\(\int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx\) [584]

Optimal result

Integrand size = 27, antiderivative size = 122 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2691, 3853, 3855, 2687, 30} \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]

[Out]

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

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 2917

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

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}& = a \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx \\ & = -\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{8} (5 a) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {a \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {1}{16} (5 a) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx \\ & = -\frac {a \cot ^7(c+d x)}{7 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{64} (5 a) \int \csc ^3(c+d x) \, dx \\ & = -\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{128} (5 a) \int \csc (c+d x) \, dx \\ & = \frac {5 a \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.76 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {15 a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {7 a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}-\frac {a \csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {15 a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {7 a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}+\frac {a \sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]

[Out]

-1/7*(a*Cot[c + d*x]^7)/d + (5*a*Csc[(c + d*x)/2]^2)/(512*d) - (15*a*Csc[(c + d*x)/2]^4)/(1024*d) + (7*a*Csc[(
c + d*x)/2]^6)/(1536*d) - (a*Csc[(c + d*x)/2]^8)/(2048*d) + (5*a*Log[Cos[(c + d*x)/2]])/(128*d) - (5*a*Log[Sin
[(c + d*x)/2]])/(128*d) - (5*a*Sec[(c + d*x)/2]^2)/(512*d) + (15*a*Sec[(c + d*x)/2]^4)/(1024*d) - (7*a*Sec[(c
+ d*x)/2]^6)/(1536*d) + (a*Sec[(c + d*x)/2]^8)/(2048*d)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.18

method	result	size
parallelrisch	\(-\frac {1765 \left (\frac {49152 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{353}+\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {179 \cos \left (3 d x +3 c \right )}{353}+\frac {397 \cos \left (5 d x +5 c \right )}{1765}+\frac {3 \cos \left (7 d x +7 c \right )}{353}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {768 \cos \left (d x +c \right )}{353}+\frac {2304 \cos \left (3 d x +3 c \right )}{1765}+\frac {768 \cos \left (5 d x +5 c \right )}{1765}+\frac {768 \cos \left (7 d x +7 c \right )}{12355}\right ) \left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a}{6291456 d}\)	\(144\)
derivativedivides	\(\frac {-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\)	\(146\)
default	\(\frac {-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\)	\(146\)
risch	\(-\frac {a \left (105 \,{\mathrm e}^{15 i \left (d x +c \right )}+2779 \,{\mathrm e}^{13 i \left (d x +c \right )}-13440 i {\mathrm e}^{10 i \left (d x +c \right )}+6265 \,{\mathrm e}^{11 i \left (d x +c \right )}-2688 i {\mathrm e}^{14 i \left (d x +c \right )}+12355 \,{\mathrm e}^{9 i \left (d x +c \right )}+8064 i {\mathrm e}^{4 i \left (d x +c \right )}+12355 \,{\mathrm e}^{7 i \left (d x +c \right )}+13440 i {\mathrm e}^{8 i \left (d x +c \right )}+6265 \,{\mathrm e}^{5 i \left (d x +c \right )}-384 i {\mathrm e}^{2 i \left (d x +c \right )}+2779 \,{\mathrm e}^{3 i \left (d x +c \right )}-8064 i {\mathrm e}^{6 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}+2688 i {\mathrm e}^{12 i \left (d x +c \right )}+384 i\right )}{1344 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}\)	\(232\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1765/6291456*(49152/353*ln(tan(1/2*d*x+1/2*c))+(sec(1/2*d*x+1/2*c)*(cos(d*x+c)+179/353*cos(3*d*x+3*c)+397/176
5*cos(5*d*x+5*c)+3/353*cos(7*d*x+7*c))*csc(1/2*d*x+1/2*c)+768/353*cos(d*x+c)+2304/1765*cos(3*d*x+3*c)+768/1765
*cos(5*d*x+5*c)+768/12355*cos(7*d*x+7*c))*sec(1/2*d*x+1/2*c)^7*csc(1/2*d*x+1/2*c)^7)*a/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (110) = 220\).

Time = 0.27 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.84 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {768 \, a \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) + 210 \, a \cos \left (d x + c\right )^{7} + 1022 \, a \cos \left (d x + c\right )^{5} - 770 \, a \cos \left (d x + c\right )^{3} + 210 \, a \cos \left (d x + c\right ) - 105 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{5376 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**9*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.03 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {7 \, a {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {768 \, a}{\tan \left (d x + c\right )^{7}}}{5376 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/5376*(7*a*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^8
- 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c)
- 1)) + 768*a/tan(d*x + c)^7)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (110) = 220\).

Time = 0.38 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.10 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 112 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1008 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1680 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1680 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {4566 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1680 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1008 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{43008 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.64 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.34 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {3\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x)))/sin(c + d*x)^9,x)

[Out]

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