\(\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\) [599]

Optimal result

Integrand size = 29, antiderivative size = 162 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-a^2 x+\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d} \]

[Out]

-a^2*x+5/8*a^2*arctanh(cos(d*x+c))/d-a^2*cot(d*x+c)/d+1/3*a^2*cot(d*x+c)^3/d-1/5*a^2*cot(d*x+c)^5/d-1/7*a^2*co
t(d*x+c)^7/d-5/8*a^2*cot(d*x+c)*csc(d*x+c)/d+5/12*a^2*cot(d*x+c)^3*csc(d*x+c)/d-1/3*a^2*cot(d*x+c)^5*csc(d*x+c
)/d

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 3554, 8, 2691, 3855, 2687, 30} \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-a^2 x \]

[In]

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

[Out]

-(a^2*x) + (5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[
c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (5*a^2*Cot[c + d*x]
^3*Csc[c + d*x])/(12*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x])/(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 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}& = \int \left (a^2 \cot ^6(c+d x)+2 a^2 \cot ^6(c+d x) \csc (c+d x)+a^2 \cot ^6(c+d x) \csc ^2(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^6(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc (c+d x) \, dx \\ & = -\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}-a^2 \int \cot ^4(c+d x) \, dx-\frac {1}{3} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac {a^2 \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}+a^2 \int \cot ^2(c+d x) \, dx+\frac {1}{4} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx \\ & = -\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}-\frac {1}{8} \left (5 a^2\right ) \int \csc (c+d x) \, dx-a^2 \int 1 \, dx \\ & = -a^2 x+\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.62 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \left (-13440 c-13440 d x-9344 \cot \left (\frac {1}{2} (c+d x)\right )-4620 \csc ^2\left (\frac {1}{2} (c+d x)\right )+8400 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8400 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4620 \sec ^2\left (\frac {1}{2} (c+d x)\right )-840 \sec ^4\left (\frac {1}{2} (c+d x)\right )+70 \sec ^6\left (\frac {1}{2} (c+d x)\right )-4624 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-\frac {15}{2} \csc ^8\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+\csc ^6\left (\frac {1}{2} (c+d x)\right ) (-70+33 \sin (c+d x))+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (840+289 \sin (c+d x))+9344 \tan \left (\frac {1}{2} (c+d x)\right )-66 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+15 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{13440 d} \]

[In]

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

[Out]

(a^2*(-13440*c - 13440*d*x - 9344*Cot[(c + d*x)/2] - 4620*Csc[(c + d*x)/2]^2 + 8400*Log[Cos[(c + d*x)/2]] - 84
00*Log[Sin[(c + d*x)/2]] + 4620*Sec[(c + d*x)/2]^2 - 840*Sec[(c + d*x)/2]^4 + 70*Sec[(c + d*x)/2]^6 - 4624*Csc
[c + d*x]^3*Sin[(c + d*x)/2]^4 - (15*Csc[(c + d*x)/2]^8*Sin[c + d*x])/2 + Csc[(c + d*x)/2]^6*(-70 + 33*Sin[c +
d*x]) + Csc[(c + d*x)/2]^4*(840 + 289*Sin[c + d*x]) + 9344*Tan[(c + d*x)/2] - 66*Sec[(c + d*x)/2]^4*Tan[(c +
d*x)/2] + 15*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(13440*d)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.88

method	result	size
parallelrisch	\(-\frac {5 \left (512 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )-\frac {13 \cos \left (3 d x +3 c \right )}{25}+\frac {43 \cos \left (5 d x +5 c \right )}{75}-\frac {73 \cos \left (7 d x +7 c \right )}{525}\right ) \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{18}+\frac {11 \cos \left (5 d x +5 c \right )}{30}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4096 d x}{5}\right ) a^{2}}{4096 d}\)	\(142\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}}{d}\)	\(173\)
default	\(\frac {a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}}{d}\)	\(173\)
risch	\(-a^{2} x +\frac {a^{2} \left (-1680 i {\mathrm e}^{12 i \left (d x +c \right )}+1155 \,{\mathrm e}^{13 i \left (d x +c \right )}+10080 i {\mathrm e}^{10 i \left (d x +c \right )}-980 \,{\mathrm e}^{11 i \left (d x +c \right )}-16240 i {\mathrm e}^{8 i \left (d x +c \right )}+2975 \,{\mathrm e}^{9 i \left (d x +c \right )}+24640 i {\mathrm e}^{6 i \left (d x +c \right )}-14448 i {\mathrm e}^{4 i \left (d x +c \right )}-2975 \,{\mathrm e}^{5 i \left (d x +c \right )}+6496 i {\mathrm e}^{2 i \left (d x +c \right )}+980 \,{\mathrm e}^{3 i \left (d x +c \right )}-1168 i-1155 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{420 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\)	\(210\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-5/4096*(512*ln(tan(1/2*d*x+1/2*c))+sec(1/2*d*x+1/2*c)^7*(cos(d*x+c)-13/25*cos(3*d*x+3*c)+43/75*cos(5*d*x+5*c)
-73/525*cos(7*d*x+7*c))*csc(1/2*d*x+1/2*c)^7+3*(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+4096/5*d*x)*a^2/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (148) = 296\).

Time = 0.29 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.99 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {2336 \, a^{2} \cos \left (d x + c\right )^{7} - 6496 \, a^{2} \cos \left (d x + c\right )^{5} + 5600 \, a^{2} \cos \left (d x + c\right )^{3} - 1680 \, a^{2} \cos \left (d x + c\right ) - 525 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 525 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (24 \, a^{2} d x \cos \left (d x + c\right )^{6} - 72 \, a^{2} d x \cos \left (d x + c\right )^{4} - 33 \, a^{2} \cos \left (d x + c\right )^{5} + 72 \, a^{2} d x \cos \left (d x + c\right )^{2} + 40 \, a^{2} \cos \left (d x + c\right )^{3} - 24 \, a^{2} d x - 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/1680*(2336*a^2*cos(d*x + c)^7 - 6496*a^2*cos(d*x + c)^5 + 5600*a^2*cos(d*x + c)^3 - 1680*a^2*cos(d*x + c) -
525*(a^2*cos(d*x + c)^6 - 3*a^2*cos(d*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2)*sin(
d*x + c) + 525*(a^2*cos(d*x + c)^6 - 3*a^2*cos(d*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*x + c)
+ 1/2)*sin(d*x + c) + 70*(24*a^2*d*x*cos(d*x + c)^6 - 72*a^2*d*x*cos(d*x + c)^4 - 33*a^2*cos(d*x + c)^5 + 72*a
^2*d*x*cos(d*x + c)^2 + 40*a^2*cos(d*x + c)^3 - 24*a^2*d*x - 15*a^2*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. \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**8*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.95 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {112 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 35 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {240 \, a^{2}}{\tan \left (d x + c\right )^{7}}}{1680 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/1680*(112*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^2 - 35*a^2*(2*(33*c
os(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)) + 240*a^2/tan(d*x + c)^7)/d

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.67 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 665 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3150 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 13440 \, {\left (d x + c\right )} a^{2} - 8400 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 8715 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {21780 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8715 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3150 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 665 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/13440*(15*a^2*tan(1/2*d*x + 1/2*c)^7 + 70*a^2*tan(1/2*d*x + 1/2*c)^6 - 21*a^2*tan(1/2*d*x + 1/2*c)^5 - 630*a
^2*tan(1/2*d*x + 1/2*c)^4 - 665*a^2*tan(1/2*d*x + 1/2*c)^3 + 3150*a^2*tan(1/2*d*x + 1/2*c)^2 - 13440*(d*x + c)
*a^2 - 8400*a^2*log(abs(tan(1/2*d*x + 1/2*c))) + 8715*a^2*tan(1/2*d*x + 1/2*c) + (21780*a^2*tan(1/2*d*x + 1/2*
c)^7 - 8715*a^2*tan(1/2*d*x + 1/2*c)^6 - 3150*a^2*tan(1/2*d*x + 1/2*c)^5 + 665*a^2*tan(1/2*d*x + 1/2*c)^4 + 63
0*a^2*tan(1/2*d*x + 1/2*c)^3 + 21*a^2*tan(1/2*d*x + 1/2*c)^2 - 70*a^2*tan(1/2*d*x + 1/2*c) - 15*a^2)/tan(1/2*d
*x + 1/2*c)^7)/d

Mupad [B] (verification not implemented)

Time = 11.00 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.17 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {19\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {15\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {15\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}-\frac {19\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {2\,a^2\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {5\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,d}-\frac {83\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {83\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^8,x)

[Out]

(19*a^2*cot(c/2 + (d*x)/2)^3)/(384*d) - (15*a^2*cot(c/2 + (d*x)/2)^2)/(64*d) + (3*a^2*cot(c/2 + (d*x)/2)^4)/(6
4*d) + (a^2*cot(c/2 + (d*x)/2)^5)/(640*d) - (a^2*cot(c/2 + (d*x)/2)^6)/(192*d) - (a^2*cot(c/2 + (d*x)/2)^7)/(8
96*d) + (15*a^2*tan(c/2 + (d*x)/2)^2)/(64*d) - (19*a^2*tan(c/2 + (d*x)/2)^3)/(384*d) - (3*a^2*tan(c/2 + (d*x)/
2)^4)/(64*d) - (a^2*tan(c/2 + (d*x)/2)^5)/(640*d) + (a^2*tan(c/2 + (d*x)/2)^6)/(192*d) + (a^2*tan(c/2 + (d*x)/
2)^7)/(896*d) - (2*a^2*atan((8*cos(c/2 + (d*x)/2) + 5*sin(c/2 + (d*x)/2))/(5*cos(c/2 + (d*x)/2) - 8*sin(c/2 +
(d*x)/2))))/d - (5*a^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(8*d) - (83*a^2*cot(c/2 + (d*x)/2))/(128*d)
+ (83*a^2*tan(c/2 + (d*x)/2))/(128*d)