\(\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx\) [583]

Optimal result

Integrand size = 27, antiderivative size = 96 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d} \]

[Out]

5/16*a*arctanh(cos(d*x+c))/d-1/7*a*cot(d*x+c)^7/d-5/16*a*cot(d*x+c)*csc(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 = 96, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2917, 2687, 30, 2691, 3855} \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

(5*a*ArcTanh[Cos[c + d*x]])/(16*d) - (a*Cot[c + d*x]^7)/(7*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 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 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 (c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx \\ & = -\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+\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 (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 \\ & = -\frac {a \cot ^7(c+d x)}{7 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 \\ & = \frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot ^7(c+d x)}{7 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 [A] (verified)

Time = 0.02 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.82 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^7(c+d x)}{7 d}-\frac {11 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {11 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

-1/7*(a*Cot[c + d*x]^7)/d - (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) + (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 = 0.38 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}}{d}\)	\(128\)
default	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}}{d}\)	\(128\)
parallelrisch	\(-\frac {5 \left (512 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {3 \cos \left (3 d x +3 c \right )}{5}+\frac {\cos \left (5 d x +5 c \right )}{5}+\frac {\cos \left (7 d x +7 c \right )}{35}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+3 \cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{6}+\frac {11 \cos \left (5 d x +5 c \right )}{10}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a}{8192 d}\)	\(133\)
risch	\(\frac {a \left (336 i {\mathrm e}^{12 i \left (d x +c \right )}+231 \,{\mathrm e}^{13 i \left (d x +c \right )}-196 \,{\mathrm e}^{11 i \left (d x +c \right )}+1680 i {\mathrm e}^{8 i \left (d x +c \right )}+595 \,{\mathrm e}^{9 i \left (d x +c \right )}+1008 i {\mathrm e}^{4 i \left (d x +c \right )}-595 \,{\mathrm e}^{5 i \left (d x +c \right )}+196 \,{\mathrm e}^{3 i \left (d x +c \right )}+48 i-231 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{168 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}\)	\(162\)
norman	\(\frac {-\frac {a}{896 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d}+\frac {3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{448 d}+\frac {a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}-\frac {a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {3 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {3 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}-\frac {3 a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{448 d}+\frac {a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {a \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{896 d}-\frac {15 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {5 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\)	\(288\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(a*(-1/6/sin(d*x+c)^6*cos(d*x+c)^7+1/24/sin(d*x+c)^4*cos(d*x+c)^7-1/16/sin(d*x+c)^2*cos(d*x+c)^7-1/16*cos(
d*x+c)^5-5/48*cos(d*x+c)^3-5/16*cos(d*x+c)-5/16*ln(csc(d*x+c)-cot(d*x+c)))-1/7*a/sin(d*x+c)^7*cos(d*x+c)^7)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (86) = 172\).

Time = 0.27 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.19 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {96 \, a \cos \left (d x + c\right )^{7} + 105 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 14 \, {\left (33 \, a \cos \left (d x + c\right )^{5} - 40 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/672*(96*a*cos(d*x + c)^7 + 105*(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)*sin(d*x + c) - 105*(a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^4 + 3*a*cos(d*x + c)^2 - a)*log(-1/2*c
os(d*x + c) + 1/2)*sin(d*x + c) + 14*(33*a*cos(d*x + c)^5 - 40*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)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**8*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.10 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {7 \, a {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {96 \, a}{\tan \left (d x + c\right )^{7}}}{672 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/672*(7*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)) - 96*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. 228 vs. \(2 (86) = 172\).

Time = 0.37 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.38 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 840 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {2178 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{2688 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/2688*(3*a*tan(1/2*d*x + 1/2*c)^7 + 7*a*tan(1/2*d*x + 1/2*c)^6 - 21*a*tan(1/2*d*x + 1/2*c)^5 - 63*a*tan(1/2*d
*x + 1/2*c)^4 + 63*a*tan(1/2*d*x + 1/2*c)^3 + 315*a*tan(1/2*d*x + 1/2*c)^2 - 840*a*log(abs(tan(1/2*d*x + 1/2*c
))) - 105*a*tan(1/2*d*x + 1/2*c) + (2178*a*tan(1/2*d*x + 1/2*c)^7 + 105*a*tan(1/2*d*x + 1/2*c)^6 - 315*a*tan(1
/2*d*x + 1/2*c)^5 - 63*a*tan(1/2*d*x + 1/2*c)^4 + 63*a*tan(1/2*d*x + 1/2*c)^3 + 21*a*tan(1/2*d*x + 1/2*c)^2 -
7*a*tan(1/2*d*x + 1/2*c) - 3*a)/tan(1/2*d*x + 1/2*c)^7)/d

Mupad [B] (verification not implemented)

Time = 11.11 (sec) , antiderivative size = 385, normalized size of antiderivative = 4.01 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a\,\left (3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}{2688\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x)))/sin(c + d*x)^8,x)

[Out]

-(a*(3*cos(c/2 + (d*x)/2)^14 - 3*sin(c/2 + (d*x)/2)^14 - 7*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^13 + 7*cos(c/
2 + (d*x)/2)^13*sin(c/2 + (d*x)/2) + 21*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 + 63*cos(c/2 + (d*x)/2)^3*s
in(c/2 + (d*x)/2)^11 - 63*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 - 315*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x
)/2)^9 + 105*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 - 105*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 315*c
os(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 + 63*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 - 63*cos(c/2 + (d*x)/
2)^11*sin(c/2 + (d*x)/2)^3 - 21*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2 + 840*log(sin(c/2 + (d*x)/2)/cos(c/
2 + (d*x)/2))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7))/(2688*d*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)