\(\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx\) [620]

Optimal result

Integrand size = 29, antiderivative size = 246 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {19 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \]

[Out]

19/256*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-5/9*a^3*cot(d*x+c)^9/d-1/11*a^3*cot(d*x+c)^11/d+19/256
*a^3*cot(d*x+c)*csc(d*x+c)/d-7/128*a^3*cot(d*x+c)*csc(d*x+c)^3/d+5/48*a^3*cot(d*x+c)^3*csc(d*x+c)^3/d-1/8*a^3*
cot(d*x+c)^5*csc(d*x+c)^3/d-3/32*a^3*cot(d*x+c)*csc(d*x+c)^5/d+3/16*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d-3/10*a^3*c
ot(d*x+c)^5*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2691, 3853, 3855, 2687, 14, 276} \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {19 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]

[In]

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

[Out]

(19*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (5*a^3*Cot[c + d*x]^9)/(9*d) - (a^3*Co
t[c + d*x]^11)/(11*d) + (19*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (7*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)/(48*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (3*a^3*Cot[c +
d*x]*Csc[c + d*x]^5)/(32*d) + (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (3*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 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 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 ^3(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^5(c+d x)+a^3 \cot ^6(c+d x) \csc ^6(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx \\ & = -\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{8} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-\frac {1}{2} \left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, 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 {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {1}{16} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {1}{16} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\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 {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}-\frac {5 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)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{64} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{32} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx \\ & = -\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-\frac {7 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)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{128} \left (5 a^3\right ) \int \csc (c+d x) \, dx-\frac {1}{128} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx \\ & = \frac {5 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 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)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{256} \left (9 a^3\right ) \int \csc (c+d x) \, dx \\ & = \frac {19 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 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)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.13 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.76 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (16853760 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (10050560+12423680 \cos (2 (c+d x))+839680 \cos (4 (c+d x))-2149120 \cos (6 (c+d x))-568320 \cos (8 (c+d x))+47360 \cos (10 (c+d x))[phone] \sin (c+d x)+5875716 \sin (3 (c+d x))+7902972 \sin (5 (c+d x))-414645 \sin (7 (c+d x))-65835 \sin (9 (c+d x)))\right )}{227082240 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

[In]

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

[Out]

(a^3*(1 + Sin[c + d*x])^3*(16853760*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x
]^10*(10050560 + 12423680*Cos[2*(c + d*x)] + 839680*Cos[4*(c + d*x)] - 2149120*Cos[6*(c + d*x)] - 568320*Cos[8
*(c + d*x)] + 47360*Cos[10*(c + d*x)] + 14477694*Sin[c + d*x] + 5875716*Sin[3*(c + d*x)] + 7902972*Sin[5*(c +
d*x)] - 414645*Sin[7*(c + d*x)] - 65835*Sin[9*(c + d*x)])))/(227082240*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])
^6)

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.73

method	result	size
parallelrisch	\(-\frac {37 \left (\frac {53932032 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{37}+\left (\sec ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (11 d x +11 c \right )+\frac {25410 \cos \left (d x +c \right )}{37}+\frac {10362 \cos \left (3 d x +3 c \right )}{37}-\frac {1023 \cos \left (5 d x +5 c \right )}{37}-\frac {2123 \cos \left (7 d x +7 c \right )}{37}-11 \cos \left (9 d x +9 c \right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {4107411 \cos \left (d x +c \right )}{1184}+\frac {1036035 \cos \left (3 d x +3 c \right )}{592}+\frac {1735503 \cos \left (5 d x +5 c \right )}{2960}-\frac {109263 \cos \left (7 d x +7 c \right )}{2368}-\frac {13167 \cos \left (9 d x +9 c \right )}{2368}\right ) \left (\csc ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{3}}{726663168 d}\)	\(179\)
risch	\(-\frac {a^{3} \left (65835 \,{\mathrm e}^{21 i \left (d x +c \right )}+2365440 i {\mathrm e}^{14 i \left (d x +c \right )}+480480 \,{\mathrm e}^{19 i \left (d x +c \right )}-6504960 i {\mathrm e}^{16 i \left (d x +c \right )}-7488327 \,{\mathrm e}^{17 i \left (d x +c \right )}-28892160 i {\mathrm e}^{8 i \left (d x +c \right )}-13778688 \,{\mathrm e}^{15 i \left (d x +c \right )}+5322240 i {\mathrm e}^{18 i \left (d x +c \right )}-20353410 \,{\mathrm e}^{13 i \left (d x +c \right )}+9123840 i {\mathrm e}^{6 i \left (d x +c \right )}-54405120 i {\mathrm e}^{12 i \left (d x +c \right )}[phone] \,{\mathrm e}^{9 i \left (d x +c \right )}-10644480 i {\mathrm e}^{10 i \left (d x +c \right )}[phone] \,{\mathrm e}^{7 i \left (d x +c \right )}+112640 i {\mathrm e}^{4 i \left (d x +c \right )}+7488327 \,{\mathrm e}^{5 i \left (d x +c \right )}+1041920 i {\mathrm e}^{2 i \left (d x +c \right )}-480480 \,{\mathrm e}^{3 i \left (d x +c \right )}-94720 i-65835 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{443520 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}-\frac {19 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}+\frac {19 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\)	\(284\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(372\)
default	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(372\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-37/726663168*(53932032/37*ln(tan(1/2*d*x+1/2*c))+sec(1/2*d*x+1/2*c)^10*(sec(1/2*d*x+1/2*c)*(cos(11*d*x+11*c)+
25410/37*cos(d*x+c)+10362/37*cos(3*d*x+3*c)-1023/37*cos(5*d*x+5*c)-2123/37*cos(7*d*x+7*c)-11*cos(9*d*x+9*c))*c
sc(1/2*d*x+1/2*c)+4107411/1184*cos(d*x+c)+1036035/592*cos(3*d*x+3*c)+1735503/2960*cos(5*d*x+5*c)-109263/2368*c
os(7*d*x+7*c)-13167/2368*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)^10)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.46 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {189440 \, a^{3} \cos \left (d x + c\right )^{11} - 1041920 \, a^{3} \cos \left (d x + c\right )^{9} + 1013760 \, a^{3} \cos \left (d x + c\right )^{7} + 65835 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 65835 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 462 \, {\left (285 \, a^{3} \cos \left (d x + c\right )^{9} - 50 \, a^{3} \cos \left (d x + c\right )^{7} - 2432 \, a^{3} \cos \left (d x + c\right )^{5} + 1330 \, a^{3} \cos \left (d x + c\right )^{3} - 285 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1774080 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/1774080*(189440*a^3*cos(d*x + c)^11 - 1041920*a^3*cos(d*x + c)^9 + 1013760*a^3*cos(d*x + c)^7 + 65835*(a^3*c
os(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)*sin(d*x + c) - 65835*(a^3*cos(d*x + c)^10 - 5*a^3*cos(d*x + c)^8 + 10*a^3*co
s(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)*sin(d*x + c) -
462*(285*a^3*cos(d*x + c)^9 - 50*a^3*cos(d*x + c)^7 - 2432*a^3*cos(d*x + c)^5 + 1330*a^3*cos(d*x + c)^3 - 285
*a^3*cos(d*x + c))*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)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**12*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.25 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2079 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2310 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {84480 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}} + \frac {2560 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}}}{1774080 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/1774080*(2079*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*c
os(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)) + 2310*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)) + 84480*(9*tan(d*x + c)^2 + 7)*a^3/tan(d*x + c)
^9 + 2560*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^3/tan(d*x + c)^11)/d

Giac [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.58 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {630 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4158 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 8470 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3465 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 40590 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 57750 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 138600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 244860 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 152460 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1053360 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 568260 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3181018 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 568260 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 152460 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 244860 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 138600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 6930 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 57750 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40590 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8470 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4158 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 630 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{14192640 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/14192640*(630*a^3*tan(1/2*d*x + 1/2*c)^11 + 4158*a^3*tan(1/2*d*x + 1/2*c)^10 + 8470*a^3*tan(1/2*d*x + 1/2*c)
^9 - 3465*a^3*tan(1/2*d*x + 1/2*c)^8 - 40590*a^3*tan(1/2*d*x + 1/2*c)^7 - 57750*a^3*tan(1/2*d*x + 1/2*c)^6 + 6
930*a^3*tan(1/2*d*x + 1/2*c)^5 + 138600*a^3*tan(1/2*d*x + 1/2*c)^4 + 244860*a^3*tan(1/2*d*x + 1/2*c)^3 + 15246
0*a^3*tan(1/2*d*x + 1/2*c)^2 - 1053360*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 568260*a^3*tan(1/2*d*x + 1/2*c) +
(3181018*a^3*tan(1/2*d*x + 1/2*c)^11 + 568260*a^3*tan(1/2*d*x + 1/2*c)^10 - 152460*a^3*tan(1/2*d*x + 1/2*c)^9
- 244860*a^3*tan(1/2*d*x + 1/2*c)^8 - 138600*a^3*tan(1/2*d*x + 1/2*c)^7 - 6930*a^3*tan(1/2*d*x + 1/2*c)^6 + 57
750*a^3*tan(1/2*d*x + 1/2*c)^5 + 40590*a^3*tan(1/2*d*x + 1/2*c)^4 + 3465*a^3*tan(1/2*d*x + 1/2*c)^3 - 8470*a^3
*tan(1/2*d*x + 1/2*c)^2 - 4158*a^3*tan(1/2*d*x + 1/2*c) - 630*a^3)/tan(1/2*d*x + 1/2*c)^11)/d

Mupad [B] (verification not implemented)

Time = 11.95 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.76 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {25\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {53\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}-\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {11\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {41\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {11\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{18432\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {53\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}+\frac {5\,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}{2048\,d}-\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {41\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{18432\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}-\frac {19\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}+\frac {41\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d}-\frac {41\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^12,x)

[Out]

(25*a^3*cot(c/2 + (d*x)/2)^6)/(6144*d) - (53*a^3*cot(c/2 + (d*x)/2)^3)/(3072*d) - (5*a^3*cot(c/2 + (d*x)/2)^4)
/(512*d) - (a^3*cot(c/2 + (d*x)/2)^5)/(2048*d) - (11*a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (41*a^3*cot(c/2 + (d
*x)/2)^7)/(14336*d) + (a^3*cot(c/2 + (d*x)/2)^8)/(4096*d) - (11*a^3*cot(c/2 + (d*x)/2)^9)/(18432*d) - (3*a^3*c
ot(c/2 + (d*x)/2)^10)/(10240*d) - (a^3*cot(c/2 + (d*x)/2)^11)/(22528*d) + (11*a^3*tan(c/2 + (d*x)/2)^2)/(1024*
d) + (53*a^3*tan(c/2 + (d*x)/2)^3)/(3072*d) + (5*a^3*tan(c/2 + (d*x)/2)^4)/(512*d) + (a^3*tan(c/2 + (d*x)/2)^5
)/(2048*d) - (25*a^3*tan(c/2 + (d*x)/2)^6)/(6144*d) - (41*a^3*tan(c/2 + (d*x)/2)^7)/(14336*d) - (a^3*tan(c/2 +
(d*x)/2)^8)/(4096*d) + (11*a^3*tan(c/2 + (d*x)/2)^9)/(18432*d) + (3*a^3*tan(c/2 + (d*x)/2)^10)/(10240*d) + (a
^3*tan(c/2 + (d*x)/2)^11)/(22528*d) - (19*a^3*log(tan(c/2 + (d*x)/2)))/(256*d) + (41*a^3*cot(c/2 + (d*x)/2))/(
1024*d) - (41*a^3*tan(c/2 + (d*x)/2))/(1024*d)