3.1.31 \(\int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx\) [31]

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

[Out]

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+23/3*cot(x)*csc(x)^2/(a^3+a^3*sin(x))

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Rubi [A]
time = 0.16, antiderivative size = 103, normalized size of antiderivative = 1.00, 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 = {2845, 3057, 2827, 3852, 3853, 3855} \begin {gather*} -\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} \end {gather*}

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 2827

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 2845

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 3057

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 3852

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 3853

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 3855

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 \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(299\) vs. \(2(103)=206\).
time = 0.63, size = 299, normalized size = 2.90 \begin {gather*} \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} \end {gather*}

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 = 0.23, size = 110, normalized size = 1.07

method	result	size
default	\(\frac {\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}-3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+27 \tan \left (\frac {x}{2}\right )-\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}}+\frac {3}{\tan \left (\frac {x}{2}\right )^{2}}-\frac {27}{\tan \left (\frac {x}{2}\right )}-92 \ln \left (\tan \left (\frac {x}{2}\right )\right )-\frac {64}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {32}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {256}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {96}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {240}{\tan \left (\frac {x}{2}\right )+1}}{8 a^{3}}\)	\(110\)
risch	\(-\frac {11684 \,{\mathrm e}^{6 i x}-12622 \,{\mathrm e}^{4 i x}-544-4370 \,{\mathrm e}^{8 i x}+1725 i {\mathrm e}^{9 i x}-8050 i {\mathrm e}^{7 i x}+13340 i {\mathrm e}^{5 i x}+5347 \,{\mathrm e}^{2 i x}-9230 i {\mathrm e}^{3 i x}+2375 i {\mathrm e}^{i x}+345 \,{\mathrm e}^{10 i x}}{15 \left ({\mathrm e}^{2 i x}-1\right )^{3} \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}+\frac {23 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{3}}-\frac {23 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{3}}\)	\(129\)
norman	\(\frac {-\frac {1}{24 a}+\frac {\tan \left (\frac {x}{2}\right )}{6 a}-\frac {23 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{12 a}+\frac {23 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{12 a}-\frac {\tan ^{10}\left (\frac {x}{2}\right )}{6 a}+\frac {\tan ^{11}\left (\frac {x}{2}\right )}{24 a}-\frac {228 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {1067 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{20 a}-\frac {611 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {1567 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8 a}-\frac {567 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{8 a}}{\tan \left (\frac {x}{2}\right )^{3} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {23 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{3}}\)	\(144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a^3*(1/3*tan(1/2*x)^3-3*tan(1/2*x)^2+27*tan(1/2*x)-1/3/tan(1/2*x)^3+3/tan(1/2*x)^2-27/tan(1/2*x)-92*ln(tan
(1/2*x))-64/5/(tan(1/2*x)+1)^5+32/(tan(1/2*x)+1)^4-256/3/(tan(1/2*x)+1)^3+96/(tan(1/2*x)+1)^2-240/(tan(1/2*x)+
1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (89) = 178\).
time = 0.44, size = 232, normalized size = 2.25 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (89) = 178\).
time = 0.35, size = 333, normalized size = 3.23 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.47, size = 128, normalized size = 1.24 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 6.69, size = 117, normalized size = 1.14 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^4*(a + a*sin(x))^3),x)

[Out]

(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)

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