3.738 \(\int \frac{\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=210 \[ -\frac{\cot ^{11}(c+d x)}{11 a^2 d}-\frac{4 \cot ^9(c+d x)}{9 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}+\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac{3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{128 a^2 d} \]

[Out]

(3*ArcTanh[Cos[c + d*x]])/(128*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - (5*Cot[c + d*x]^7)/(7*a^2*d) - (4*Cot[c
+ d*x]^9)/(9*a^2*d) - Cot[c + d*x]^11/(11*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(128*a^2*d) + (Cot[c + d*x]*
Csc[c + d*x]^3)/(64*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(80*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x]^7)/(40*a^
2*d) + (Cot[c + d*x]^3*Csc[c + d*x]^7)/(5*a^2*d)

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Rubi [A]  time = 0.432113, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2875, 2873, 2607, 270, 2611, 3768, 3770} \[ -\frac{\cot ^{11}(c+d x)}{11 a^2 d}-\frac{4 \cot ^9(c+d x)}{9 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}+\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac{3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{128 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

(3*ArcTanh[Cos[c + d*x]])/(128*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - (5*Cot[c + d*x]^7)/(7*a^2*d) - (4*Cot[c
+ d*x]^9)/(9*a^2*d) - Cot[c + d*x]^11/(11*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(128*a^2*d) + (Cot[c + d*x]*
Csc[c + d*x]^3)/(64*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(80*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x]^7)/(40*a^
2*d) + (Cot[c + d*x]^3*Csc[c + d*x]^7)/(5*a^2*d)

Rule 2875

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[((g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^n)/(a - b*Sin[e +
f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]

Rule 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

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{\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \cot ^4(c+d x) \csc ^8(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \cot ^4(c+d x) \csc ^6(c+d x)-2 a^2 \cot ^4(c+d x) \csc ^7(c+d x)+a^2 \cot ^4(c+d x) \csc ^8(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \cot ^4(c+d x) \csc ^6(c+d x) \, dx}{a^2}+\frac{\int \cot ^4(c+d x) \csc ^8(c+d x) \, dx}{a^2}-\frac{2 \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx}{a^2}\\ &=\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}+\frac{3 \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx}{5 a^2}+\frac{\operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^2 d}\\ &=-\frac{3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac{3 \int \csc ^7(c+d x) \, dx}{40 a^2}+\frac{\operatorname{Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}\\ &=-\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{4 \cot ^9(c+d x)}{9 a^2 d}-\frac{\cot ^{11}(c+d x)}{11 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac{3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac{\int \csc ^5(c+d x) \, dx}{16 a^2}\\ &=-\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{4 \cot ^9(c+d x)}{9 a^2 d}-\frac{\cot ^{11}(c+d x)}{11 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac{3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac{3 \int \csc ^3(c+d x) \, dx}{64 a^2}\\ &=-\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{4 \cot ^9(c+d x)}{9 a^2 d}-\frac{\cot ^{11}(c+d x)}{11 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac{3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac{3 \int \csc (c+d x) \, dx}{128 a^2}\\ &=\frac{3 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{4 \cot ^9(c+d x)}{9 a^2 d}-\frac{\cot ^{11}(c+d x)}{11 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac{3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}\\ \end{align*}

Mathematica [A]  time = 4.03829, size = 186, normalized size = 0.89 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4 \left (2661120 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )+\cot (c+d x) \csc ^{10}(c+d x) (2457378 \sin (c+d x)+5907132 \sin (3 (c+d x))+656964 \sin (5 (c+d x))-121275 \sin (7 (c+d x))+10395 \sin (9 (c+d x))-5752832 \cos (2 (c+d x))+346112 \cos (4 (c+d x))+583168 \cos (6 (c+d x))-104448 \cos (8 (c+d x))+8704 \cos (10 (c+d x))-5402624)\right )}{113541120 a^2 d (\sin (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(2661120*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + Cot[c + d*
x]*Csc[c + d*x]^10*(-5402624 - 5752832*Cos[2*(c + d*x)] + 346112*Cos[4*(c + d*x)] + 583168*Cos[6*(c + d*x)] -
104448*Cos[8*(c + d*x)] + 8704*Cos[10*(c + d*x)] + 2457378*Sin[c + d*x] + 5907132*Sin[3*(c + d*x)] + 656964*Si
n[5*(c + d*x)] - 121275*Sin[7*(c + d*x)] + 10395*Sin[9*(c + d*x)])))/(113541120*a^2*d*(1 + Sin[c + d*x])^2)

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Maple [B]  time = 0.239, size = 436, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*csc(d*x+c)^12/(a+a*sin(d*x+c))^2,x)

[Out]

1/22528/d/a^2*tan(1/2*d*x+1/2*c)^11-1/5120/d/a^2*tan(1/2*d*x+1/2*c)^10+7/18432/d/a^2*tan(1/2*d*x+1/2*c)^9-1/20
48/d/a^2*tan(1/2*d*x+1/2*c)^8+3/14336/d/a^2*tan(1/2*d*x+1/2*c)^7+1/1024/d/a^2*tan(1/2*d*x+1/2*c)^6-27/10240/d/
a^2*tan(1/2*d*x+1/2*c)^5+1/256/d/a^2*tan(1/2*d*x+1/2*c)^4-11/3072/d/a^2*tan(1/2*d*x+1/2*c)^3-1/512/d/a^2*tan(1
/2*d*x+1/2*c)^2+19/1024/d/a^2*tan(1/2*d*x+1/2*c)+1/5120/d/a^2/tan(1/2*d*x+1/2*c)^10-3/14336/d/a^2/tan(1/2*d*x+
1/2*c)^7-19/1024/d/a^2/tan(1/2*d*x+1/2*c)-1/22528/d/a^2/tan(1/2*d*x+1/2*c)^11+1/2048/d/a^2/tan(1/2*d*x+1/2*c)^
8+27/10240/d/a^2/tan(1/2*d*x+1/2*c)^5-1/256/d/a^2/tan(1/2*d*x+1/2*c)^4-7/18432/d/a^2/tan(1/2*d*x+1/2*c)^9-3/12
8/d/a^2*ln(tan(1/2*d*x+1/2*c))-1/1024/d/a^2/tan(1/2*d*x+1/2*c)^6+11/3072/d/a^2/tan(1/2*d*x+1/2*c)^3+1/512/d/a^
2/tan(1/2*d*x+1/2*c)^2

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Maxima [B]  time = 1.04086, size = 640, normalized size = 3.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^12/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/7096320*((131670*sin(d*x + c)/(cos(d*x + c) + 1) - 13860*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 25410*sin(d*x
+ c)^3/(cos(d*x + c) + 1)^3 + 27720*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 18711*sin(d*x + c)^5/(cos(d*x + c)
+ 1)^5 + 6930*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1485*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 3465*sin(d*x +
c)^8/(cos(d*x + c) + 1)^8 + 2695*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 1386*sin(d*x + c)^10/(cos(d*x + c) + 1)
^10 + 315*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/a^2 - 166320*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 + (1386
*sin(d*x + c)/(cos(d*x + c) + 1) - 2695*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3465*sin(d*x + c)^3/(cos(d*x + c
) + 1)^3 - 1485*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6930*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 18711*sin(d*x
+ c)^6/(cos(d*x + c) + 1)^6 - 27720*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 25410*sin(d*x + c)^8/(cos(d*x + c)
+ 1)^8 + 13860*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 131670*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 315)*(cos(
d*x + c) + 1)^11/(a^2*sin(d*x + c)^11))/d

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Fricas [A]  time = 1.27652, size = 910, normalized size = 4.33 \begin{align*} -\frac{34816 \, \cos \left (d x + c\right )^{11} - 191488 \, \cos \left (d x + c\right )^{9} + 430848 \, \cos \left (d x + c\right )^{7} - 354816 \, \cos \left (d x + c\right )^{5} - 10395 \,{\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 10395 \,{\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 1386 \,{\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \,{\left (a^{2} d \cos \left (d x + c\right )^{10} - 5 \, a^{2} d \cos \left (d x + c\right )^{8} + 10 \, a^{2} d \cos \left (d x + c\right )^{6} - 10 \, a^{2} d \cos \left (d x + c\right )^{4} + 5 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^12/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/887040*(34816*cos(d*x + c)^11 - 191488*cos(d*x + c)^9 + 430848*cos(d*x + c)^7 - 354816*cos(d*x + c)^5 - 103
95*(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1)*log(1/2
*cos(d*x + c) + 1/2)*sin(d*x + c) + 10395*(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x
+ c)^4 + 5*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 1386*(15*cos(d*x + c)^9 - 70*cos(d
*x + c)^7 + 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos(d*x + c))*sin(d*x + c))/((a^2*d*cos(d*x + c)^10 -
5*a^2*d*cos(d*x + c)^8 + 10*a^2*d*cos(d*x + c)^6 - 10*a^2*d*cos(d*x + c)^4 + 5*a^2*d*cos(d*x + c)^2 - a^2*d)*s
in(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**12/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.36872, size = 487, normalized size = 2.32 \begin{align*} -\frac{\frac{166320 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{502266 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 131670 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 13860 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 25410 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 27720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 18711 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 6930 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1485 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 3465 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2695 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1386 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 315}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11}} - \frac{315 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 1386 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 2695 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 3465 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 1485 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 6930 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 18711 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 27720 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 25410 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 13860 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 131670 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{22}}}{7096320 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^12/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/7096320*(166320*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (502266*tan(1/2*d*x + 1/2*c)^11 - 131670*tan(1/2*d*x +
1/2*c)^10 + 13860*tan(1/2*d*x + 1/2*c)^9 + 25410*tan(1/2*d*x + 1/2*c)^8 - 27720*tan(1/2*d*x + 1/2*c)^7 + 1871
1*tan(1/2*d*x + 1/2*c)^6 - 6930*tan(1/2*d*x + 1/2*c)^5 - 1485*tan(1/2*d*x + 1/2*c)^4 + 3465*tan(1/2*d*x + 1/2*
c)^3 - 2695*tan(1/2*d*x + 1/2*c)^2 + 1386*tan(1/2*d*x + 1/2*c) - 315)/(a^2*tan(1/2*d*x + 1/2*c)^11) - (315*a^2
0*tan(1/2*d*x + 1/2*c)^11 - 1386*a^20*tan(1/2*d*x + 1/2*c)^10 + 2695*a^20*tan(1/2*d*x + 1/2*c)^9 - 3465*a^20*t
an(1/2*d*x + 1/2*c)^8 + 1485*a^20*tan(1/2*d*x + 1/2*c)^7 + 6930*a^20*tan(1/2*d*x + 1/2*c)^6 - 18711*a^20*tan(1
/2*d*x + 1/2*c)^5 + 27720*a^20*tan(1/2*d*x + 1/2*c)^4 - 25410*a^20*tan(1/2*d*x + 1/2*c)^3 - 13860*a^20*tan(1/2
*d*x + 1/2*c)^2 + 131670*a^20*tan(1/2*d*x + 1/2*c))/a^22)/d