\(\int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [746]

Optimal result

Integrand size = 29, antiderivative size = 97 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {x}{a^3}+\frac {13 \text {arctanh}(\cos (c+d x))}{8 a^3 d}+\frac {\cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d} \]

[Out]

x/a^3+13/8*arctanh(cos(d*x+c))/a^3/d+cot(d*x+c)/a^3/d+cot(d*x+c)^3/a^3/d-11/8*cot(d*x+c)*csc(d*x+c)/a^3/d-1/4*
cot(d*x+c)*csc(d*x+c)^3/a^3/d

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2954, 2952, 3554, 8, 2691, 3855, 2687, 30, 3853} \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13 \text {arctanh}(\cos (c+d x))}{8 a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}+\frac {\cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {x}{a^3} \]

[In]

Int[(Cos[c + d*x]^3*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]

[Out]

x/a^3 + (13*ArcTanh[Cos[c + d*x]])/(8*a^3*d) + Cot[c + d*x]/(a^3*d) + Cot[c + d*x]^3/(a^3*d) - (11*Cot[c + d*x
]*Csc[c + d*x])/(8*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

Rule 2687

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 2691

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 2952

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 2954

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 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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*} \text {integral}& = \frac {\int \cot ^2(c+d x) \csc ^3(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \cot ^2(c+d x)+3 a^3 \cot ^2(c+d x) \csc (c+d x)-3 a^3 \cot ^2(c+d x) \csc ^2(c+d x)+a^3 \cot ^2(c+d x) \csc ^3(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \cot ^2(c+d x) \, dx}{a^3}+\frac {\int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^2(c+d x) \csc (c+d x) \, dx}{a^3}-\frac {3 \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx}{a^3} \\ & = \frac {\cot (c+d x)}{a^3 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {\int \csc ^3(c+d x) \, dx}{4 a^3}+\frac {\int 1 \, dx}{a^3}-\frac {3 \int \csc (c+d x) \, dx}{2 a^3}-\frac {3 \text {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = \frac {x}{a^3}+\frac {3 \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {\cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {\int \csc (c+d x) \, dx}{8 a^3} \\ & = \frac {x}{a^3}+\frac {13 \text {arctanh}(\cos (c+d x))}{8 a^3 d}+\frac {\cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.50 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.70 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (-22 \csc ^2\left (\frac {1}{2} (c+d x)\right )+22 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+8 \left (8 c+8 d x+13 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-13 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-8 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (-1+4 \sin (c+d x))\right )}{64 a^3 d (1+\sin (c+d x))^3} \]

[In]

Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(-22*Csc[(c + d*x)/2]^2 + 22*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4
+ 8*(8*c + 8*d*x + 13*Log[Cos[(c + d*x)/2]] - 13*Log[Sin[(c + d*x)/2]] - 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4)
+ Csc[(c + d*x)/2]^4*(-1 + 4*Sin[c + d*x])))/(64*a^3*d*(1 + Sin[c + d*x])^3)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.28

method	result	size
parallelrisch	\(\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+64 d x -8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-104 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d \,a^{3}}\)	\(124\)
derivativedivides	\(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-26 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{3}}\)	\(136\)
default	\(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-26 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{3}}\)	\(136\)
risch	\(\frac {x}{a^{3}}+\frac {11 \,{\mathrm e}^{7 i \left (d x +c \right )}-19 \,{\mathrm e}^{5 i \left (d x +c \right )}-16 i {\mathrm e}^{6 i \left (d x +c \right )}-19 \,{\mathrm e}^{3 i \left (d x +c \right )}+11 \,{\mathrm e}^{i \left (d x +c \right )}+16 i {\mathrm e}^{2 i \left (d x +c \right )}}{4 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {13 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{3}}-\frac {13 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{3}}\)	\(137\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/64*(tan(1/2*d*x+1/2*c)^4-cot(1/2*d*x+1/2*c)^4-8*tan(1/2*d*x+1/2*c)^3+8*cot(1/2*d*x+1/2*c)^3+24*tan(1/2*d*x+1
/2*c)^2-24*cot(1/2*d*x+1/2*c)^2+64*d*x-8*tan(1/2*d*x+1/2*c)+8*cot(1/2*d*x+1/2*c)-104*ln(tan(1/2*d*x+1/2*c)))/d
/a^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {16 \, d x \cos \left (d x + c\right )^{4} - 32 \, d x \cos \left (d x + c\right )^{2} + 22 \, \cos \left (d x + c\right )^{3} + 16 \, d x + 13 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 13 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 16 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 26 \, \cos \left (d x + c\right )}{16 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/16*(16*d*x*cos(d*x + c)^4 - 32*d*x*cos(d*x + c)^2 + 22*cos(d*x + c)^3 + 16*d*x + 13*(cos(d*x + c)^4 - 2*cos(
d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) - 13*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c)
+ 1/2) + 16*cos(d*x + c)*sin(d*x + c) - 26*cos(d*x + c))/(a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3
*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**5/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (91) = 182\).

Time = 0.31 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.25 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{3}} - \frac {128 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {104 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a^{3} \sin \left (d x + c\right )^{4}}}{64 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/64*((8*sin(d*x + c)/(cos(d*x + c) + 1) - 24*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 8*sin(d*x + c)^3/(cos(d*x
+ c) + 1)^3 - sin(d*x + c)^4/(cos(d*x + c) + 1)^4)/a^3 - 128*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 + 10
4*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - (8*sin(d*x + c)/(cos(d*x + c) + 1) - 24*sin(d*x + c)^2/(cos(d*x +
c) + 1)^2 + 8*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1)*(cos(d*x + c) + 1)^4/(a^3*sin(d*x + c)^4))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.71 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {192 \, {\left (d x + c\right )}}{a^{3}} - \frac {312 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {3 \, {\left (a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{12}}}{192 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/192*(192*(d*x + c)/a^3 - 312*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 + (650*tan(1/2*d*x + 1/2*c)^4 + 24*tan(1/2*d
*x + 1/2*c)^3 - 72*tan(1/2*d*x + 1/2*c)^2 + 24*tan(1/2*d*x + 1/2*c) - 3)/(a^3*tan(1/2*d*x + 1/2*c)^4) + 3*(a^9
*tan(1/2*d*x + 1/2*c)^4 - 8*a^9*tan(1/2*d*x + 1/2*c)^3 + 24*a^9*tan(1/2*d*x + 1/2*c)^2 - 8*a^9*tan(1/2*d*x + 1
/2*c))/a^12)/d

Mupad [B] (verification not implemented)

Time = 10.77 (sec) , antiderivative size = 315, normalized size of antiderivative = 3.25 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+128\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-13\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{13\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+104\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^5*(a + a*sin(c + d*x))^3),x)

[Out]

-(cos(c/2 + (d*x)/2)^8 - sin(c/2 + (d*x)/2)^8 + 8*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^7 - 8*cos(c/2 + (d*x)/
2)^7*sin(c/2 + (d*x)/2) - 24*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^6 + 8*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x
)/2)^5 - 8*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^3 + 24*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^2 + 128*atan
((8*cos(c/2 + (d*x)/2) - 13*sin(c/2 + (d*x)/2))/(13*cos(c/2 + (d*x)/2) + 8*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x
)/2)^4*sin(c/2 + (d*x)/2)^4 + 104*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d
*x)/2)^4)/(64*a^3*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^4)