\(\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx\) [619]

Optimal result

Integrand size = 29, antiderivative size = 228 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {33 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}+\frac {33 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \]

[Out]

33/256*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-1/3*a^3*cot(d*x+c)^9/d+33/256*a^3*cot(d*x+c)*csc(d*x+c
)/d-29/128*a^3*cot(d*x+c)*csc(d*x+c)^3/d+5/16*a^3*cot(d*x+c)^3*csc(d*x+c)^3/d-3/8*a^3*cot(d*x+c)^5*csc(d*x+c)^
3/d-1/32*a^3*cot(d*x+c)*csc(d*x+c)^5/d+1/16*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d-1/10*a^3*cot(d*x+c)^5*csc(d*x+c)^5
/d

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2687, 30, 2691, 3853, 3855, 14} \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {33 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {33 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(33*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/(3*d) + (33*a^3*C
ot[c + d*x]*Csc[c + d*x])/(256*d) - (29*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c
+ d*x]^3)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (
a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*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 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 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}& = \int \left (a^3 \cot ^6(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^4(c+d x)+a^3 \cot ^6(c+d x) \csc ^5(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx \\ & = -\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{2} a^3 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx-\frac {1}{8} \left (15 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {1}{16} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {1}{16} \left (15 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{32} a^3 \int \csc ^5(c+d x) \, dx-\frac {1}{64} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}+\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-\frac {29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{128} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{128} \left (15 a^3\right ) \int \csc (c+d x) \, dx \\ & = \frac {15 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}+\frac {33 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{256} \left (3 a^3\right ) \int \csc (c+d x) \, dx \\ & = \frac {33 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}+\frac {33 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.35 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.60 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (51200 \cot \left (\frac {1}{2} (c+d x)\right )+13860 \csc ^2\left (\frac {1}{2} (c+d x)\right )+55440 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-55440 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-13860 \sec ^2\left (\frac {1}{2} (c+d x)\right )+19320 \sec ^4\left (\frac {1}{2} (c+d x)\right )-5250 \sec ^6\left (\frac {1}{2} (c+d x)\right )+315 \sec ^8\left (\frac {1}{2} (c+d x)\right )+42 \sec ^{10}\left (\frac {1}{2} (c+d x)\right )+164800 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+3840 \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+\csc ^6\left (\frac {1}{2} (c+d x)\right ) (5250-60 \sin (c+d x))-14 \csc ^{10}\left (\frac {1}{2} (c+d x)\right ) (3+10 \sin (c+d x))+5 \csc ^8\left (\frac {1}{2} (c+d x)\right ) (-63+172 \sin (c+d x))-20 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (966+515 \sin (c+d x))-51200 \tan \left (\frac {1}{2} (c+d x)\right )-1720 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+280 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{430080 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(1 + Sin[c + d*x])^3*(51200*Cot[(c + d*x)/2] + 13860*Csc[(c + d*x)/2]^2 + 55440*Log[Cos[(c + d*x)/2]] - 5
5440*Log[Sin[(c + d*x)/2]] - 13860*Sec[(c + d*x)/2]^2 + 19320*Sec[(c + d*x)/2]^4 - 5250*Sec[(c + d*x)/2]^6 + 3
15*Sec[(c + d*x)/2]^8 + 42*Sec[(c + d*x)/2]^10 + 164800*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 3840*Csc[c + d*x]^
5*Sin[(c + d*x)/2]^6 + Csc[(c + d*x)/2]^6*(5250 - 60*Sin[c + d*x]) - 14*Csc[(c + d*x)/2]^10*(3 + 10*Sin[c + d*
x]) + 5*Csc[(c + d*x)/2]^8*(-63 + 172*Sin[c + d*x]) - 20*Csc[(c + d*x)/2]^4*(966 + 515*Sin[c + d*x]) - 51200*T
an[(c + d*x)/2] - 1720*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2] + 280*Sec[(c + d*x)/2]^8*Tan[(c + d*x)/2]))/(430080
*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(-\frac {2749 \left (\frac {4325376 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2749}+\left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {666 \cos \left (3 d x +3 c \right )}{2749}+\frac {1154 \cos \left (5 d x +5 c \right )}{13745}-\frac {705 \cos \left (7 d x +7 c \right )}{5498}-\frac {33 \cos \left (9 d x +9 c \right )}{5498}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {12800 \cos \left (d x +c \right )}{2749}+\frac {7168 \cos \left (3 d x +3 c \right )}{2749}+\frac {13312 \cos \left (5 d x +5 c \right )}{19243}+\frac {256 \cos \left (7 d x +7 c \right )}{19243}-\frac {1280 \cos \left (9 d x +9 c \right )}{57729}\right ) \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{3}}{33554432 d}\)	\(168\)
risch	\(-\frac {a^{3} \left (3465 \,{\mathrm e}^{19 i \left (d x +c \right )}+74025 \,{\mathrm e}^{17 i \left (d x +c \right )}-107520 i {\mathrm e}^{14 i \left (d x +c \right )}-48468 \,{\mathrm e}^{15 i \left (d x +c \right )}+241920 i {\mathrm e}^{16 i \left (d x +c \right )}-139860 \,{\mathrm e}^{13 i \left (d x +c \right )}+376320 i {\mathrm e}^{8 i \left (d x +c \right )}-577290 \,{\mathrm e}^{11 i \left (d x +c \right )}-26880 i {\mathrm e}^{18 i \left (d x +c \right )}-577290 \,{\mathrm e}^{9 i \left (d x +c \right )}-660480 i {\mathrm e}^{6 i \left (d x +c \right )}-139860 \,{\mathrm e}^{7 i \left (d x +c \right )}+967680 i {\mathrm e}^{12 i \left (d x +c \right )}-48468 \,{\mathrm e}^{5 i \left (d x +c \right )}-806400 i {\mathrm e}^{10 i \left (d x +c \right )}+74025 \,{\mathrm e}^{3 i \left (d x +c \right )}+46080 i {\mathrm e}^{4 i \left (d x +c \right )}+3465 \,{\mathrm e}^{i \left (d x +c \right )}-37120 i {\mathrm e}^{2 i \left (d x +c \right )}+6400 i\right )}{13440 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}-\frac {33 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}+\frac {33 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\)	\(284\)
derivativedivides	\(\frac {-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(334\)
default	\(\frac {-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(334\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-2749/33554432*(4325376/2749*ln(tan(1/2*d*x+1/2*c))+csc(1/2*d*x+1/2*c)^9*(sec(1/2*d*x+1/2*c)*(cos(d*x+c)+666/2
749*cos(3*d*x+3*c)+1154/13745*cos(5*d*x+5*c)-705/5498*cos(7*d*x+7*c)-33/5498*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c
)+12800/2749*cos(d*x+c)+7168/2749*cos(3*d*x+3*c)+13312/19243*cos(5*d*x+5*c)+256/19243*cos(7*d*x+7*c)-1280/5772
9*cos(9*d*x+9*c))*sec(1/2*d*x+1/2*c)^9)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {6930 \, a^{3} \cos \left (d x + c\right )^{9} + 21420 \, a^{3} \cos \left (d x + c\right )^{7} - 59136 \, a^{3} \cos \left (d x + c\right )^{5} + 32340 \, a^{3} \cos \left (d x + c\right )^{3} - 6930 \, a^{3} \cos \left (d x + c\right ) - 3465 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3465 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2560 \, {\left (5 \, a^{3} \cos \left (d x + c\right )^{9} - 12 \, a^{3} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{53760 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/53760*(6930*a^3*cos(d*x + c)^9 + 21420*a^3*cos(d*x + c)^7 - 59136*a^3*cos(d*x + c)^5 + 32340*a^3*cos(d*x +
c)^3 - 6930*a^3*cos(d*x + c) - 3465*(a^3*cos(d*x + c)^10 - 5*a^3*cos(d*x + c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a
^3*cos(d*x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) + 3465*(a^3*cos(d*x + c)^10 - 5*a^
3*cos(d*x + c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a^3*cos(d*x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*
x + c) + 1/2) + 2560*(5*a^3*cos(d*x + c)^9 - 12*a^3*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c)^10 - 5*d*cos
(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**11*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.25 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {21 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 210 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {7680 \, a^{3}}{\tan \left (d x + c\right )^{7}} + \frac {2560 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{53760 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/53760*(21*a^3*(2*(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))/(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) -
15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 210*a^3*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55
*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 +
1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 7680*a^3/tan(d*x + c)^7 + 2560*(9*tan(d*x + c)^2 +
7)*a^3/tan(d*x + c)^9)/d

Giac [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.56 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {42 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3570 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3360 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10500 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 55440 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 31920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {162382 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 31920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 10500 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 16800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3360 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3570 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 525 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 42 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{430080 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/430080*(42*a^3*tan(1/2*d*x + 1/2*c)^10 + 280*a^3*tan(1/2*d*x + 1/2*c)^9 + 525*a^3*tan(1/2*d*x + 1/2*c)^8 - 6
00*a^3*tan(1/2*d*x + 1/2*c)^7 - 3570*a^3*tan(1/2*d*x + 1/2*c)^6 - 3360*a^3*tan(1/2*d*x + 1/2*c)^5 + 5880*a^3*t
an(1/2*d*x + 1/2*c)^4 + 16800*a^3*tan(1/2*d*x + 1/2*c)^3 + 10500*a^3*tan(1/2*d*x + 1/2*c)^2 - 55440*a^3*log(ab
s(tan(1/2*d*x + 1/2*c))) - 31920*a^3*tan(1/2*d*x + 1/2*c) + (162382*a^3*tan(1/2*d*x + 1/2*c)^10 + 31920*a^3*ta
n(1/2*d*x + 1/2*c)^9 - 10500*a^3*tan(1/2*d*x + 1/2*c)^8 - 16800*a^3*tan(1/2*d*x + 1/2*c)^7 - 5880*a^3*tan(1/2*
d*x + 1/2*c)^6 + 3360*a^3*tan(1/2*d*x + 1/2*c)^5 + 3570*a^3*tan(1/2*d*x + 1/2*c)^4 + 600*a^3*tan(1/2*d*x + 1/2
*c)^3 - 525*a^3*tan(1/2*d*x + 1/2*c)^2 - 280*a^3*tan(1/2*d*x + 1/2*c) - 42*a^3)/tan(1/2*d*x + 1/2*c)^10)/d

Mupad [B] (verification not implemented)

Time = 11.20 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.73 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}-\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {25\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {17\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {33\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}+\frac {19\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {19\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^11,x)

[Out]

(a^3*cot(c/2 + (d*x)/2)^5)/(128*d) - (5*a^3*cot(c/2 + (d*x)/2)^3)/(128*d) - (7*a^3*cot(c/2 + (d*x)/2)^4)/(512*
d) - (25*a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (17*a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) + (5*a^3*cot(c/2 + (d*x)/
2)^7)/(3584*d) - (5*a^3*cot(c/2 + (d*x)/2)^8)/(4096*d) - (a^3*cot(c/2 + (d*x)/2)^9)/(1536*d) - (a^3*cot(c/2 +
(d*x)/2)^10)/(10240*d) + (25*a^3*tan(c/2 + (d*x)/2)^2)/(1024*d) + (5*a^3*tan(c/2 + (d*x)/2)^3)/(128*d) + (7*a^
3*tan(c/2 + (d*x)/2)^4)/(512*d) - (a^3*tan(c/2 + (d*x)/2)^5)/(128*d) - (17*a^3*tan(c/2 + (d*x)/2)^6)/(2048*d)
- (5*a^3*tan(c/2 + (d*x)/2)^7)/(3584*d) + (5*a^3*tan(c/2 + (d*x)/2)^8)/(4096*d) + (a^3*tan(c/2 + (d*x)/2)^9)/(
1536*d) + (a^3*tan(c/2 + (d*x)/2)^10)/(10240*d) - (33*a^3*log(tan(c/2 + (d*x)/2)))/(256*d) + (19*a^3*cot(c/2 +
(d*x)/2))/(256*d) - (19*a^3*tan(c/2 + (d*x)/2))/(256*d)