Integrand size = 13, antiderivative size = 103 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {23 \text {arctanh}(\cos (x))}{2 a^3}-\frac {136 \cot (x)}{5 a^3}-\frac {136 \cot ^3(x)}{15 a^3}+\frac {23 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )} \] Output:
23/2*arctanh(cos(x))/a^3-136/5*cot(x)/a^3-136/15*cot(x)^3/a^3+23/2*cot(x)* csc(x)/a^3+1/5*cot(x)*csc(x)^2/(a+a*sin(x))^3+13/15*cot(x)*csc(x)^2/a/(a+a *sin(x))^2+23*cot(x)*csc(x)^2/(3*a^3+3*a^3*sin(x))
Leaf count is larger than twice the leaf count of optimal. \(299\) vs. \(2(103)=206\).
Time = 1.75 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.90 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (48 \sin \left (\frac {x}{2}\right )-5 \cos \left (\frac {x}{2}\right ) \left (1+\cot \left (\frac {x}{2}\right )\right )^5 \sin ^2\left (\frac {x}{2}\right )+45 \left (1+\cot \left (\frac {x}{2}\right )\right )^5 \sin ^3\left (\frac {x}{2}\right )-24 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+352 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2-176 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+2752 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4-400 \cot \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+1380 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-1380 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+400 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5 \tan \left (\frac {x}{2}\right )-45 \cos ^3\left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^5+5 \cos ^2\left (\frac {x}{2}\right ) \sin \left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^5\right )}{120 a^3 (1+\sin (x))^3} \] Input:
Integrate[Csc[x]^4/(a + a*Sin[x])^3,x]
Output:
((Cos[x/2] + Sin[x/2])*(48*Sin[x/2] - 5*Cos[x/2]*(1 + Cot[x/2])^5*Sin[x/2] ^2 + 45*(1 + Cot[x/2])^5*Sin[x/2]^3 - 24*(Cos[x/2] + Sin[x/2]) + 352*Sin[x /2]*(Cos[x/2] + Sin[x/2])^2 - 176*(Cos[x/2] + Sin[x/2])^3 + 2752*Sin[x/2]* (Cos[x/2] + Sin[x/2])^4 - 400*Cot[x/2]*(Cos[x/2] + Sin[x/2])^5 + 1380*Log[ Cos[x/2]]*(Cos[x/2] + Sin[x/2])^5 - 1380*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2 ])^5 + 400*(Cos[x/2] + Sin[x/2])^5*Tan[x/2] - 45*Cos[x/2]^3*(1 + Tan[x/2]) ^5 + 5*Cos[x/2]^2*Sin[x/2]*(1 + Tan[x/2])^5))/(120*a^3*(1 + Sin[x])^3)
Time = 0.92 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.13, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 3245, 3042, 3457, 3042, 3457, 27, 3042, 3227, 3042, 4254, 2009, 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.
| \(\displaystyle \int \frac {\csc ^4(x)}{(a \sin (x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^4 (a \sin (x)+a)^3}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle \frac {\int \frac {\csc ^4(x) (8 a-5 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {8 a-5 a \sin (x)}{\sin (x)^4 (\sin (x) a+a)^2}dx}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \frac {\csc ^4(x) \left (63 a^2-52 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {63 a^2-52 a^2 \sin (x)}{\sin (x)^4 (\sin (x) a+a)}dx}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\int 3 \csc ^4(x) \left (136 a^3-115 a^3 \sin (x)\right )dx}{a^2}+\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \csc ^4(x) \left (136 a^3-115 a^3 \sin (x)\right )dx}{a^2}+\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {136 a^3-115 a^3 \sin (x)}{\sin (x)^4}dx}{a^2}+\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {\frac {3 \left (136 a^3 \int \csc ^4(x)dx-115 a^3 \int \csc ^3(x)dx\right )}{a^2}+\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (136 a^3 \int \csc (x)^4dx-115 a^3 \int \csc (x)^3dx\right )}{a^2}+\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-136 a^3 \int \left (\cot ^2(x)+1\right )d\cot (x)-115 a^3 \int \csc (x)^3dx\right )}{a^2}+\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-115 a^3 \int \csc (x)^3dx-136 a^3 \left (\frac {\cot ^3(x)}{3}+\cot (x)\right )\right )}{a^2}+\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-115 a^3 \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )-136 a^3 \left (\frac {\cot ^3(x)}{3}+\cot (x)\right )\right )}{a^2}+\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-115 a^3 \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )-136 a^3 \left (\frac {\cot ^3(x)}{3}+\cot (x)\right )\right )}{a^2}+\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\frac {115 a^2 \cot (x) \csc ^2(x)}{a \sin (x)+a}+\frac {3 \left (-115 a^3 \left (-\frac {1}{2} \text {arctanh}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)\right )-136 a^3 \left (\frac {\cot ^3(x)}{3}+\cot (x)\right )\right )}{a^2}}{3 a^2}+\frac {13 a \cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}\) |
Input:
Int[Csc[x]^4/(a + a*Sin[x])^3,x]
Output:
(Cot[x]*Csc[x]^2)/(5*(a + a*Sin[x])^3) + ((13*a*Cot[x]*Csc[x]^2)/(3*(a + a *Sin[x])^2) + ((3*(-136*a^3*(Cot[x] + Cot[x]^3/3) - 115*a^3*(-1/2*ArcTanh[ Cos[x]] - (Cot[x]*Csc[x])/2)))/a^2 + (115*a^2*Cot[x]*Csc[x]^2)/(a + a*Sin[ x]))/(3*a^2))/(5*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 0.64 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04
| method | result | size |
| parallelrisch | \(\frac {-1380 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5} \ln \left (\tan \left (\frac {x}{2}\right )\right )+5 \tan \left (\frac {x}{2}\right )^{8}-20 \tan \left (\frac {x}{2}\right )^{7}+230 \tan \left (\frac {x}{2}\right )^{6}-8505 \tan \left (\frac {x}{2}\right )^{4}-5 \cot \left (\frac {x}{2}\right )^{3}-27360 \tan \left (\frac {x}{2}\right )^{3}+20 \cot \left (\frac {x}{2}\right )^{2}-36660 \tan \left (\frac {x}{2}\right )^{2}-230 \cot \left (\frac {x}{2}\right )-23505 \tan \left (\frac {x}{2}\right )-6402}{120 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(107\) |
| default | \(\frac {\frac {\tan \left (\frac {x}{2}\right )^{3}}{3}-3 \tan \left (\frac {x}{2}\right )^{2}+27 \tan \left (\frac {x}{2}\right )-\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}}+\frac {3}{\tan \left (\frac {x}{2}\right )^{2}}-\frac {27}{\tan \left (\frac {x}{2}\right )}-92 \ln \left (\tan \left (\frac {x}{2}\right )\right )-\frac {64}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {32}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {256}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {96}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {240}{\tan \left (\frac {x}{2}\right )+1}}{8 a^{3}}\) | \(110\) |
| risch | \(-\frac {-544+5347 \,{\mathrm e}^{2 i x}-4370 \,{\mathrm e}^{8 i x}+11684 \,{\mathrm e}^{6 i x}+345 \,{\mathrm e}^{10 i x}-12622 \,{\mathrm e}^{4 i x}+1725 i {\mathrm e}^{9 i x}-8050 i {\mathrm e}^{7 i x}+13340 i {\mathrm e}^{5 i x}-9230 i {\mathrm e}^{3 i x}+2375 i {\mathrm e}^{i x}}{15 \left ({\mathrm e}^{2 i x}-1\right )^{3} \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}+\frac {23 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{3}}-\frac {23 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{3}}\) | \(129\) |
| norman | \(\frac {-\frac {1}{24 a}+\frac {\tan \left (\frac {x}{2}\right )}{6 a}-\frac {23 \tan \left (\frac {x}{2}\right )^{2}}{12 a}+\frac {23 \tan \left (\frac {x}{2}\right )^{9}}{12 a}-\frac {\tan \left (\frac {x}{2}\right )^{10}}{6 a}+\frac {\tan \left (\frac {x}{2}\right )^{11}}{24 a}-\frac {228 \tan \left (\frac {x}{2}\right )^{6}}{a}-\frac {1067 \tan \left (\frac {x}{2}\right )^{3}}{20 a}-\frac {611 \tan \left (\frac {x}{2}\right )^{5}}{2 a}-\frac {1567 \tan \left (\frac {x}{2}\right )^{4}}{8 a}-\frac {567 \tan \left (\frac {x}{2}\right )^{7}}{8 a}}{\tan \left (\frac {x}{2}\right )^{3} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {23 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{3}}\) | \(144\) |
Input:
int(csc(x)^4/(a+a*sin(x))^3,x,method=_RETURNVERBOSE)
Output:
1/120*(-1380*(tan(1/2*x)+1)^5*ln(tan(1/2*x))+5*tan(1/2*x)^8-20*tan(1/2*x)^ 7+230*tan(1/2*x)^6-8505*tan(1/2*x)^4-5*cot(1/2*x)^3-27360*tan(1/2*x)^3+20* cot(1/2*x)^2-36660*tan(1/2*x)^2-230*cot(1/2*x)-23505*tan(1/2*x)-6402)/a^3/ (tan(1/2*x)+1)^5
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (89) = 178\).
Time = 0.11 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.23 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {1088 \, \cos \left (x\right )^{6} + 2574 \, \cos \left (x\right )^{5} - 2428 \, \cos \left (x\right )^{4} - 5338 \, \cos \left (x\right )^{3} + 1372 \, \cos \left (x\right )^{2} + 345 \, {\left (\cos \left (x\right )^{6} - 2 \, \cos \left (x\right )^{5} - 6 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{3} + 9 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 345 \, {\left (\cos \left (x\right )^{6} - 2 \, \cos \left (x\right )^{5} - 6 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{3} + 9 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, {\left (544 \, \cos \left (x\right )^{5} - 743 \, \cos \left (x\right )^{4} - 1957 \, \cos \left (x\right )^{3} + 712 \, \cos \left (x\right )^{2} + 1398 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) + 2784 \, \cos \left (x\right ) - 12}{60 \, {\left (a^{3} \cos \left (x\right )^{6} - 2 \, a^{3} \cos \left (x\right )^{5} - 6 \, a^{3} \cos \left (x\right )^{4} + 4 \, a^{3} \cos \left (x\right )^{3} + 9 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} - {\left (a^{3} \cos \left (x\right )^{5} + 3 \, a^{3} \cos \left (x\right )^{4} - 3 \, a^{3} \cos \left (x\right )^{3} - 7 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(csc(x)^4/(a+a*sin(x))^3,x, algorithm="fricas")
Output:
1/60*(1088*cos(x)^6 + 2574*cos(x)^5 - 2428*cos(x)^4 - 5338*cos(x)^3 + 1372 *cos(x)^2 + 345*(cos(x)^6 - 2*cos(x)^5 - 6*cos(x)^4 + 4*cos(x)^3 + 9*cos(x )^2 - (cos(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7*cos(x)^2 + 2*cos(x) + 4)*sin (x) - 2*cos(x) - 4)*log(1/2*cos(x) + 1/2) - 345*(cos(x)^6 - 2*cos(x)^5 - 6 *cos(x)^4 + 4*cos(x)^3 + 9*cos(x)^2 - (cos(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7*cos(x)^2 + 2*cos(x) + 4)*sin(x) - 2*cos(x) - 4)*log(-1/2*cos(x) + 1/2) + 2*(544*cos(x)^5 - 743*cos(x)^4 - 1957*cos(x)^3 + 712*cos(x)^2 + 1398*co s(x) + 6)*sin(x) + 2784*cos(x) - 12)/(a^3*cos(x)^6 - 2*a^3*cos(x)^5 - 6*a^ 3*cos(x)^4 + 4*a^3*cos(x)^3 + 9*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 - (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)*sin(x))
\[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {\int \frac {\csc ^{4}{\left (x \right )}}{\sin ^{3}{\left (x \right )} + 3 \sin ^{2}{\left (x \right )} + 3 \sin {\left (x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(csc(x)**4/(a+a*sin(x))**3,x)
Output:
Integral(csc(x)**4/(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. 232 vs. \(2 (89) = 178\).
Time = 0.04 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.25 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {\frac {20 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {230 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {4777 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {15785 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {22390 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {14940 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {4005 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - 5}{120 \, {\left (\frac {a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {10 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {a^{3} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} + \frac {\frac {81 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {9 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {\sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{24 \, a^{3}} - \frac {23 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{3}} \] Input:
integrate(csc(x)^4/(a+a*sin(x))^3,x, algorithm="maxima")
Output:
1/120*(20*sin(x)/(cos(x) + 1) - 230*sin(x)^2/(cos(x) + 1)^2 - 4777*sin(x)^ 3/(cos(x) + 1)^3 - 15785*sin(x)^4/(cos(x) + 1)^4 - 22390*sin(x)^5/(cos(x) + 1)^5 - 14940*sin(x)^6/(cos(x) + 1)^6 - 4005*sin(x)^7/(cos(x) + 1)^7 - 5) /(a^3*sin(x)^3/(cos(x) + 1)^3 + 5*a^3*sin(x)^4/(cos(x) + 1)^4 + 10*a^3*sin (x)^5/(cos(x) + 1)^5 + 10*a^3*sin(x)^6/(cos(x) + 1)^6 + 5*a^3*sin(x)^7/(co s(x) + 1)^7 + a^3*sin(x)^8/(cos(x) + 1)^8) + 1/24*(81*sin(x)/(cos(x) + 1) - 9*sin(x)^2/(cos(x) + 1)^2 + sin(x)^3/(cos(x) + 1)^3)/a^3 - 23/2*log(sin( x)/(cos(x) + 1))/a^3
Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.24 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=-\frac {23 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{3}} + \frac {506 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 81 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{24 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{3}} - \frac {2 \, {\left (225 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 810 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 1160 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 760 \, \tan \left (\frac {1}{2} \, x\right ) + 197\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} + \frac {a^{6} \tan \left (\frac {1}{2} \, x\right )^{3} - 9 \, a^{6} \tan \left (\frac {1}{2} \, x\right )^{2} + 81 \, a^{6} \tan \left (\frac {1}{2} \, x\right )}{24 \, a^{9}} \] Input:
integrate(csc(x)^4/(a+a*sin(x))^3,x, algorithm="giac")
Output:
-23/2*log(abs(tan(1/2*x)))/a^3 + 1/24*(506*tan(1/2*x)^3 - 81*tan(1/2*x)^2 + 9*tan(1/2*x) - 1)/(a^3*tan(1/2*x)^3) - 2/15*(225*tan(1/2*x)^4 + 810*tan( 1/2*x)^3 + 1160*tan(1/2*x)^2 + 760*tan(1/2*x) + 197)/(a^3*(tan(1/2*x) + 1) ^5) + 1/24*(a^6*tan(1/2*x)^3 - 9*a^6*tan(1/2*x)^2 + 81*a^6*tan(1/2*x))/a^9
Time = 17.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {27\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a^3}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^3}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a^3}-\frac {23\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a^3}-\frac {\frac {267\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7}{8}+\frac {249\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{2}+\frac {2239\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{12}+\frac {3157\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{24}+\frac {4777\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{120}+\frac {23\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{12}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{6}+\frac {1}{24}}{a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \] Input:
int(1/(sin(x)^4*(a + a*sin(x))^3),x)
Output:
(27*tan(x/2))/(8*a^3) - (3*tan(x/2)^2)/(8*a^3) + tan(x/2)^3/(24*a^3) - (23 *log(tan(x/2)))/(2*a^3) - ((23*tan(x/2)^2)/12 - tan(x/2)/6 + (4777*tan(x/2 )^3)/120 + (3157*tan(x/2)^4)/24 + (2239*tan(x/2)^5)/12 + (249*tan(x/2)^6)/ 2 + (267*tan(x/2)^7)/8 + 1/24)/(a^3*tan(x/2)^3*(tan(x/2) + 1)^5)
Time = 0.19 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.99 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {-345 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{5}-690 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4}-345 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3}+169 \cos \left (x \right ) \sin \left (x \right )^{5}+537 \cos \left (x \right ) \sin \left (x \right )^{4}+494 \cos \left (x \right ) \sin \left (x \right )^{3}+95 \cos \left (x \right ) \sin \left (x \right )^{2}-15 \cos \left (x \right ) \sin \left (x \right )+10 \cos \left (x \right )+345 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{6}+1035 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{5}+1035 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4}+345 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3}+919 \sin \left (x \right )^{6}+1868 \sin \left (x \right )^{5}+707 \sin \left (x \right )^{4}-399 \sin \left (x \right )^{3}-110 \sin \left (x \right )^{2}+25 \sin \left (x \right )-10}{30 \sin \left (x \right )^{3} a^{3} \left (\cos \left (x \right ) \sin \left (x \right )^{2}+2 \cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right )-\sin \left (x \right )^{3}-3 \sin \left (x \right )^{2}-3 \sin \left (x \right )-1\right )} \] Input:
int(csc(x)^4/(a+a*sin(x))^3,x)
Output:
( - 345*cos(x)*log(tan(x/2))*sin(x)**5 - 690*cos(x)*log(tan(x/2))*sin(x)** 4 - 345*cos(x)*log(tan(x/2))*sin(x)**3 + 169*cos(x)*sin(x)**5 + 537*cos(x) *sin(x)**4 + 494*cos(x)*sin(x)**3 + 95*cos(x)*sin(x)**2 - 15*cos(x)*sin(x) + 10*cos(x) + 345*log(tan(x/2))*sin(x)**6 + 1035*log(tan(x/2))*sin(x)**5 + 1035*log(tan(x/2))*sin(x)**4 + 345*log(tan(x/2))*sin(x)**3 + 919*sin(x)* *6 + 1868*sin(x)**5 + 707*sin(x)**4 - 399*sin(x)**3 - 110*sin(x)**2 + 25*s in(x) - 10)/(30*sin(x)**3*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))