frac {76 a^2 \cot (x) \csc (x)}{a \sin (x)+a}}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\frac {152 a^3 \int 1d\cot (x)+195 a^3 \int \csc (x)^3dx}{a^2}+\frac {76 a^2 \cot (x) \csc (x)}{a \sin (x)+a}}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {195 a^3 \int \csc (x)^3dx+152 a^3 \cot (x)}{a^2}+\frac {76 a^2 \cot (x) \csc (x)}{a \sin (x)+a}}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {\frac {195 a^3 \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )+152 a^3 \cot (x)}{a^2}+\frac {76 a^2 \cot (x) \csc (x)}{a \sin (x)+a}}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {195 a^3 \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )+152 a^3 \cot (x)}{a^2}+\frac {76 a^2 \cot (x) \csc (x)}{a \sin (x)+a}}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\frac {76 a^2 \cot (x) \csc (x)}{a \sin (x)+a}+\frac {195 a^3 \left (-\frac {1}{2} \text {arctanh}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)\right )+152 a^3 \cot (x)}{a^2}}{3 a^2}+\frac {11 a \cot (x) \csc (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}\) |
Input:
Int[Csc[x]^3/(a + a*Sin[x])^3,x]
Output:
(Cot[x]*Csc[x])/(5*(a + a*Sin[x])^3) + ((11*a*Cot[x]*Csc[x])/(3*(a + a*Sin [x])^2) + ((152*a^3*Cot[x] + 195*a^3*(-1/2*ArcTanh[Cos[x]] - (Cot[x]*Csc[x ])/2))/a^2 + (76*a^2*Cot[x]*Csc[x])/(a + a*Sin[x]))/(3*a^2))/(5*a^2)
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^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[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[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, 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] && (IntegerQ[2*n] || EqQ[c, 0])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
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 = 0.54 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.06
| method | result | size |
| parallelrisch | \(\frac {780 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5} \ln \left (\tan \left (\frac {x}{2}\right )\right )+15 \tan \left (\frac {x}{2}\right )^{7}-105 \tan \left (\frac {x}{2}\right )^{6}+4680 \tan \left (\frac {x}{2}\right )^{4}+15060 \tan \left (\frac {x}{2}\right )^{3}-15 \cot \left (\frac {x}{2}\right )^{2}+20260 \tan \left (\frac {x}{2}\right )^{2}+105 \cot \left (\frac {x}{2}\right )+12980 \tan \left (\frac {x}{2}\right )+3532}{120 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(91\) |
| default | \(\frac {\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}-6 \tan \left (\frac {x}{2}\right )+\frac {32}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {16}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {112}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {40}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {80}{\tan \left (\frac {x}{2}\right )+1}-\frac {1}{2 \tan \left (\frac {x}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {x}{2}\right )}+26 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{4 a^{3}}\) | \(94\) |
| risch | \(\frac {975 i {\mathrm e}^{7 i x}+195 \,{\mathrm e}^{8 i x}-3575 i {\mathrm e}^{5 i x}-2275 \,{\mathrm e}^{6 i x}+3805 i {\mathrm e}^{3 i x}+4329 \,{\mathrm e}^{4 i x}-1325 i {\mathrm e}^{i x}-2673 \,{\mathrm e}^{2 i x}+304}{15 \left ({\mathrm e}^{2 i x}-1\right )^{2} \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}+\frac {13 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{3}}-\frac {13 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{3}}\) | \(114\) |
| norman | \(\frac {\frac {39 \tan \left (\frac {x}{2}\right )^{6}}{a}-\frac {1}{8 a}+\frac {7 \tan \left (\frac {x}{2}\right )}{8 a}-\frac {7 \tan \left (\frac {x}{2}\right )^{8}}{8 a}+\frac {\tan \left (\frac {x}{2}\right )^{9}}{8 a}+\frac {251 \tan \left (\frac {x}{2}\right )^{5}}{2 a}+\frac {649 \tan \left (\frac {x}{2}\right )^{3}}{6 a}+\frac {883 \tan \left (\frac {x}{2}\right )^{2}}{30 a}+\frac {1013 \tan \left (\frac {x}{2}\right )^{4}}{6 a}}{\tan \left (\frac {x}{2}\right )^{2} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {13 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{3}}\) | \(122\) |
Input:
int(csc(x)^3/(a+a*sin(x))^3,x,method=_RETURNVERBOSE)
Output:
1/120*(780*(tan(1/2*x)+1)^5*ln(tan(1/2*x))+15*tan(1/2*x)^7-105*tan(1/2*x)^ 6+4680*tan(1/2*x)^4+15060*tan(1/2*x)^3-15*cot(1/2*x)^2+20260*tan(1/2*x)^2+ 105*cot(1/2*x)+12980*tan(1/2*x)+3532)/a^3/(tan(1/2*x)+1)^5
Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (74) = 148\).
Time = 0.16 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.21 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {608 \, \cos \left (x\right )^{5} - 826 \, \cos \left (x\right )^{4} - 2174 \, \cos \left (x\right )^{3} + 784 \, \cos \left (x\right )^{2} - 195 \, {\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 195 \, {\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2 \, {\left (304 \, \cos \left (x\right )^{4} + 717 \, \cos \left (x\right )^{3} - 370 \, \cos \left (x\right )^{2} - 762 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) + 1536 \, \cos \left (x\right ) + 12}{60 \, {\left (a^{3} \cos \left (x\right )^{5} + 3 \, a^{3} \cos \left (x\right )^{4} - 3 \, a^{3} \cos \left (x\right )^{3} - 7 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{4} - 2 \, a^{3} \cos \left (x\right )^{3} - 5 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(csc(x)^3/(a+a*sin(x))^3,x, algorithm="fricas")
Output:
1/60*(608*cos(x)^5 - 826*cos(x)^4 - 2174*cos(x)^3 + 784*cos(x)^2 - 195*(co s(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7*cos(x)^2 + (cos(x)^4 - 2*cos(x)^3 - 5 *cos(x)^2 + 2*cos(x) + 4)*sin(x) + 2*cos(x) + 4)*log(1/2*cos(x) + 1/2) + 1 95*(cos(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7*cos(x)^2 + (cos(x)^4 - 2*cos(x) ^3 - 5*cos(x)^2 + 2*cos(x) + 4)*sin(x) + 2*cos(x) + 4)*log(-1/2*cos(x) + 1 /2) - 2*(304*cos(x)^4 + 717*cos(x)^3 - 370*cos(x)^2 - 762*cos(x) + 6)*sin( x) + 1536*cos(x) + 12)/(a^3*cos(x)^5 + 3*a^3*cos(x)^4 - 3*a^3*cos(x)^3 - 7 *a^3*cos(x)^2 + 2*a^3*cos(x) + 4*a^3 + (a^3*cos(x)^4 - 2*a^3*cos(x)^3 - 5* a^3*cos(x)^2 + 2*a^3*cos(x) + 4*a^3)*sin(x))
\[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {\int \frac {\csc ^{3}{\left (x \right )}}{\sin ^{3}{\left (x \right )} + 3 \sin ^{2}{\left (x \right )} + 3 \sin {\left (x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(csc(x)**3/(a+a*sin(x))**3,x)
Output:
Integral(csc(x)**3/(sin(x)**3 + 3*sin(x)**2 + 3*sin(x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (74) = 148\).
Time = 0.04 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.43 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {\frac {105 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2782 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {9410 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {13645 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {9285 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {2580 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - 15}{120 \, {\left (\frac {a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {5 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {10 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {5 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac {\frac {12 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{3}} + \frac {13 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{3}} \] Input:
integrate(csc(x)^3/(a+a*sin(x))^3,x, algorithm="maxima")
Output:
1/120*(105*sin(x)/(cos(x) + 1) + 2782*sin(x)^2/(cos(x) + 1)^2 + 9410*sin(x )^3/(cos(x) + 1)^3 + 13645*sin(x)^4/(cos(x) + 1)^4 + 9285*sin(x)^5/(cos(x) + 1)^5 + 2580*sin(x)^6/(cos(x) + 1)^6 - 15)/(a^3*sin(x)^2/(cos(x) + 1)^2 + 5*a^3*sin(x)^3/(cos(x) + 1)^3 + 10*a^3*sin(x)^4/(cos(x) + 1)^4 + 10*a^3* sin(x)^5/(cos(x) + 1)^5 + 5*a^3*sin(x)^6/(cos(x) + 1)^6 + a^3*sin(x)^7/(co s(x) + 1)^7) - 1/8*(12*sin(x)/(cos(x) + 1) - sin(x)^2/(cos(x) + 1)^2)/a^3 + 13/2*log(sin(x)/(cos(x) + 1))/a^3
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.27 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {13 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{3}} - \frac {78 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, x\right ) + 1}{8 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{2}} + \frac {a^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{6}} + \frac {2 \, {\left (150 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 525 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 745 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 485 \, \tan \left (\frac {1}{2} \, x\right ) + 127\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \] Input:
integrate(csc(x)^3/(a+a*sin(x))^3,x, algorithm="giac")
Output:
13/2*log(abs(tan(1/2*x)))/a^3 - 1/8*(78*tan(1/2*x)^2 - 12*tan(1/2*x) + 1)/ (a^3*tan(1/2*x)^2) + 1/8*(a^3*tan(1/2*x)^2 - 12*a^3*tan(1/2*x))/a^6 + 2/15 *(150*tan(1/2*x)^4 + 525*tan(1/2*x)^3 + 745*tan(1/2*x)^2 + 485*tan(1/2*x) + 127)/(a^3*(tan(1/2*x) + 1)^5)
Time = 17.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.13 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^3}-\frac {3\,\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a^3}+\frac {13\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a^3}+\frac {\frac {43\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{2}+\frac {619\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{8}+\frac {2729\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{24}+\frac {941\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{12}+\frac {1391\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{60}+\frac {7\,\mathrm {tan}\left (\frac {x}{2}\right )}{8}-\frac {1}{8}}{a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \] Input:
int(1/(sin(x)^3*(a + a*sin(x))^3),x)
Output:
tan(x/2)^2/(8*a^3) - (3*tan(x/2))/(2*a^3) + (13*log(tan(x/2)))/(2*a^3) + ( (7*tan(x/2))/8 + (1391*tan(x/2)^2)/60 + (941*tan(x/2)^3)/12 + (2729*tan(x/ 2)^4)/24 + (619*tan(x/2)^5)/8 + (43*tan(x/2)^6)/2 - 1/8)/(a^3*tan(x/2)^2*( tan(x/2) + 1)^5)
Time = 0.17 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.22 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {390 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4}+780 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3}+390 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2}-193 \cos \left (x \right ) \sin \left (x \right )^{4}-604 \cos \left (x \right ) \sin \left (x \right )^{3}-543 \cos \left (x \right ) \sin \left (x \right )^{2}-90 \cos \left (x \right ) \sin \left (x \right )+30 \cos \left (x \right )-390 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{5}-1170 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4}-1170 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3}-390 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2}-1023 \sin \left (x \right )^{5}-2071 \sin \left (x \right )^{4}-769 \sin \left (x \right )^{3}+453 \sin \left (x \right )^{2}+120 \sin \left (x \right )-30}{60 \sin \left (x \right )^{2} a^{3} \left (\cos \left (x \right ) \sin \left (x \right )^{2}+2 \cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right )-\sin \left (x \right )^{3}-3 \sin \left (x \right )^{2}-3 \sin \left (x \right )-1\right )} \] Input:
int(csc(x)^3/(a+a*sin(x))^3,x)
Output:
(390*cos(x)*log(tan(x/2))*sin(x)**4 + 780*cos(x)*log(tan(x/2))*sin(x)**3 + 390*cos(x)*log(tan(x/2))*sin(x)**2 - 193*cos(x)*sin(x)**4 - 604*cos(x)*si n(x)**3 - 543*cos(x)*sin(x)**2 - 90*cos(x)*sin(x) + 30*cos(x) - 390*log(ta n(x/2))*sin(x)**5 - 1170*log(tan(x/2))*sin(x)**4 - 1170*log(tan(x/2))*sin( x)**3 - 390*log(tan(x/2))*sin(x)**2 - 1023*sin(x)**5 - 2071*sin(x)**4 - 76 9*sin(x)**3 + 453*sin(x)**2 + 120*sin(x) - 30)/(60*sin(x)**2*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))