\(\int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [107]

Optimal result

Integrand size = 21, antiderivative size = 145 \[ \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {13 \cot ^7(c+d x)}{7 a^3 d}+\frac {7 \cot ^9(c+d x)}{3 a^3 d}+\frac {15 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {4 \cot ^{13}(c+d x)}{13 a^3 d}-\frac {\csc ^9(c+d x)}{3 a^3 d}+\frac {7 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {4 \csc ^{13}(c+d x)}{13 a^3 d} \]

[Out]

3/5*cot(d*x+c)^5/a^3/d+13/7*cot(d*x+c)^7/a^3/d+7/3*cot(d*x+c)^9/a^3/d+15/11*cot(d*x+c)^11/a^3/d+4/13*cot(d*x+c
)^13/a^3/d-1/3*csc(d*x+c)^9/a^3/d+7/11*csc(d*x+c)^11/a^3/d-4/13*csc(d*x+c)^13/a^3/d

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2954, 2952, 2687, 276, 2686, 14} \[ \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {4 \cot ^{13}(c+d x)}{13 a^3 d}+\frac {15 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {7 \cot ^9(c+d x)}{3 a^3 d}+\frac {13 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^{13}(c+d x)}{13 a^3 d}+\frac {7 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {\csc ^9(c+d x)}{3 a^3 d} \]

[In]

Int[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]

[Out]

(3*Cot[c + d*x]^5)/(5*a^3*d) + (13*Cot[c + d*x]^7)/(7*a^3*d) + (7*Cot[c + d*x]^9)/(3*a^3*d) + (15*Cot[c + d*x]
^11)/(11*a^3*d) + (4*Cot[c + d*x]^13)/(13*a^3*d) - Csc[c + d*x]^9/(3*a^3*d) + (7*Csc[c + d*x]^11)/(11*a^3*d) -
(4*Csc[c + d*x]^13)/(13*a^3*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 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 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 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot ^3(c+d x) \csc ^5(c+d x)}{(-a-a \cos (c+d x))^3} \, dx \\ & = -\frac {\int (-a+a \cos (c+d x))^3 \cot ^3(c+d x) \csc ^{11}(c+d x) \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \cot ^6(c+d x) \csc ^8(c+d x)+3 a^3 \cot ^5(c+d x) \csc ^9(c+d x)-3 a^3 \cot ^4(c+d x) \csc ^{10}(c+d x)+a^3 \cot ^3(c+d x) \csc ^{11}(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \cot ^6(c+d x) \csc ^8(c+d x) \, dx}{a^3}+\frac {\int \cot ^3(c+d x) \csc ^{11}(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^5(c+d x) \csc ^9(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^4(c+d x) \csc ^{10}(c+d x) \, dx}{a^3} \\ & = -\frac {\text {Subst}\left (\int x^{10} \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int x^6 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^8 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^4 \left (1+x^2\right )^4 \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = -\frac {\text {Subst}\left (\int \left (-x^{10}+x^{12}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int \left (x^6+3 x^8+3 x^{10}+x^{12}\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^8-2 x^{10}+x^{12}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^4+4 x^6+6 x^8+4 x^{10}+x^{12}\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = \frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {13 \cot ^7(c+d x)}{7 a^3 d}+\frac {7 \cot ^9(c+d x)}{3 a^3 d}+\frac {15 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {4 \cot ^{13}(c+d x)}{13 a^3 d}-\frac {\csc ^9(c+d x)}{3 a^3 d}+\frac {7 \csc ^{11}(c+d x)}{11 a^3 d}-\frac {4 \csc ^{13}(c+d x)}{13 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.10 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.83 \[ \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\csc (c) \csc ^7(c+d x) \sec ^3(c+d x) (49201152 \sin (c)-6336512 \sin (d x)-2764580 \sin (c+d x)-1382290 \sin (2 (c+d x))+1275960 \sin (3 (c+d x))+1336720 \sin (4 (c+d x))-60760 \sin (5 (c+d x))-524055 \sin (6 (c+d x))-167090 \sin (7 (c+d x))+60760 \sin (8 (c+d x))+45570 \sin (9 (c+d x))+7595 \sin (10 (c+d x))[phone] \sin (2 c+d x)-23668736 \sin (c+2 d x)[phone] \sin (3 c+2 d x)-6537216 \sin (2 c+3 d x)-6848512 \sin (3 c+4 d x)+311296 \sin (4 c+5 d x)+2684928 \sin (5 c+6 d x)+856064 \sin (6 c+7 d x)-311296 \sin (7 c+8 d x)-233472 \sin (8 c+9 d x)-38912 \sin (9 c+10 d x))}{984023040 a^3 d (1+\sec (c+d x))^3} \]

[In]

Integrate[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]

[Out]

-1/984023040*(Csc[c]*Csc[c + d*x]^7*Sec[c + d*x]^3*(49201152*Sin[c] - 6336512*Sin[d*x] - 2764580*Sin[c + d*x]
- 1382290*Sin[2*(c + d*x)] + 1275960*Sin[3*(c + d*x)] + 1336720*Sin[4*(c + d*x)] - 60760*Sin[5*(c + d*x)] - 52
4055*Sin[6*(c + d*x)] - 167090*Sin[7*(c + d*x)] + 60760*Sin[8*(c + d*x)] + 45570*Sin[9*(c + d*x)] + 7595*Sin[1
0*(c + d*x)] + 20500480*Sin[2*c + d*x] - 23668736*Sin[c + 2*d*x] + 30750720*Sin[3*c + 2*d*x] - 6537216*Sin[2*c
+ 3*d*x] - 6848512*Sin[3*c + 4*d*x] + 311296*Sin[4*c + 5*d*x] + 2684928*Sin[5*c + 6*d*x] + 856064*Sin[6*c + 7
*d*x] - 311296*Sin[7*c + 8*d*x] - 233472*Sin[8*c + 9*d*x] - 38912*Sin[9*c + 10*d*x]))/(a^3*d*(1 + Sec[c + d*x]
)^3)

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94

method	result	size
parallelrisch	\(\frac {-1155 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}-5460 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-5005 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+17160 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+42042 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-210210 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2145 \cot \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {28 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5}+7 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-56\right )}{15375360 a^{3} d}\)	\(136\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{13}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{1024 d \,a^{3}}\)	\(138\)
default	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{13}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{1024 d \,a^{3}}\)	\(138\)
risch	\(\frac {32 i \left (15015 \,{\mathrm e}^{12 i \left (d x +c \right )}+10010 \,{\mathrm e}^{11 i \left (d x +c \right )}+24024 \,{\mathrm e}^{10 i \left (d x +c \right )}+3094 \,{\mathrm e}^{9 i \left (d x +c \right )}+11557 \,{\mathrm e}^{8 i \left (d x +c \right )}+3192 \,{\mathrm e}^{7 i \left (d x +c \right )}+3344 \,{\mathrm e}^{6 i \left (d x +c \right )}-152 \,{\mathrm e}^{5 i \left (d x +c \right )}-1311 \,{\mathrm e}^{4 i \left (d x +c \right )}-418 \,{\mathrm e}^{3 i \left (d x +c \right )}+152 \,{\mathrm e}^{2 i \left (d x +c \right )}+114 \,{\mathrm e}^{i \left (d x +c \right )}+19\right )}{15015 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{13} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{7}}\)	\(170\)

[In]

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

[Out]

1/15375360*(-1155*tan(1/2*d*x+1/2*c)^13-5460*tan(1/2*d*x+1/2*c)^11-5005*tan(1/2*d*x+1/2*c)^9+17160*tan(1/2*d*x
+1/2*c)^7+42042*tan(1/2*d*x+1/2*c)^5-210210*tan(1/2*d*x+1/2*c)-2145*cot(1/2*d*x+1/2*c)*(cot(1/2*d*x+1/2*c)^6+2
8/5*cot(1/2*d*x+1/2*c)^4+7*cot(1/2*d*x+1/2*c)^2-56))/a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.48 \[ \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {304 \, \cos \left (d x + c\right )^{10} + 912 \, \cos \left (d x + c\right )^{9} - 152 \, \cos \left (d x + c\right )^{8} - 2888 \, \cos \left (d x + c\right )^{7} - 1862 \, \cos \left (d x + c\right )^{6} + 2926 \, \cos \left (d x + c\right )^{5} + 3325 \, \cos \left (d x + c\right )^{4} - 665 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )^{2} + 210 \, \cos \left (d x + c\right ) + 70}{15015 \, {\left (a^{3} d \cos \left (d x + c\right )^{9} + 3 \, a^{3} d \cos \left (d x + c\right )^{8} - 8 \, a^{3} d \cos \left (d x + c\right )^{6} - 6 \, a^{3} d \cos \left (d x + c\right )^{5} + 6 \, a^{3} d \cos \left (d x + c\right )^{4} + 8 \, a^{3} d \cos \left (d x + c\right )^{3} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

1/15015*(304*cos(d*x + c)^10 + 912*cos(d*x + c)^9 - 152*cos(d*x + c)^8 - 2888*cos(d*x + c)^7 - 1862*cos(d*x +
c)^6 + 2926*cos(d*x + c)^5 + 3325*cos(d*x + c)^4 - 665*cos(d*x + c)^3 - 35*cos(d*x + c)^2 + 210*cos(d*x + c) +
70)/((a^3*d*cos(d*x + c)^9 + 3*a^3*d*cos(d*x + c)^8 - 8*a^3*d*cos(d*x + c)^6 - 6*a^3*d*cos(d*x + c)^5 + 6*a^3
*d*cos(d*x + c)^4 + 8*a^3*d*cos(d*x + c)^3 - 3*a^3*d*cos(d*x + c) - a^3*d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**8/(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.48 \[ \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {\frac {210210 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {42042 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {17160 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {5005 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {5460 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {1155 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}}}{a^{3}} + \frac {429 \, {\left (\frac {28 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {280 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{3} \sin \left (d x + c\right )^{7}}}{15375360 \, d} \]

[In]

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

[Out]

-1/15375360*((210210*sin(d*x + c)/(cos(d*x + c) + 1) - 42042*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 17160*sin(d
*x + c)^7/(cos(d*x + c) + 1)^7 + 5005*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 5460*sin(d*x + c)^11/(cos(d*x + c)
+ 1)^11 + 1155*sin(d*x + c)^13/(cos(d*x + c) + 1)^13)/a^3 + 429*(28*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 35*
sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 280*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5)*(cos(d*x + c) + 1)^7/(a^3*s
in(d*x + c)^7))/d

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {429 \, {\left (280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 28 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} - \frac {1155 \, a^{36} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5460 \, a^{36} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 5005 \, a^{36} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 17160 \, a^{36} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 42042 \, a^{36} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 210210 \, a^{36} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{39}}}{15375360 \, d} \]

[In]

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

[Out]

1/15375360*(429*(280*tan(1/2*d*x + 1/2*c)^6 - 35*tan(1/2*d*x + 1/2*c)^4 - 28*tan(1/2*d*x + 1/2*c)^2 - 5)/(a^3*
tan(1/2*d*x + 1/2*c)^7) - (1155*a^36*tan(1/2*d*x + 1/2*c)^13 + 5460*a^36*tan(1/2*d*x + 1/2*c)^11 + 5005*a^36*t
an(1/2*d*x + 1/2*c)^9 - 17160*a^36*tan(1/2*d*x + 1/2*c)^7 - 42042*a^36*tan(1/2*d*x + 1/2*c)^5 + 210210*a^36*ta
n(1/2*d*x + 1/2*c))/a^39)/d

Mupad [B] (verification not implemented)

Time = 14.48 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.72 \[ \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {2145\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+12012\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+15015\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-120120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+210210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-42042\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-17160\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+5005\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+5460\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+1155\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}}{15375360\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

[In]

int(1/(sin(c + d*x)^8*(a + a/cos(c + d*x))^3),x)

[Out]

-(2145*cos(c/2 + (d*x)/2)^20 + 1155*sin(c/2 + (d*x)/2)^20 + 5460*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^18 +
5005*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^16 - 17160*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^14 - 42042*cos
(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^12 + 210210*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^8 - 120120*cos(c/2 +
(d*x)/2)^14*sin(c/2 + (d*x)/2)^6 + 15015*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^4 + 12012*cos(c/2 + (d*x)/2
)^18*sin(c/2 + (d*x)/2)^2)/(15375360*a^3*d*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^7)