Integrand size = 13, antiderivative size = 64 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=-\frac {7 \text {arctanh}(\cos (x))}{2 a^2}+\frac {16 \cot (x)}{3 a^2}-\frac {7 \cot (x) \csc (x)}{2 a^2}+\frac {8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc (x)}{3 (a+a \sin (x))^2} \] Output:
-7/2*arctanh(cos(x))/a^2+16/3*cot(x)/a^2-7/2*cot(x)*csc(x)/a^2+8/3*cot(x)* csc(x)/a^2/(1+sin(x))+1/3*cot(x)*csc(x)/(a+a*sin(x))^2
Leaf count is larger than twice the leaf count of optimal. \(203\) vs. \(2(64)=128\).
Time = 0.88 (sec) , antiderivative size = 203, normalized size of antiderivative = 3.17 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-16 \sin \left (\frac {x}{2}\right )-3 \left (1+\cot \left (\frac {x}{2}\right )\right )^3 \sin \left (\frac {x}{2}\right )+8 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-160 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2+24 \cot \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-84 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+84 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-24 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3 \tan \left (\frac {x}{2}\right )+3 \cos \left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^3\right )}{24 a^2 (1+\sin (x))^2} \] Input:
Integrate[Csc[x]^3/(a + a*Sin[x])^2,x]
Output:
((Cos[x/2] + Sin[x/2])*(-16*Sin[x/2] - 3*(1 + Cot[x/2])^3*Sin[x/2] + 8*(Co s[x/2] + Sin[x/2]) - 160*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 + 24*Cot[x/2]*(C os[x/2] + Sin[x/2])^3 - 84*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^3 + 84*Log[ Sin[x/2]]*(Cos[x/2] + Sin[x/2])^3 - 24*(Cos[x/2] + Sin[x/2])^3*Tan[x/2] + 3*Cos[x/2]*(1 + Tan[x/2])^3))/(24*a^2*(1 + Sin[x])^2)
Time = 0.64 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {3042, 3245, 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.
| \(\displaystyle \int \frac {\csc ^3(x)}{(a \sin (x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^3 (a \sin (x)+a)^2}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle \frac {\int \frac {\csc ^3(x) (5 a-3 a \sin (x))}{\sin (x) a+a}dx}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 a-3 a \sin (x)}{\sin (x)^3 (\sin (x) a+a)}dx}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \csc ^3(x) \left (21 a^2-16 a^2 \sin (x)\right )dx}{a^2}+\frac {8 \cot (x) \csc (x)}{\sin (x)+1}}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {21 a^2-16 a^2 \sin (x)}{\sin (x)^3}dx}{a^2}+\frac {8 \cot (x) \csc (x)}{\sin (x)+1}}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {21 a^2 \int \csc ^3(x)dx-16 a^2 \int \csc ^2(x)dx}{a^2}+\frac {8 \cot (x) \csc (x)}{\sin (x)+1}}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {21 a^2 \int \csc (x)^3dx-16 a^2 \int \csc (x)^2dx}{a^2}+\frac {8 \cot (x) \csc (x)}{\sin (x)+1}}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {16 a^2 \int 1d\cot (x)+21 a^2 \int \csc (x)^3dx}{a^2}+\frac {8 \cot (x) \csc (x)}{\sin (x)+1}}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {21 a^2 \int \csc (x)^3dx+16 a^2 \cot (x)}{a^2}+\frac {8 \cot (x) \csc (x)}{\sin (x)+1}}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {21 a^2 \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )+16 a^2 \cot (x)}{a^2}+\frac {8 \cot (x) \csc (x)}{\sin (x)+1}}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {21 a^2 \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )+16 a^2 \cot (x)}{a^2}+\frac {8 \cot (x) \csc (x)}{\sin (x)+1}}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {21 a^2 \left (-\frac {1}{2} \text {arctanh}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)\right )+16 a^2 \cot (x)}{a^2}+\frac {8 \cot (x) \csc (x)}{\sin (x)+1}}{3 a^2}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}\) |
Input:
Int[Csc[x]^3/(a + a*Sin[x])^2,x]
Output:
(Cot[x]*Csc[x])/(3*(a + a*Sin[x])^2) + ((16*a^2*Cot[x] + 21*a^2*(-1/2*ArcT anh[Cos[x]] - (Cot[x]*Csc[x])/2))/a^2 + (8*Cot[x]*Csc[x])/(1 + Sin[x]))/(3 *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 = 0.46 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.16
| method | result | size |
| default | \(\frac {\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}-4 \tan \left (\frac {x}{2}\right )+\frac {16}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {8}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {32}{\tan \left (\frac {x}{2}\right )+1}-\frac {1}{2 \tan \left (\frac {x}{2}\right )^{2}}+\frac {4}{\tan \left (\frac {x}{2}\right )}+14 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{4 a^{2}}\) | \(74\) |
| parallelrisch | \(\frac {84 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3} \ln \left (\tan \left (\frac {x}{2}\right )\right )+3 \tan \left (\frac {x}{2}\right )^{5}-15 \tan \left (\frac {x}{2}\right )^{4}-3 \cot \left (\frac {x}{2}\right )^{2}+336 \tan \left (\frac {x}{2}\right )^{2}+15 \cot \left (\frac {x}{2}\right )+570 \tan \left (\frac {x}{2}\right )+302}{24 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\) | \(75\) |
| risch | \(\frac {63 i {\mathrm e}^{5 i x}+21 \,{\mathrm e}^{6 i x}-126 i {\mathrm e}^{3 i x}-98 \,{\mathrm e}^{4 i x}+75 i {\mathrm e}^{i x}+97 \,{\mathrm e}^{2 i x}-32}{3 \left ({\mathrm e}^{2 i x}-1\right )^{2} \left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}-\frac {7 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{2}}+\frac {7 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{2}}\) | \(99\) |
| norman | \(\frac {\frac {14 \tan \left (\frac {x}{2}\right )^{4}}{a}-\frac {1}{8 a}+\frac {5 \tan \left (\frac {x}{2}\right )}{8 a}-\frac {5 \tan \left (\frac {x}{2}\right )^{6}}{8 a}+\frac {\tan \left (\frac {x}{2}\right )^{7}}{8 a}+\frac {95 \tan \left (\frac {x}{2}\right )^{3}}{4 a}+\frac {151 \tan \left (\frac {x}{2}\right )^{2}}{12 a}}{\tan \left (\frac {x}{2}\right )^{2} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {7 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{2}}\) | \(100\) |
Input:
int(csc(x)^3/(a+a*sin(x))^2,x,method=_RETURNVERBOSE)
Output:
1/4/a^2*(1/2*tan(1/2*x)^2-4*tan(1/2*x)+16/3/(tan(1/2*x)+1)^3-8/(tan(1/2*x) +1)^2+32/(tan(1/2*x)+1)-1/2/tan(1/2*x)^2+4/tan(1/2*x)+14*ln(tan(1/2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (54) = 108\).
Time = 0.09 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.44 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=-\frac {64 \, \cos \left (x\right )^{4} + 86 \, \cos \left (x\right )^{3} - 54 \, \cos \left (x\right )^{2} + 21 \, {\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} + 2 \, \cos \left (x\right )^{2} - \cos \left (x\right ) - 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 21 \, {\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} + 2 \, \cos \left (x\right )^{2} - \cos \left (x\right ) - 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, {\left (32 \, \cos \left (x\right )^{3} - 11 \, \cos \left (x\right )^{2} - 38 \, \cos \left (x\right ) + 2\right )} \sin \left (x\right ) - 80 \, \cos \left (x\right ) - 4}{12 \, {\left (a^{2} \cos \left (x\right )^{4} - a^{2} \cos \left (x\right )^{3} - 3 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) + 2 \, a^{2} - {\left (a^{2} \cos \left (x\right )^{3} + 2 \, a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(csc(x)^3/(a+a*sin(x))^2,x, algorithm="fricas")
Output:
-1/12*(64*cos(x)^4 + 86*cos(x)^3 - 54*cos(x)^2 + 21*(cos(x)^4 - cos(x)^3 - 3*cos(x)^2 - (cos(x)^3 + 2*cos(x)^2 - cos(x) - 2)*sin(x) + cos(x) + 2)*lo g(1/2*cos(x) + 1/2) - 21*(cos(x)^4 - cos(x)^3 - 3*cos(x)^2 - (cos(x)^3 + 2 *cos(x)^2 - cos(x) - 2)*sin(x) + cos(x) + 2)*log(-1/2*cos(x) + 1/2) + 2*(3 2*cos(x)^3 - 11*cos(x)^2 - 38*cos(x) + 2)*sin(x) - 80*cos(x) - 4)/(a^2*cos (x)^4 - a^2*cos(x)^3 - 3*a^2*cos(x)^2 + a^2*cos(x) + 2*a^2 - (a^2*cos(x)^3 + 2*a^2*cos(x)^2 - a^2*cos(x) - 2*a^2)*sin(x))
\[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {\int \frac {\csc ^{3}{\left (x \right )}}{\sin ^{2}{\left (x \right )} + 2 \sin {\left (x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(csc(x)**3/(a+a*sin(x))**2,x)
Output:
Integral(csc(x)**3/(sin(x)**2 + 2*sin(x) + 1), x)/a**2
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (54) = 108\).
Time = 0.03 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.42 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {\frac {15 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {239 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {405 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {216 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - 3}{24 \, {\left (\frac {a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} - \frac {\frac {8 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{2}} + \frac {7 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{2}} \] Input:
integrate(csc(x)^3/(a+a*sin(x))^2,x, algorithm="maxima")
Output:
1/24*(15*sin(x)/(cos(x) + 1) + 239*sin(x)^2/(cos(x) + 1)^2 + 405*sin(x)^3/ (cos(x) + 1)^3 + 216*sin(x)^4/(cos(x) + 1)^4 - 3)/(a^2*sin(x)^2/(cos(x) + 1)^2 + 3*a^2*sin(x)^3/(cos(x) + 1)^3 + 3*a^2*sin(x)^4/(cos(x) + 1)^4 + a^2 *sin(x)^5/(cos(x) + 1)^5) - 1/8*(8*sin(x)/(cos(x) + 1) - sin(x)^2/(cos(x) + 1)^2)/a^2 + 7/2*log(sin(x)/(cos(x) + 1))/a^2
Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.45 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {7 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{2}} + \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a^{2} \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{4}} - \frac {42 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, x\right ) + 1}{8 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2}} + \frac {2 \, {\left (12 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 21 \, \tan \left (\frac {1}{2} \, x\right ) + 11\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \] Input:
integrate(csc(x)^3/(a+a*sin(x))^2,x, algorithm="giac")
Output:
7/2*log(abs(tan(1/2*x)))/a^2 + 1/8*(a^2*tan(1/2*x)^2 - 8*a^2*tan(1/2*x))/a ^4 - 1/8*(42*tan(1/2*x)^2 - 8*tan(1/2*x) + 1)/(a^2*tan(1/2*x)^2) + 2/3*(12 *tan(1/2*x)^2 + 21*tan(1/2*x) + 11)/(a^2*(tan(1/2*x) + 1)^3)
Time = 17.38 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.73 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {36\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\frac {135\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{2}+\frac {239\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{6}+\frac {5\,\mathrm {tan}\left (\frac {x}{2}\right )}{2}-\frac {1}{2}}{4\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+12\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+12\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+4\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^2}+\frac {7\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a^2} \] Input:
int(1/(sin(x)^3*(a + a*sin(x))^2),x)
Output:
((5*tan(x/2))/2 + (239*tan(x/2)^2)/6 + (135*tan(x/2)^3)/2 + 36*tan(x/2)^4 - 1/2)/(4*a^2*tan(x/2)^2 + 12*a^2*tan(x/2)^3 + 12*a^2*tan(x/2)^4 + 4*a^2*t an(x/2)^5) - tan(x/2)/a^2 + tan(x/2)^2/(8*a^2) + (7*log(tan(x/2)))/(2*a^2)
Time = 0.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.17 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {84 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3}+84 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2}-89 \cos \left (x \right ) \sin \left (x \right )^{3}-133 \cos \left (x \right ) \sin \left (x \right )^{2}-24 \cos \left (x \right ) \sin \left (x \right )+12 \cos \left (x \right )-84 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4}-168 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3}-84 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2}-167 \sin \left (x \right )^{4}-122 \sin \left (x \right )^{3}+109 \sin \left (x \right )^{2}+36 \sin \left (x \right )-12}{24 \sin \left (x \right )^{2} a^{2} \left (\cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right )-\sin \left (x \right )^{2}-2 \sin \left (x \right )-1\right )} \] Input:
int(csc(x)^3/(a+a*sin(x))^2,x)
Output:
(84*cos(x)*log(tan(x/2))*sin(x)**3 + 84*cos(x)*log(tan(x/2))*sin(x)**2 - 8 9*cos(x)*sin(x)**3 - 133*cos(x)*sin(x)**2 - 24*cos(x)*sin(x) + 12*cos(x) - 84*log(tan(x/2))*sin(x)**4 - 168*log(tan(x/2))*sin(x)**3 - 84*log(tan(x/2 ))*sin(x)**2 - 167*sin(x)**4 - 122*sin(x)**3 + 109*sin(x)**2 + 36*sin(x) - 12)/(24*sin(x)**2*a**2*(cos(x)*sin(x) + cos(x) - sin(x)**2 - 2*sin(x) - 1 ))