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

Optimal. Leaf size=134 \[ \frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \tanh ^{-1}(\cos (c+d x))}{128 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d} \]

[Out]

5/128*arctanh(cos(d*x+c))/a/d+1/7*cot(d*x+c)^7/a/d+5/128*cot(d*x+c)*csc(d*x+c)/a/d-5/64*cot(d*x+c)*csc(d*x+c)^
3/a/d+5/48*cot(d*x+c)^3*csc(d*x+c)^3/a/d-1/8*cot(d*x+c)^5*csc(d*x+c)^3/a/d

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2839, 2611, 3768, 3770, 2607, 30} \[ \frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \tanh ^{-1}(\cos (c+d x))}{128 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*ArcTanh[Cos[c + d*x]])/(128*a*d) + Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(128*a*d) - (5*Co
t[c + d*x]*Csc[c + d*x]^3)/(64*a*d) + (5*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*a*d) - (Cot[c + d*x]^5*Csc[c + d*x
]^3)/(8*a*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^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 {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^6(c+d x) \csc ^2(c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \csc ^3(c+d x) \, dx}{a}\\ &=-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx}{8 a}-\frac {\operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac {5 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{16 a}\\ &=\frac {\cot ^7(c+d x)}{7 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \int \csc ^3(c+d x) \, dx}{64 a}\\ &=\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \int \csc (c+d x) \, dx}{128 a}\\ &=\frac {5 \tanh ^{-1}(\cos (c+d x))}{128 a d}+\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 1.00, size = 291, normalized size = 2.17 \[ \frac {\csc ^8(c+d x) \left (5376 \sin (2 (c+d x))+5376 \sin (4 (c+d x))+2304 \sin (6 (c+d x))+384 \sin (8 (c+d x))-24710 \cos (c+d x)-12530 \cos (3 (c+d x))-5558 \cos (5 (c+d x))-210 \cos (7 (c+d x))-3675 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-5880 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2940 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-840 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3675 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+5880 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2940 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+840 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{344064 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c + d*x]^8*(-24710*Cos[c + d*x] - 12530*Cos[3*(c + d*x)] - 5558*Cos[5*(c + d*x)] - 210*Cos[7*(c + d*x)] +
3675*Log[Cos[(c + d*x)/2]] - 5880*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 2940*Cos[4*(c + d*x)]*Log[Cos[(c +
d*x)/2]] - 840*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 105*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 3675*Log
[Sin[(c + d*x)/2]] + 5880*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 2940*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]]
+ 840*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 105*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 5376*Sin[2*(c + d
*x)] + 5376*Sin[4*(c + d*x)] + 2304*Sin[6*(c + d*x)] + 384*Sin[8*(c + d*x)]))/(344064*a*d)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 216, normalized size = 1.61 \[ \frac {768 \, \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 210 \, \cos \left (d x + c\right )^{7} - 1022 \, \cos \left (d x + c\right )^{5} + 770 \, \cos \left (d x + c\right )^{3} + 105 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 105 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 210 \, \cos \left (d x + c\right )}{5376 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/5376*(768*cos(d*x + c)^7*sin(d*x + c) - 210*cos(d*x + c)^7 - 1022*cos(d*x + c)^5 + 770*cos(d*x + c)^3 + 105*
(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) - 10
5*(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2) -
210*cos(d*x + c))/(a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^6 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^2 +
a*d)

________________________________________________________________________________________

giac [B]  time = 0.26, size = 274, normalized size = 2.04 \[ -\frac {\frac {1680 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {21 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 48 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 112 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 336 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 168 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1008 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 336 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1680 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{8}} - \frac {4566 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1008 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{43008 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/43008*(1680*log(abs(tan(1/2*d*x + 1/2*c)))/a - (21*a^7*tan(1/2*d*x + 1/2*c)^8 - 48*a^7*tan(1/2*d*x + 1/2*c)
^7 - 112*a^7*tan(1/2*d*x + 1/2*c)^6 + 336*a^7*tan(1/2*d*x + 1/2*c)^5 + 168*a^7*tan(1/2*d*x + 1/2*c)^4 - 1008*a
^7*tan(1/2*d*x + 1/2*c)^3 + 336*a^7*tan(1/2*d*x + 1/2*c)^2 + 1680*a^7*tan(1/2*d*x + 1/2*c))/a^8 - (4566*tan(1/
2*d*x + 1/2*c)^8 - 1680*tan(1/2*d*x + 1/2*c)^7 - 336*tan(1/2*d*x + 1/2*c)^6 + 1008*tan(1/2*d*x + 1/2*c)^5 - 16
8*tan(1/2*d*x + 1/2*c)^4 - 336*tan(1/2*d*x + 1/2*c)^3 + 112*tan(1/2*d*x + 1/2*c)^2 + 48*tan(1/2*d*x + 1/2*c) -
21)/(a*tan(1/2*d*x + 1/2*c)^8))/d

________________________________________________________________________________________

maple [B]  time = 0.56, size = 322, normalized size = 2.40 \[ \frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2048 a d}-\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{896 a d}-\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 a d}+\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{256 a d}-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d}+\frac {1}{384 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {5}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d}-\frac {1}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{896 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{2048 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {1}{256 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {3}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c)),x)

[Out]

1/2048/a/d*tan(1/2*d*x+1/2*c)^8-1/896/a/d*tan(1/2*d*x+1/2*c)^7-1/384/a/d*tan(1/2*d*x+1/2*c)^6+1/128/a/d*tan(1/
2*d*x+1/2*c)^5+1/256/a/d*tan(1/2*d*x+1/2*c)^4-3/128/a/d*tan(1/2*d*x+1/2*c)^3+1/128/a/d*tan(1/2*d*x+1/2*c)^2+5/
128/a/d*tan(1/2*d*x+1/2*c)+1/384/a/d/tan(1/2*d*x+1/2*c)^6-5/128/a/d/tan(1/2*d*x+1/2*c)-5/128/a/d*ln(tan(1/2*d*
x+1/2*c))-1/128/a/d/tan(1/2*d*x+1/2*c)^5+1/896/a/d/tan(1/2*d*x+1/2*c)^7-1/128/a/d/tan(1/2*d*x+1/2*c)^2-1/2048/
a/d/tan(1/2*d*x+1/2*c)^8-1/256/a/d/tan(1/2*d*x+1/2*c)^4+3/128/a/d/tan(1/2*d*x+1/2*c)^3

________________________________________________________________________________________

maxima [B]  time = 0.33, size = 354, normalized size = 2.64 \[ \frac {\frac {\frac {1680 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {336 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1008 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {168 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {336 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {112 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {48 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {21 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}{a} - \frac {1680 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {48 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {112 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {336 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {168 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1680 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 21\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{8}}{a \sin \left (d x + c\right )^{8}}}{43008 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/43008*((1680*sin(d*x + c)/(cos(d*x + c) + 1) + 336*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1008*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 + 168*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 336*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 11
2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 48*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 21*sin(d*x + c)^8/(cos(d*x +
c) + 1)^8)/a - 1680*log(sin(d*x + c)/(cos(d*x + c) + 1))/a + (48*sin(d*x + c)/(cos(d*x + c) + 1) + 112*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 - 336*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 168*sin(d*x + c)^4/(cos(d*x + c) + 1)
^4 + 1008*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 336*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 1680*sin(d*x + c)^7/
(cos(d*x + c) + 1)^7 - 21)*(cos(d*x + c) + 1)^8/(a*sin(d*x + c)^8))/d

________________________________________________________________________________________

mupad [B]  time = 11.39, size = 435, normalized size = 3.25 \[ -\frac {21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-21\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+48\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-168\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+168\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1680\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{43008\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^9*(a + a*sin(c + d*x))),x)

[Out]

-(21*cos(c/2 + (d*x)/2)^16 - 21*sin(c/2 + (d*x)/2)^16 + 48*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^15 - 48*cos(c
/2 + (d*x)/2)^15*sin(c/2 + (d*x)/2) + 112*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^14 - 336*cos(c/2 + (d*x)/2)^
3*sin(c/2 + (d*x)/2)^13 - 168*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^12 + 1008*cos(c/2 + (d*x)/2)^5*sin(c/2 +
(d*x)/2)^11 - 336*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 - 1680*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^9
+ 1680*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^7 + 336*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^6 - 1008*cos(
c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^5 + 168*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 + 336*cos(c/2 + (d*x)/
2)^13*sin(c/2 + (d*x)/2)^3 - 112*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^2 + 1680*log(sin(c/2 + (d*x)/2)/cos(
c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8)/(43008*a*d*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^
8)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**9/(a+a*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________