\(\int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [738]

Optimal result

Integrand size = 29, antiderivative size = 210 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{128 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d} \]

[Out]

3/128*arctanh(cos(d*x+c))/a^2/d-2/5*cot(d*x+c)^5/a^2/d-5/7*cot(d*x+c)^7/a^2/d-4/9*cot(d*x+c)^9/a^2/d-1/11*cot(
d*x+c)^11/a^2/d+3/128*cot(d*x+c)*csc(d*x+c)/a^2/d+1/64*cot(d*x+c)*csc(d*x+c)^3/a^2/d+1/80*cot(d*x+c)*csc(d*x+c
)^5/a^2/d-3/40*cot(d*x+c)*csc(d*x+c)^7/a^2/d+1/5*cot(d*x+c)^3*csc(d*x+c)^7/a^2/d

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2954, 2952, 2687, 276, 2691, 3853, 3855} \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{128 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{128 a^2 d} \]

[In]

Int[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

(3*ArcTanh[Cos[c + d*x]])/(128*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - (5*Cot[c + d*x]^7)/(7*a^2*d) - (4*Cot[c
+ d*x]^9)/(9*a^2*d) - Cot[c + d*x]^11/(11*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(128*a^2*d) + (Cot[c + d*x]*
Csc[c + d*x]^3)/(64*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(80*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x]^7)/(40*a^
2*d) + (Cot[c + d*x]^3*Csc[c + d*x]^7)/(5*a^2*d)

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 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 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 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}& = \frac {\int \cot ^4(c+d x) \csc ^8(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \cot ^4(c+d x) \csc ^6(c+d x)-2 a^2 \cot ^4(c+d x) \csc ^7(c+d x)+a^2 \cot ^4(c+d x) \csc ^8(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \cot ^4(c+d x) \csc ^6(c+d x) \, dx}{a^2}+\frac {\int \cot ^4(c+d x) \csc ^8(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx}{a^2} \\ & = \frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}+\frac {3 \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx}{5 a^2}+\frac {\text {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\text {Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^2 d} \\ & = -\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {3 \int \csc ^7(c+d x) \, dx}{40 a^2}+\frac {\text {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\text {Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d} \\ & = -\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {\int \csc ^5(c+d x) \, dx}{16 a^2} \\ & = -\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {3 \int \csc ^3(c+d x) \, dx}{64 a^2} \\ & = -\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {3 \int \csc (c+d x) \, dx}{128 a^2} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{128 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.75 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (2661120 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\cot (c+d x) \csc ^{10}(c+d x) (-5402624-5752832 \cos (2 (c+d x))+346112 \cos (4 (c+d x))+583168 \cos (6 (c+d x))-104448 \cos (8 (c+d x))+8704 \cos (10 (c+d x))+2457378 \sin (c+d x)+5907132 \sin (3 (c+d x))+656964 \sin (5 (c+d x))-121275 \sin (7 (c+d x))+10395 \sin (9 (c+d x)))\right )}{113541120 a^2 d (1+\sin (c+d x))^2} \]

[In]

Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(2661120*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + Cot[c + d*
x]*Csc[c + d*x]^10*(-5402624 - 5752832*Cos[2*(c + d*x)] + 346112*Cos[4*(c + d*x)] + 583168*Cos[6*(c + d*x)] -
104448*Cos[8*(c + d*x)] + 8704*Cos[10*(c + d*x)] + 2457378*Sin[c + d*x] + 5907132*Sin[3*(c + d*x)] + 656964*Si
n[5*(c + d*x)] - 121275*Sin[7*(c + d*x)] + 10395*Sin[9*(c + d*x)])))/(113541120*a^2*d*(1 + Sin[c + d*x])^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.30

method	result	size
risch	\(-\frac {10395 \,{\mathrm e}^{21 i \left (d x +c \right )}-110880 \,{\mathrm e}^{19 i \left (d x +c \right )}-8279040 i {\mathrm e}^{14 i \left (d x +c \right )}+535689 \,{\mathrm e}^{17 i \left (d x +c \right )}+2365440 i {\mathrm e}^{16 i \left (d x +c \right )}+6564096 \,{\mathrm e}^{15 i \left (d x +c \right )}-2534400 i {\mathrm e}^{8 i \left (d x +c \right )}+8364510 \,{\mathrm e}^{13 i \left (d x +c \right )}-506880 i {\mathrm e}^{6 i \left (d x +c \right )}-12536832 i {\mathrm e}^{12 i \left (d x +c \right )}-8364510 \,{\mathrm e}^{9 i \left (d x +c \right )}-20579328 i {\mathrm e}^{10 i \left (d x +c \right )}-6564096 \,{\mathrm e}^{7 i \left (d x +c \right )}+957440 i {\mathrm e}^{4 i \left (d x +c \right )}-535689 \,{\mathrm e}^{5 i \left (d x +c \right )}-191488 i {\mathrm e}^{2 i \left (d x +c \right )}+110880 \,{\mathrm e}^{3 i \left (d x +c \right )}+17408 i-10395 \,{\mathrm e}^{i \left (d x +c \right )}}{221760 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d \,a^{2}}\)	\(272\)
derivativedivides	\(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {27 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {22 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+38 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {38}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {27}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {22}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {3}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}-\frac {7}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}}{2048 d \,a^{2}}\)	\(304\)
default	\(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {27 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {22 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+38 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {38}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {27}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {22}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {3}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}-\frac {7}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}}{2048 d \,a^{2}}\)	\(304\)
parallelrisch	\(\frac {-315 \left (\cot ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1386 \left (\cot ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1386 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2695 \left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2695 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3465 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1485 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1485 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6930 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6930 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18711 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-18711 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-27720 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+27720 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+25410 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-25410 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+13860 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-13860 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-131670 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-166320 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+131670 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{7096320 d \,a^{2}}\)	\(304\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^12/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/221760*(10395*exp(21*I*(d*x+c))-110880*exp(19*I*(d*x+c))-8279040*I*exp(14*I*(d*x+c))+535689*exp(17*I*(d*x+c
))+2365440*I*exp(16*I*(d*x+c))+6564096*exp(15*I*(d*x+c))-2534400*I*exp(8*I*(d*x+c))+8364510*exp(13*I*(d*x+c))-
506880*I*exp(6*I*(d*x+c))-12536832*I*exp(12*I*(d*x+c))-8364510*exp(9*I*(d*x+c))-20579328*I*exp(10*I*(d*x+c))-6
564096*exp(7*I*(d*x+c))+957440*I*exp(4*I*(d*x+c))-535689*exp(5*I*(d*x+c))-191488*I*exp(2*I*(d*x+c))+110880*exp
(3*I*(d*x+c))+17408*I-10395*exp(I*(d*x+c)))/d/a^2/(exp(2*I*(d*x+c))-1)^11-3/128/d/a^2*ln(exp(I*(d*x+c))-1)+3/1
28/d/a^2*ln(exp(I*(d*x+c))+1)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.54 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {34816 \, \cos \left (d x + c\right )^{11} - 191488 \, \cos \left (d x + c\right )^{9} + 430848 \, \cos \left (d x + c\right )^{7} - 354816 \, \cos \left (d x + c\right )^{5} - 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 1386 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, {\left (a^{2} d \cos \left (d x + c\right )^{10} - 5 \, a^{2} d \cos \left (d x + c\right )^{8} + 10 \, a^{2} d \cos \left (d x + c\right )^{6} - 10 \, a^{2} d \cos \left (d x + c\right )^{4} + 5 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^12/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/887040*(34816*cos(d*x + c)^11 - 191488*cos(d*x + c)^9 + 430848*cos(d*x + c)^7 - 354816*cos(d*x + c)^5 - 103
95*(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1)*log(1/2
*cos(d*x + c) + 1/2)*sin(d*x + c) + 10395*(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x
+ c)^4 + 5*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 1386*(15*cos(d*x + c)^9 - 70*cos(d
*x + c)^7 + 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos(d*x + c))*sin(d*x + c))/((a^2*d*cos(d*x + c)^10 -
5*a^2*d*cos(d*x + c)^8 + 10*a^2*d*cos(d*x + c)^6 - 10*a^2*d*cos(d*x + c)^4 + 5*a^2*d*cos(d*x + c)^2 - a^2*d)*s
in(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**12/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (190) = 380\).

Time = 0.22 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.26 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {131670 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {13860 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {25410 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {27720 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {18711 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {6930 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1485 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3465 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {2695 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {1386 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{2}} - \frac {166320 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {1386 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2695 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3465 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1485 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6930 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {18711 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {27720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {25410 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {13860 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {131670 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - 315\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{11}}{a^{2} \sin \left (d x + c\right )^{11}}}{7096320 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^12/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/7096320*((131670*sin(d*x + c)/(cos(d*x + c) + 1) - 13860*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 25410*sin(d*x
+ c)^3/(cos(d*x + c) + 1)^3 + 27720*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 18711*sin(d*x + c)^5/(cos(d*x + c)
+ 1)^5 + 6930*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1485*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 3465*sin(d*x +
c)^8/(cos(d*x + c) + 1)^8 + 2695*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 1386*sin(d*x + c)^10/(cos(d*x + c) + 1)
^10 + 315*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/a^2 - 166320*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 + (1386
*sin(d*x + c)/(cos(d*x + c) + 1) - 2695*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3465*sin(d*x + c)^3/(cos(d*x + c
) + 1)^3 - 1485*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6930*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 18711*sin(d*x
+ c)^6/(cos(d*x + c) + 1)^6 - 27720*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 25410*sin(d*x + c)^8/(cos(d*x + c)
+ 1)^8 + 13860*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 131670*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 315)*(cos(
d*x + c) + 1)^11/(a^2*sin(d*x + c)^11))/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.72 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {166320 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {502266 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 131670 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 25410 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 27720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 18711 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 6930 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1485 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2695 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1386 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 315}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}} - \frac {315 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1386 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 2695 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3465 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1485 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6930 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 18711 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27720 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 25410 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 13860 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 131670 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{22}}}{7096320 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^12/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/7096320*(166320*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (502266*tan(1/2*d*x + 1/2*c)^11 - 131670*tan(1/2*d*x +
1/2*c)^10 + 13860*tan(1/2*d*x + 1/2*c)^9 + 25410*tan(1/2*d*x + 1/2*c)^8 - 27720*tan(1/2*d*x + 1/2*c)^7 + 1871
1*tan(1/2*d*x + 1/2*c)^6 - 6930*tan(1/2*d*x + 1/2*c)^5 - 1485*tan(1/2*d*x + 1/2*c)^4 + 3465*tan(1/2*d*x + 1/2*
c)^3 - 2695*tan(1/2*d*x + 1/2*c)^2 + 1386*tan(1/2*d*x + 1/2*c) - 315)/(a^2*tan(1/2*d*x + 1/2*c)^11) - (315*a^2
0*tan(1/2*d*x + 1/2*c)^11 - 1386*a^20*tan(1/2*d*x + 1/2*c)^10 + 2695*a^20*tan(1/2*d*x + 1/2*c)^9 - 3465*a^20*t
an(1/2*d*x + 1/2*c)^8 + 1485*a^20*tan(1/2*d*x + 1/2*c)^7 + 6930*a^20*tan(1/2*d*x + 1/2*c)^6 - 18711*a^20*tan(1
/2*d*x + 1/2*c)^5 + 27720*a^20*tan(1/2*d*x + 1/2*c)^4 - 25410*a^20*tan(1/2*d*x + 1/2*c)^3 - 13860*a^20*tan(1/2
*d*x + 1/2*c)^2 + 131670*a^20*tan(1/2*d*x + 1/2*c))/a^22)/d

Mupad [B] (verification not implemented)

Time = 17.56 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.76 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}-315\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}+1386\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}-1386\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2695\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+3465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}-1485\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}+18711\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-27720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+25410\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+13860\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-131670\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+131670\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-13860\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-25410\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+27720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-18711\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+1485\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2695\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+166320\,\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 )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{7096320\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^12*(a + a*sin(c + d*x))^2),x)

[Out]

-(315*cos(c/2 + (d*x)/2)^22 - 315*sin(c/2 + (d*x)/2)^22 + 1386*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^21 - 1386
*cos(c/2 + (d*x)/2)^21*sin(c/2 + (d*x)/2) - 2695*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^20 + 3465*cos(c/2 + (
d*x)/2)^3*sin(c/2 + (d*x)/2)^19 - 1485*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^18 - 6930*cos(c/2 + (d*x)/2)^5*
sin(c/2 + (d*x)/2)^17 + 18711*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^16 - 27720*cos(c/2 + (d*x)/2)^7*sin(c/2
+ (d*x)/2)^15 + 25410*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^14 + 13860*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/
2)^13 - 131670*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^12 + 131670*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^1
0 - 13860*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^9 - 25410*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^8 + 2772
0*cos(c/2 + (d*x)/2)^15*sin(c/2 + (d*x)/2)^7 - 18711*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^6 + 6930*cos(c/2
+ (d*x)/2)^17*sin(c/2 + (d*x)/2)^5 + 1485*cos(c/2 + (d*x)/2)^18*sin(c/2 + (d*x)/2)^4 - 3465*cos(c/2 + (d*x)/2
)^19*sin(c/2 + (d*x)/2)^3 + 2695*cos(c/2 + (d*x)/2)^20*sin(c/2 + (d*x)/2)^2 + 166320*log(sin(c/2 + (d*x)/2)/co
s(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^11)/(7096320*a^2*d*cos(c/2 + (d*x)/2)^11*sin(c/2 +
(d*x)/2)^11)