Integrand size = 13, antiderivative size = 65 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {5 \text {arctanh}(\cos (x))}{a^2}-\frac {4 \cot (x)}{a^2}-\frac {\cot ^3(x)}{3 a^2}+\frac {\cot (x) \csc (x)}{a^2}-\frac {\cos (x)}{3 a^2 (1+\sin (x))^2}-\frac {13 \cos (x)}{3 a^2 (1+\sin (x))} \] Output:
5*arctanh(cos(x))/a^2-4*cot(x)/a^2-1/3*cot(x)^3/a^2+cot(x)*csc(x)/a^2-1/3* cos(x)/a^2/(1+sin(x))^2-13/3*cos(x)/a^2/(1+sin(x))
Leaf count is larger than twice the leaf count of optimal. \(238\) vs. \(2(65)=130\).
Time = 3.63 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.66 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-\cos \left (\frac {x}{2}\right ) \left (1+\cot \left (\frac {x}{2}\right )\right )^3+16 \sin \left (\frac {x}{2}\right )+6 \left (1+\cot \left (\frac {x}{2}\right )\right )^3 \sin \left (\frac {x}{2}\right )-8 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+208 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2-44 \cot \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+120 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-120 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+44 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3 \tan \left (\frac {x}{2}\right )-6 \cos \left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^3+\sin \left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^3\right )}{24 a^2 (1+\sin (x))^2} \] Input:
Integrate[Csc[x]^4/(a + a*Sin[x])^2,x]
Output:
((Cos[x/2] + Sin[x/2])*(-(Cos[x/2]*(1 + Cot[x/2])^3) + 16*Sin[x/2] + 6*(1 + Cot[x/2])^3*Sin[x/2] - 8*(Cos[x/2] + Sin[x/2]) + 208*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 - 44*Cot[x/2]*(Cos[x/2] + Sin[x/2])^3 + 120*Log[Cos[x/2]]*(C os[x/2] + Sin[x/2])^3 - 120*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^3 + 44*(Co s[x/2] + Sin[x/2])^3*Tan[x/2] - 6*Cos[x/2]*(1 + Tan[x/2])^3 + Sin[x/2]*(1 + Tan[x/2])^3))/(24*a^2*(1 + Sin[x])^2)
Time = 0.69 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.29, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {3042, 3245, 27, 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.
| \(\displaystyle \int \frac {\csc ^4(x)}{(a \sin (x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^4 (a \sin (x)+a)^2}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle \frac {\int \frac {2 \csc ^4(x) (3 a-2 a \sin (x))}{\sin (x) a+a}dx}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {\csc ^4(x) (3 a-2 a \sin (x))}{\sin (x) a+a}dx}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \frac {3 a-2 a \sin (x)}{\sin (x)^4 (\sin (x) a+a)}dx}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {2 \left (\frac {\int 3 \csc ^4(x) \left (6 a^2-5 a^2 \sin (x)\right )dx}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {3 \int \csc ^4(x) \left (6 a^2-5 a^2 \sin (x)\right )dx}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {3 \int \frac {6 a^2-5 a^2 \sin (x)}{\sin (x)^4}dx}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {2 \left (\frac {3 \left (6 a^2 \int \csc ^4(x)dx-5 a^2 \int \csc ^3(x)dx\right )}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {3 \left (6 a^2 \int \csc (x)^4dx-5 a^2 \int \csc (x)^3dx\right )}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {2 \left (\frac {3 \left (-6 a^2 \int \left (\cot ^2(x)+1\right )d\cot (x)-5 a^2 \int \csc (x)^3dx\right )}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {3 \left (-5 a^2 \int \csc (x)^3dx-6 a^2 \left (\frac {\cot ^3(x)}{3}+\cot (x)\right )\right )}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {2 \left (\frac {3 \left (-5 a^2 \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )-6 a^2 \left (\frac {\cot ^3(x)}{3}+\cot (x)\right )\right )}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {3 \left (-5 a^2 \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )-6 a^2 \left (\frac {\cot ^3(x)}{3}+\cot (x)\right )\right )}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {2 \left (\frac {3 \left (-5 a^2 \left (-\frac {1}{2} \text {arctanh}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)\right )-6 a^2 \left (\frac {\cot ^3(x)}{3}+\cot (x)\right )\right )}{a^2}+\frac {5 \cot (x) \csc ^2(x)}{\sin (x)+1}\right )}{3 a^2}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}\) |
Input:
Int[Csc[x]^4/(a + a*Sin[x])^2,x]
Output:
(Cot[x]*Csc[x]^2)/(3*(a + a*Sin[x])^2) + (2*((3*(-6*a^2*(Cot[x] + Cot[x]^3 /3) - 5*a^2*(-1/2*ArcTanh[Cos[x]] - (Cot[x]*Csc[x])/2)))/a^2 + (5*Cot[x]*C sc[x]^2)/(1 + Sin[x])))/(3*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 = 0.51 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.37
| method | result | size |
| parallelrisch | \(\frac {-120 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3} \ln \left (\tan \left (\frac {x}{2}\right )\right )+\tan \left (\frac {x}{2}\right )^{6}-3 \tan \left (\frac {x}{2}\right )^{5}+30 \tan \left (\frac {x}{2}\right )^{4}-\cot \left (\frac {x}{2}\right )^{3}+3 \cot \left (\frac {x}{2}\right )^{2}-510 \tan \left (\frac {x}{2}\right )^{2}-30 \cot \left (\frac {x}{2}\right )-870 \tan \left (\frac {x}{2}\right )-460}{24 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\) | \(89\) |
| default | \(\frac {\frac {\tan \left (\frac {x}{2}\right )^{3}}{3}-2 \tan \left (\frac {x}{2}\right )^{2}+15 \tan \left (\frac {x}{2}\right )-\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {x}{2}\right )^{2}}-\frac {15}{\tan \left (\frac {x}{2}\right )}-40 \ln \left (\tan \left (\frac {x}{2}\right )\right )-\frac {32}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {80}{\tan \left (\frac {x}{2}\right )+1}}{8 a^{2}}\) | \(90\) |
| risch | \(-\frac {2 \left (-85 \,{\mathrm e}^{6 i x}+45 i {\mathrm e}^{7 i x}+153 \,{\mathrm e}^{4 i x}-135 i {\mathrm e}^{5 i x}+15 \,{\mathrm e}^{8 i x}-99 \,{\mathrm e}^{2 i x}+155 i {\mathrm e}^{3 i x}+24-57 i {\mathrm e}^{i x}\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3} \left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}-\frac {5 \ln \left ({\mathrm e}^{i x}-1\right )}{a^{2}}+\frac {5 \ln \left ({\mathrm e}^{i x}+1\right )}{a^{2}}\) | \(114\) |
| norman | \(\frac {-\frac {1}{24 a}+\frac {\tan \left (\frac {x}{2}\right )}{8 a}-\frac {5 \tan \left (\frac {x}{2}\right )^{2}}{4 a}+\frac {5 \tan \left (\frac {x}{2}\right )^{7}}{4 a}-\frac {\tan \left (\frac {x}{2}\right )^{8}}{8 a}+\frac {\tan \left (\frac {x}{2}\right )^{9}}{24 a}-\frac {115 \tan \left (\frac {x}{2}\right )^{3}}{6 a}-\frac {145 \tan \left (\frac {x}{2}\right )^{4}}{4 a}-\frac {85 \tan \left (\frac {x}{2}\right )^{5}}{4 a}}{\tan \left (\frac {x}{2}\right )^{3} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\) | \(122\) |
Input:
int(csc(x)^4/(a+a*sin(x))^2,x,method=_RETURNVERBOSE)
Output:
1/24*(-120*(tan(1/2*x)+1)^3*ln(tan(1/2*x))+tan(1/2*x)^6-3*tan(1/2*x)^5+30* tan(1/2*x)^4-cot(1/2*x)^3+3*cot(1/2*x)^2-510*tan(1/2*x)^2-30*cot(1/2*x)-87 0*tan(1/2*x)-460)/a^2/(tan(1/2*x)+1)^3
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (59) = 118\).
Time = 0.08 (sec) , antiderivative size = 266, normalized size of antiderivative = 4.09 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=-\frac {48 \, \cos \left (x\right )^{5} - 18 \, \cos \left (x\right )^{4} - 108 \, \cos \left (x\right )^{3} + 22 \, \cos \left (x\right )^{2} - 15 \, {\left (\cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 4 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} + \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 4 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} + \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2 \, {\left (24 \, \cos \left (x\right )^{4} + 33 \, \cos \left (x\right )^{3} - 21 \, \cos \left (x\right )^{2} - 32 \, \cos \left (x\right ) - 1\right )} \sin \left (x\right ) + 62 \, \cos \left (x\right ) - 2}{6 \, {\left (a^{2} \cos \left (x\right )^{5} + 2 \, a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{3} - 4 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) + 2 \, a^{2} + {\left (a^{2} \cos \left (x\right )^{4} - a^{2} \cos \left (x\right )^{3} - 3 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(csc(x)^4/(a+a*sin(x))^2,x, algorithm="fricas")
Output:
-1/6*(48*cos(x)^5 - 18*cos(x)^4 - 108*cos(x)^3 + 22*cos(x)^2 - 15*(cos(x)^ 5 + 2*cos(x)^4 - 2*cos(x)^3 - 4*cos(x)^2 + (cos(x)^4 - cos(x)^3 - 3*cos(x) ^2 + cos(x) + 2)*sin(x) + cos(x) + 2)*log(1/2*cos(x) + 1/2) + 15*(cos(x)^5 + 2*cos(x)^4 - 2*cos(x)^3 - 4*cos(x)^2 + (cos(x)^4 - cos(x)^3 - 3*cos(x)^ 2 + cos(x) + 2)*sin(x) + cos(x) + 2)*log(-1/2*cos(x) + 1/2) - 2*(24*cos(x) ^4 + 33*cos(x)^3 - 21*cos(x)^2 - 32*cos(x) - 1)*sin(x) + 62*cos(x) - 2)/(a ^2*cos(x)^5 + 2*a^2*cos(x)^4 - 2*a^2*cos(x)^3 - 4*a^2*cos(x)^2 + a^2*cos(x ) + 2*a^2 + (a^2*cos(x)^4 - a^2*cos(x)^3 - 3*a^2*cos(x)^2 + a^2*cos(x) + 2 *a^2)*sin(x))
\[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {\int \frac {\csc ^{4}{\left (x \right )}}{\sin ^{2}{\left (x \right )} + 2 \sin {\left (x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(csc(x)**4/(a+a*sin(x))**2,x)
Output:
Integral(csc(x)**4/(sin(x)**2 + 2*sin(x) + 1), x)/a**2
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (59) = 118\).
Time = 0.03 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.74 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {\frac {3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {30 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {342 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {561 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {285 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - 1}{24 \, {\left (\frac {a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {3 \, a^{2} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {a^{2} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}\right )}} + \frac {\frac {45 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {6 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {\sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{24 \, a^{2}} - \frac {5 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \] Input:
integrate(csc(x)^4/(a+a*sin(x))^2,x, algorithm="maxima")
Output:
1/24*(3*sin(x)/(cos(x) + 1) - 30*sin(x)^2/(cos(x) + 1)^2 - 342*sin(x)^3/(c os(x) + 1)^3 - 561*sin(x)^4/(cos(x) + 1)^4 - 285*sin(x)^5/(cos(x) + 1)^5 - 1)/(a^2*sin(x)^3/(cos(x) + 1)^3 + 3*a^2*sin(x)^4/(cos(x) + 1)^4 + 3*a^2*s in(x)^5/(cos(x) + 1)^5 + a^2*sin(x)^6/(cos(x) + 1)^6) + 1/24*(45*sin(x)/(c os(x) + 1) - 6*sin(x)^2/(cos(x) + 1)^2 + sin(x)^3/(cos(x) + 1)^3)/a^2 - 5* log(sin(x)/(cos(x) + 1))/a^2
Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.75 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=-\frac {5 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{2}} + \frac {110 \, \tan \left (\frac {1}{2} \, x\right )^{6} + 45 \, \tan \left (\frac {1}{2} \, x\right )^{5} - 231 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 232 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 30 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{24 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right )\right )}^{3} a^{2}} + \frac {a^{4} \tan \left (\frac {1}{2} \, x\right )^{3} - 6 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 45 \, a^{4} \tan \left (\frac {1}{2} \, x\right )}{24 \, a^{6}} \] Input:
integrate(csc(x)^4/(a+a*sin(x))^2,x, algorithm="giac")
Output:
-5*log(abs(tan(1/2*x)))/a^2 + 1/24*(110*tan(1/2*x)^6 + 45*tan(1/2*x)^5 - 2 31*tan(1/2*x)^4 - 232*tan(1/2*x)^3 - 30*tan(1/2*x)^2 + 3*tan(1/2*x) - 1)/( (tan(1/2*x)^2 + tan(1/2*x))^3*a^2) + 1/24*(a^4*tan(1/2*x)^3 - 6*a^4*tan(1/ 2*x)^2 + 45*a^4*tan(1/2*x))/a^6
Time = 17.33 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.55 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {15\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a^2}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4\,a^2}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a^2}-\frac {5\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^2}-\frac {\frac {95\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{8}+\frac {187\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{8}+\frac {57\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{4}+\frac {5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{8}+\frac {1}{24}}{a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \] Input:
int(1/(sin(x)^4*(a + a*sin(x))^2),x)
Output:
(15*tan(x/2))/(8*a^2) - tan(x/2)^2/(4*a^2) + tan(x/2)^3/(24*a^2) - (5*log( tan(x/2)))/a^2 - ((5*tan(x/2)^2)/4 - tan(x/2)/8 + (57*tan(x/2)^3)/4 + (187 *tan(x/2)^4)/8 + (95*tan(x/2)^5)/8 + 1/24)/(a^2*tan(x/2)^3*(tan(x/2) + 1)^ 3)
Time = 0.17 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.35 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {-30 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4}-30 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3}+33 \cos \left (x \right ) \sin \left (x \right )^{4}+51 \cos \left (x \right ) \sin \left (x \right )^{3}+12 \cos \left (x \right ) \sin \left (x \right )^{2}-2 \cos \left (x \right ) \sin \left (x \right )+2 \cos \left (x \right )+30 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{5}+60 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4}+30 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3}+63 \sin \left (x \right )^{5}+48 \sin \left (x \right )^{4}-39 \sin \left (x \right )^{3}-14 \sin \left (x \right )^{2}+4 \sin \left (x \right )-2}{6 \sin \left (x \right )^{3} a^{2} \left (\cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right )-\sin \left (x \right )^{2}-2 \sin \left (x \right )-1\right )} \] Input:
int(csc(x)^4/(a+a*sin(x))^2,x)
Output:
( - 30*cos(x)*log(tan(x/2))*sin(x)**4 - 30*cos(x)*log(tan(x/2))*sin(x)**3 + 33*cos(x)*sin(x)**4 + 51*cos(x)*sin(x)**3 + 12*cos(x)*sin(x)**2 - 2*cos( x)*sin(x) + 2*cos(x) + 30*log(tan(x/2))*sin(x)**5 + 60*log(tan(x/2))*sin(x )**4 + 30*log(tan(x/2))*sin(x)**3 + 63*sin(x)**5 + 48*sin(x)**4 - 39*sin(x )**3 - 14*sin(x)**2 + 4*sin(x) - 2)/(6*sin(x)**3*a**2*(cos(x)*sin(x) + cos (x) - sin(x)**2 - 2*sin(x) - 1))