\(\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx\) [582]

Optimal result

Integrand size = 25, antiderivative size = 128 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-a x+\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot (c+d x)}{d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d} \]

[Out]

-a*x+5/16*a*arctanh(cos(d*x+c))/d-a*cot(d*x+c)/d+1/3*a*cot(d*x+c)^3/d-1/5*a*cot(d*x+c)^5/d-5/16*a*cot(d*x+c)*c
sc(d*x+c)/d+5/24*a*cot(d*x+c)^3*csc(d*x+c)/d-1/6*a*cot(d*x+c)^5*csc(d*x+c)/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2917, 2691, 3855, 3554, 8} \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-a x \]

[In]

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

[Out]

-(a*x) + (5*a*ArcTanh[Cos[c + d*x]])/(16*d) - (a*Cot[c + d*x])/d + (a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^
5)/(5*d) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a*Cot[c + d*x]
^5*Csc[c + d*x])/(6*d)

Rule 8

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

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 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 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) \, dx+a \int \cot ^6(c+d x) \csc (c+d x) \, dx \\ & = -\frac {a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {1}{6} (5 a) \int \cot ^4(c+d x) \csc (c+d x) \, dx-a \int \cot ^4(c+d x) \, dx \\ & = \frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {1}{8} (5 a) \int \cot ^2(c+d x) \csc (c+d x) \, dx+a \int \cot ^2(c+d x) \, dx \\ & = -\frac {a \cot (c+d x)}{d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {1}{16} (5 a) \int \csc (c+d x) \, dx-a \int 1 \, dx \\ & = -a x+\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot (c+d x)}{d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.17 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.51 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {11 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {11 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]

[In]

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

[Out]

(-11*a*Csc[(c + d*x)/2]^2)/(64*d) + (a*Csc[(c + d*x)/2]^4)/(32*d) - (a*Csc[(c + d*x)/2]^6)/(384*d) - (a*Cot[c
+ d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) + (5*a*Log[Cos[(c + d*x)/2]])/(16*d) - (5*a*
Log[Sin[(c + d*x)/2]])/(16*d) + (11*a*Sec[(c + d*x)/2]^2)/(64*d) - (a*Sec[(c + d*x)/2]^4)/(32*d) + (a*Sec[(c +
d*x)/2]^6)/(384*d)

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(-\frac {5 \left (\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {3 \left (\cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{18}+\frac {11 \cos \left (5 d x +5 c \right )}{30}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {\left (\sec ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )-\frac {\cos \left (3 d x +3 c \right )}{2}+\frac {23 \cos \left (5 d x +5 c \right )}{50}\right ) \left (\csc ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+\frac {16 d x}{5}\right ) a}{16 d}\)	\(128\)
derivativedivides	\(\frac {a \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(147\)
default	\(\frac {a \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(147\)
risch	\(-a x +\frac {a \left (165 \,{\mathrm e}^{11 i \left (d x +c \right )}+25 \,{\mathrm e}^{9 i \left (d x +c \right )}-720 i {\mathrm e}^{10 i \left (d x +c \right )}+450 \,{\mathrm e}^{7 i \left (d x +c \right )}+2160 i {\mathrm e}^{8 i \left (d x +c \right )}+450 \,{\mathrm e}^{5 i \left (d x +c \right )}-3680 i {\mathrm e}^{6 i \left (d x +c \right )}+25 \,{\mathrm e}^{3 i \left (d x +c \right )}+3360 i {\mathrm e}^{4 i \left (d x +c \right )}+165 \,{\mathrm e}^{i \left (d x +c \right )}-1488 i {\mathrm e}^{2 i \left (d x +c \right )}+368 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}\)	\(190\)
norman	\(\frac {-\frac {a}{384 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d}+\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}+\frac {a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {59 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {59 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {3 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}+\frac {a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {15 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {5 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\)	\(284\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-5/16*(ln(tan(1/2*d*x+1/2*c))+3/512*(cos(d*x+c)+1/18*cos(3*d*x+3*c)+11/30*cos(5*d*x+5*c))*sec(1/2*d*x+1/2*c)^6
*csc(1/2*d*x+1/2*c)^6+1/48*sec(1/2*d*x+1/2*c)^5*(cos(d*x+c)-1/2*cos(3*d*x+3*c)+23/50*cos(5*d*x+5*c))*csc(1/2*d
*x+1/2*c)^5+16/5*d*x)*a/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (116) = 232\).

Time = 0.28 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.98 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {480 \, a d x \cos \left (d x + c\right )^{6} - 1440 \, a d x \cos \left (d x + c\right )^{4} - 330 \, a \cos \left (d x + c\right )^{5} + 1440 \, a d x \cos \left (d x + c\right )^{2} + 400 \, a \cos \left (d x + c\right )^{3} - 480 \, a d x - 150 \, a \cos \left (d x + c\right ) - 75 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 75 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (23 \, a \cos \left (d x + c\right )^{5} - 35 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/480*(480*a*d*x*cos(d*x + c)^6 - 1440*a*d*x*cos(d*x + c)^4 - 330*a*cos(d*x + c)^5 + 1440*a*d*x*cos(d*x + c)^
2 + 400*a*cos(d*x + c)^3 - 480*a*d*x - 150*a*cos(d*x + c) - 75*(a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^4 + 3*a*co
s(d*x + c)^2 - a)*log(1/2*cos(d*x + c) + 1/2) + 75*(a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^4 + 3*a*cos(d*x + c)^2
- a)*log(-1/2*cos(d*x + c) + 1/2) - 32*(23*a*cos(d*x + c)^5 - 35*a*cos(d*x + c)^3 + 15*a*cos(d*x + c))*sin(d*
x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**7*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {32 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a - 5 \, a {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/480*(32*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a - 5*a*(2*(33*cos(d*x
+ c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) + 15*
log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.62 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1920 \, {\left (d x + c\right )} a - 600 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 1320 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {1470 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1320 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1920*(5*a*tan(1/2*d*x + 1/2*c)^6 + 12*a*tan(1/2*d*x + 1/2*c)^5 - 45*a*tan(1/2*d*x + 1/2*c)^4 - 140*a*tan(1/2
*d*x + 1/2*c)^3 + 225*a*tan(1/2*d*x + 1/2*c)^2 - 1920*(d*x + c)*a - 600*a*log(abs(tan(1/2*d*x + 1/2*c))) + 132
0*a*tan(1/2*d*x + 1/2*c) + (1470*a*tan(1/2*d*x + 1/2*c)^6 - 1320*a*tan(1/2*d*x + 1/2*c)^5 - 225*a*tan(1/2*d*x
+ 1/2*c)^4 + 140*a*tan(1/2*d*x + 1/2*c)^3 + 45*a*tan(1/2*d*x + 1/2*c)^2 - 12*a*tan(1/2*d*x + 1/2*c) - 5*a)/tan
(1/2*d*x + 1/2*c)^6)/d

Mupad [B] (verification not implemented)

Time = 11.00 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.23 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {11\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {5\,a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{16\,d}-\frac {11\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {2\,a\,\mathrm {atan}\left (\frac {16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-16\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {15\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {7\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {3\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x)))/sin(c + d*x)^7,x)

[Out]

(11*a*tan(c/2 + (d*x)/2))/(16*d) - (5*a*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(16*d) - (11*a*cot(c/2 + (
d*x)/2))/(16*d) - (2*a*atan((16*cos(c/2 + (d*x)/2) + 5*sin(c/2 + (d*x)/2))/(5*cos(c/2 + (d*x)/2) - 16*sin(c/2
+ (d*x)/2))))/d - (15*a*cot(c/2 + (d*x)/2)^2)/(128*d) + (7*a*cot(c/2 + (d*x)/2)^3)/(96*d) + (3*a*cot(c/2 + (d*
x)/2)^4)/(128*d) - (a*cot(c/2 + (d*x)/2)^5)/(160*d) - (a*cot(c/2 + (d*x)/2)^6)/(384*d) + (15*a*tan(c/2 + (d*x)
/2)^2)/(128*d) - (7*a*tan(c/2 + (d*x)/2)^3)/(96*d) - (3*a*tan(c/2 + (d*x)/2)^4)/(128*d) + (a*tan(c/2 + (d*x)/2
)^5)/(160*d) + (a*tan(c/2 + (d*x)/2)^6)/(384*d)