\(\int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx\) [721]

Optimal result

Integrand size = 29, antiderivative size = 194 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\cos (c+d x))}{256 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d} \]

[Out]

-3/256*arctanh(cos(d*x+c))/a/d-1/7*cot(d*x+c)^7/a/d-2/9*cot(d*x+c)^9/a/d-1/11*cot(d*x+c)^11/a/d-3/256*cot(d*x+
c)*csc(d*x+c)/a/d-1/128*cot(d*x+c)*csc(d*x+c)^3/a/d+1/32*cot(d*x+c)*csc(d*x+c)^5/a/d-1/16*cot(d*x+c)^3*csc(d*x
+c)^5/a/d+1/10*cot(d*x+c)^5*csc(d*x+c)^5/a/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2918, 2687, 276, 2691, 3853, 3855} \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\cos (c+d x))}{256 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d} \]

[In]

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

[Out]

(-3*ArcTanh[Cos[c + d*x]])/(256*a*d) - Cot[c + d*x]^7/(7*a*d) - (2*Cot[c + d*x]^9)/(9*a*d) - Cot[c + d*x]^11/(
11*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(256*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(128*a*d) + (Cot[c + d*x]*Cs
c[c + d*x]^5)/(32*a*d) - (Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*a*d) + (Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*a*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 2918

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 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 ^6(c+d x) \csc ^5(c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \csc ^6(c+d x) \, dx}{a} \\ & = \frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {\int \cot ^4(c+d x) \csc ^5(c+d x) \, dx}{2 a}+\frac {\text {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = -\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac {3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{16 a}+\frac {\text {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = -\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {\int \csc ^5(c+d x) \, dx}{32 a} \\ & = -\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {3 \int \csc ^3(c+d x) \, dx}{128 a} \\ & = -\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac {3 \int \csc (c+d x) \, dx}{256 a} \\ & = -\frac {3 \text {arctanh}(\cos (c+d x))}{256 a d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {2 \cot ^9(c+d x)}{9 a d}-\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{256 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac {\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.48 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-2661120 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (6840320+9973760 \cos (2 (c+d x))+3543040 \cos (4 (c+d x))+343040 \cos (6 (c+d x))-61440 \cos (8 (c+d x))+5120 \cos (10 (c+d x))-3219678 \sin (c+d x)-2608452 \sin (3 (c+d x))-2181564 \sin (5 (c+d x))-121275 \sin (7 (c+d x))+10395 \sin (9 (c+d x)))\right )}{227082240 a d (1+\sin (c+d x))} \]

[In]

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

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-2661120*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d
*x]*Csc[c + d*x]^10*(6840320 + 9973760*Cos[2*(c + d*x)] + 3543040*Cos[4*(c + d*x)] + 343040*Cos[6*(c + d*x)] -
61440*Cos[8*(c + d*x)] + 5120*Cos[10*(c + d*x)] - 3219678*Sin[c + d*x] - 2608452*Sin[3*(c + d*x)] - 2181564*S
in[5*(c + d*x)] - 121275*Sin[7*(c + d*x)] + 10395*Sin[9*(c + d*x)])))/(227082240*a*d*(1 + Sin[c + d*x]))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.62 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.40

method	result	size
risch	\(\frac {10395 \,{\mathrm e}^{21 i \left (d x +c \right )}-110880 \,{\mathrm e}^{19 i \left (d x +c \right )}[phone] i {\mathrm e}^{14 i \left (d x +c \right )}-2302839 \,{\mathrm e}^{17 i \left (d x +c \right )}+4730880 i {\mathrm e}^{16 i \left (d x +c \right )}-4790016 \,{\mathrm e}^{15 i \left (d x +c \right )}[phone] i {\mathrm e}^{8 i \left (d x +c \right )}-5828130 \,{\mathrm e}^{13 i \left (d x +c \right )}+3041280 i {\mathrm e}^{6 i \left (d x +c \right )}[phone] i {\mathrm e}^{12 i \left (d x +c \right )}+5828130 \,{\mathrm e}^{9 i \left (d x +c \right )}[phone] i {\mathrm e}^{10 i \left (d x +c \right )}+4790016 \,{\mathrm e}^{7 i \left (d x +c \right )}+563200 i {\mathrm e}^{4 i \left (d x +c \right )}+2302839 \,{\mathrm e}^{5 i \left (d x +c \right )}-112640 i {\mathrm e}^{2 i \left (d x +c \right )}+110880 \,{\mathrm e}^{3 i \left (d x +c \right )}+10240 i-10395 \,{\mathrm e}^{i \left (d x +c \right )}}{443520 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d a}\)	\(272\)
derivativedivides	\(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )+\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )-4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}+\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}-\frac {10}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}}{2048 d a}\)	\(302\)
default	\(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )+\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )-4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}+\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}-\frac {10}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}}{2048 d a}\)	\(302\)
parallelrisch	\(\frac {630 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-630 \left (\cot ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1386 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1386 \left (\cot ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-770 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+770 \left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3465 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4950 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4950 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6930 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6930 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6930 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6930 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-27720 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+27720 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+23100 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-23100 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-13860 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+13860 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+166320 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-69300 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+69300 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{14192640 d a}\)	\(304\)

[In]

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

[Out]

1/443520*(10395*exp(21*I*(d*x+c))-110880*exp(19*I*(d*x+c))[phone]*I*exp(14*I*(d*x+c))-2302839*exp(17*I*(d*x+
c))+4730880*I*exp(16*I*(d*x+c))-4790016*exp(15*I*(d*x+c))[phone]*I*exp(8*I*(d*x+c))-5828130*exp(13*I*(d*x+c)
)+3041280*I*exp(6*I*(d*x+c))[phone]*I*exp(12*I*(d*x+c))+5828130*exp(9*I*(d*x+c))[phone]*I*exp(10*I*(d*x+c)
)+4790016*exp(7*I*(d*x+c))+563200*I*exp(4*I*(d*x+c))+2302839*exp(5*I*(d*x+c))-112640*I*exp(2*I*(d*x+c))+110880
*exp(3*I*(d*x+c))+10240*I-10395*exp(I*(d*x+c)))/d/a/(exp(2*I*(d*x+c))-1)^11-3/256/d/a*ln(exp(I*(d*x+c))+1)+3/2
56/d/a*ln(exp(I*(d*x+c))-1)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.56 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {20480 \, \cos \left (d x + c\right )^{11} - 112640 \, \cos \left (d x + c\right )^{9} + 253440 \, \cos \left (d x + c\right )^{7} - 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 1386 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1774080 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

1/1774080*(20480*cos(d*x + c)^11 - 112640*cos(d*x + c)^9 + 253440*cos(d*x + c)^7 - 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) + 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*d*cos(d*x + c)^10 - 5*a*d*cos(d*x + c)^8 + 10*
a*d*cos(d*x + c)^6 - 10*a*d*cos(d*x + c)^4 + 5*a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (176) = 352\).

Time = 0.22 (sec) , antiderivative size = 475, normalized size of antiderivative = 2.45 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {69300 \, \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 {23100 \, \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 {6930 \, \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 {4950 \, \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 {770 \, \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 {630 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a} - \frac {166320 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {1386 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {770 \, \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 {4950 \, \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 {6930 \, \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 {23100 \, \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 {69300 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - 630\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{11}}{a \sin \left (d x + c\right )^{11}}}{14192640 \, d} \]

[In]

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

[Out]

-1/14192640*((69300*sin(d*x + c)/(cos(d*x + c) + 1) + 13860*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 23100*sin(d*
x + c)^3/(cos(d*x + c) + 1)^3 + 27720*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6930*sin(d*x + c)^5/(cos(d*x + c)
+ 1)^5 - 6930*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 4950*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 3465*sin(d*x +
c)^8/(cos(d*x + c) + 1)^8 + 770*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 1386*sin(d*x + c)^10/(cos(d*x + c) + 1)^
10 - 630*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/a - 166320*log(sin(d*x + c)/(cos(d*x + c) + 1))/a - (1386*sin(
d*x + c)/(cos(d*x + c) + 1) + 770*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 3465*sin(d*x + c)^3/(cos(d*x + c) + 1)
^3 + 4950*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6930*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 6930*sin(d*x + c)^6
/(cos(d*x + c) + 1)^6 + 27720*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 23100*sin(d*x + c)^8/(cos(d*x + c) + 1)^8
+ 13860*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 69300*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 630)*(cos(d*x + c)
+ 1)^11/(a*sin(d*x + c)^11))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (176) = 352\).

Time = 0.37 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {166320 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {630 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1386 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 770 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3465 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 4950 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6930 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27720 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 23100 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 13860 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 69300 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{11}} - \frac {502266 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 69300 \, \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} + 23100 \, \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} + 6930 \, \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} - 4950 \, \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} - 770 \, \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 ) + 630}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{14192640 \, d} \]

[In]

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

[Out]

1/14192640*(166320*log(abs(tan(1/2*d*x + 1/2*c)))/a + (630*a^10*tan(1/2*d*x + 1/2*c)^11 - 1386*a^10*tan(1/2*d*
x + 1/2*c)^10 - 770*a^10*tan(1/2*d*x + 1/2*c)^9 + 3465*a^10*tan(1/2*d*x + 1/2*c)^8 - 4950*a^10*tan(1/2*d*x + 1
/2*c)^7 + 6930*a^10*tan(1/2*d*x + 1/2*c)^6 + 6930*a^10*tan(1/2*d*x + 1/2*c)^5 - 27720*a^10*tan(1/2*d*x + 1/2*c
)^4 + 23100*a^10*tan(1/2*d*x + 1/2*c)^3 - 13860*a^10*tan(1/2*d*x + 1/2*c)^2 - 69300*a^10*tan(1/2*d*x + 1/2*c))
/a^11 - (502266*tan(1/2*d*x + 1/2*c)^11 - 69300*tan(1/2*d*x + 1/2*c)^10 - 13860*tan(1/2*d*x + 1/2*c)^9 + 23100
*tan(1/2*d*x + 1/2*c)^8 - 27720*tan(1/2*d*x + 1/2*c)^7 + 6930*tan(1/2*d*x + 1/2*c)^6 + 6930*tan(1/2*d*x + 1/2*
c)^5 - 4950*tan(1/2*d*x + 1/2*c)^4 + 3465*tan(1/2*d*x + 1/2*c)^3 - 770*tan(1/2*d*x + 1/2*c)^2 - 1386*tan(1/2*d
*x + 1/2*c) + 630)/(a*tan(1/2*d*x + 1/2*c)^11))/d

Mupad [B] (verification not implemented)

Time = 17.10 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.98 \[ \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {630\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}-630\,{\cos \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 )-770\,{\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}-4950\,{\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}+6930\,{\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}+23100\,{\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}-69300\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+69300\,{\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 )}^9-23100\,{\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 )}^7-6930\,{\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+4950\,{\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+770\,{\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}}{14192640\,a\,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))),x)

[Out]

(630*sin(c/2 + (d*x)/2)^22 - 630*cos(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) - 770*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 - 4950*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^18 + 6930*cos(c/2 + (d*x)/2)^5*si
n(c/2 + (d*x)/2)^17 + 6930*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 + 23100*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 - 69300*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^12 + 69300*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^10 + 1
3860*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^9 - 23100*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^8 + 27720*cos
(c/2 + (d*x)/2)^15*sin(c/2 + (d*x)/2)^7 - 6930*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 + 4950*cos(c/2 + (d*x)/2)^18*sin(c/2 + (d*x)/2)^4 - 3465*cos(c/2 + (d*x)/2)^19*s
in(c/2 + (d*x)/2)^3 + 770*cos(c/2 + (d*x)/2)^20*sin(c/2 + (d*x)/2)^2 + 166320*log(sin(c/2 + (d*x)/2)/cos(c/2 +
(d*x)/2))*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^11)/(14192640*a*d*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)
^11)