3.31 \(\int \frac{\csc ^4(x)}{(a+a \sin (x))^3} \, dx\)

Optimal. Leaf size=103 \[ -\frac{136 \cot ^3(x)}{15 a^3}-\frac{136 \cot (x)}{5 a^3}+\frac{23 \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{23 \cot (x) \csc (x)}{2 a^3}+\frac{23 \cot (x) \csc ^2(x)}{3 \left (a^3 \sin (x)+a^3\right )}+\frac{13 \cot (x) \csc ^2(x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3} \]

[Out]

(23*ArcTanh[Cos[x]])/(2*a^3) - (136*Cot[x])/(5*a^3) - (136*Cot[x]^3)/(15*a^3) + (23*Cot[x]*Csc[x])/(2*a^3) + (
Cot[x]*Csc[x]^2)/(5*(a + a*Sin[x])^3) + (13*Cot[x]*Csc[x]^2)/(15*a*(a + a*Sin[x])^2) + (23*Cot[x]*Csc[x]^2)/(3
*(a^3 + a^3*Sin[x]))

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Rubi [A]  time = 0.244845, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2766, 2978, 2748, 3767, 3768, 3770} \[ -\frac{136 \cot ^3(x)}{15 a^3}-\frac{136 \cot (x)}{5 a^3}+\frac{23 \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{23 \cot (x) \csc (x)}{2 a^3}+\frac{23 \cot (x) \csc ^2(x)}{3 \left (a^3 \sin (x)+a^3\right )}+\frac{13 \cot (x) \csc ^2(x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^4/(a + a*Sin[x])^3,x]

[Out]

(23*ArcTanh[Cos[x]])/(2*a^3) - (136*Cot[x])/(5*a^3) - (136*Cot[x]^3)/(15*a^3) + (23*Cot[x]*Csc[x])/(2*a^3) + (
Cot[x]*Csc[x]^2)/(5*(a + a*Sin[x])^3) + (13*Cot[x]*Csc[x]^2)/(15*a*(a + a*Sin[x])^2) + (23*Cot[x]*Csc[x]^2)/(3
*(a^3 + a^3*Sin[x]))

Rule 2766

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(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] + Dis
t[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] && (IntegersQ[2*m, 2*n] || (IntegerQ
[m] && EqQ[c, 0]))

Rule 2978

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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] + Dist[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])

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

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] + Dist[(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]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\csc ^4(x)}{(a+a \sin (x))^3} \, dx &=\frac{\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac{\int \frac{\csc ^4(x) (8 a-5 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\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{\int \frac{\csc ^4(x) \left (63 a^2-52 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\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 )}+\frac{\int \csc ^4(x) \left (408 a^3-345 a^3 \sin (x)\right ) \, dx}{15 a^6}\\ &=\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 )}-\frac{23 \int \csc ^3(x) \, dx}{a^3}+\frac{136 \int \csc ^4(x) \, dx}{5 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 )}-\frac{23 \int \csc (x) \, dx}{2 a^3}-\frac{136 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{5 a^3}\\ &=\frac{23 \tanh ^{-1}(\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 )}\\ \end{align*}

Mathematica [B]  time = 0.852753, size = 299, normalized size = 2.9 \[ \frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (48 \sin \left (\frac{x}{2}\right )-45 \cos ^3\left (\frac{x}{2}\right ) \left (\tan \left (\frac{x}{2}\right )+1\right )^5+2752 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4-176 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3+352 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2-24 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )+45 \sin ^3\left (\frac{x}{2}\right ) \left (\cot \left (\frac{x}{2}\right )+1\right )^5+5 \sin \left (\frac{x}{2}\right ) \cos ^2\left (\frac{x}{2}\right ) \left (\tan \left (\frac{x}{2}\right )+1\right )^5+1380 \log \left (\cos \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5-1380 \log \left (\sin \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5+400 \tan \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5-5 \sin ^2\left (\frac{x}{2}\right ) \cos \left (\frac{x}{2}\right ) \left (\cot \left (\frac{x}{2}\right )+1\right )^5-400 \cot \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5\right )}{120 a^3 (\sin (x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^4/(a + a*Sin[x])^3,x]

[Out]

((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)

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Maple [A]  time = 0.071, size = 141, normalized size = 1.4 \begin{align*}{\frac{1}{24\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{3}{8\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{27}{8\,{a}^{3}}\tan \left ({\frac{x}{2}} \right ) }-{\frac{8}{5\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}+4\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{4}}}-{\frac{32}{3\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+12\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{2}}}-30\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) }}-{\frac{1}{24\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{3}{8\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{27}{8\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{23}{2\,{a}^{3}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^4/(a+a*sin(x))^3,x)

[Out]

1/24/a^3*tan(1/2*x)^3-3/8/a^3*tan(1/2*x)^2+27/8/a^3*tan(1/2*x)-8/5/a^3/(tan(1/2*x)+1)^5+4/a^3/(tan(1/2*x)+1)^4
-32/3/a^3/(tan(1/2*x)+1)^3+12/a^3/(tan(1/2*x)+1)^2-30/a^3/(tan(1/2*x)+1)-1/24/a^3/tan(1/2*x)^3+3/8/a^3/tan(1/2
*x)^2-27/8/a^3/tan(1/2*x)-23/2/a^3*ln(tan(1/2*x))

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Maxima [B]  time = 1.74956, size = 313, normalized size = 3.04 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^4/(a+a*sin(x))^3,x, algorithm="maxima")

[Out]

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/(c
os(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/(cos(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

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Fricas [B]  time = 1.51376, size = 1026, normalized size = 9.96 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^4/(a+a*sin(x))^3,x, algorithm="fricas")

[Out]

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)*s
in(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*cos(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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**4/(a+a*sin(x))**3,x)

[Out]

Integral(csc(x)**4/(sin(x)**3 + 3*sin(x)**2 + 3*sin(x) + 1), x)/a**3

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Giac [A]  time = 1.31258, size = 173, normalized size = 1.68 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^4/(a+a*sin(x))^3,x, algorithm="giac")

[Out]

-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