\(\int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx\) [621]

Optimal result

Integrand size = 29, antiderivative size = 270 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {41 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d} \]

[Out]

41/1024*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-7/9*a^3*cot(d*x+c)^9/d-3/11*a^3*cot(d*x+c)^11/d+41/10
24*a^3*cot(d*x+c)*csc(d*x+c)/d+41/1536*a^3*cot(d*x+c)*csc(d*x+c)^3/d-35/384*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*cot(d*x+c)^5*csc(d*x+c)^5/d-1/64*a^3*cot(d*x+c)*csc(d*x+c)^7/d+1/24*
a^3*cot(d*x+c)^3*csc(d*x+c)^7/d-1/12*a^3*cot(d*x+c)^5*csc(d*x+c)^7/d

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2687, 14, 2691, 3853, 3855, 276} \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {41 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

(41*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (7*a^3*Cot[c + d*x]^9)/(9*d) - (3*a^3
*Cot[c + d*x]^11)/(11*d) + (41*a^3*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (41*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/
(1536*d) - (35*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(384*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) - (a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^3*Cot[c + d*x]^3*Csc[c
+ d*x]^7)/(24*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^7)/(12*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 ^4(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^6(c+d x)+a^3 \cot ^6(c+d x) \csc ^7(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx \\ & = -\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 ^7(c+d x)}{12 d}-\frac {1}{12} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^7(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 ) \, 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 )^2 \, dx,x,-\cot (c+d x)\right )}{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 {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}+\frac {1}{8} a^3 \int \cot ^2(c+d x) \csc ^7(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+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 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 {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{64} a^3 \int \csc ^7(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 {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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 {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{384} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx-\frac {1}{128} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {9 a^3 \cot (c+d x) \csc (c+d x)}{256 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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 {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{512} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{256} \left (9 a^3\right ) \int \csc (c+d x) \, dx \\ & = \frac {9 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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 {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {\left (5 a^3\right ) \int \csc (c+d x) \, dx}{1024} \\ & = \frac {41 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {7 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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 {a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.13 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.73 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (72737280 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{11}(c+d x) (91311066+62609778 \cos (2 (c+d x))[phone] \cos (4 (c+d x))-23426403 \cos (6 (c+d x))-1799490 \cos (8 (c+d x))+142065 \cos (10 (c+d x))[phone] \sin (c+d x)[phone] \sin (3 (c+d x))[phone] \sin (5 (c+d x))+2206720 \sin (7 (c+d x))-1530880 \sin (9 (c+d x))+117760 \sin (11 (c+d x)))\right )}{1816657920 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(1 + Sin[c + d*x])^3*(72737280*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x
]^11*(91311066 + 62609778*Cos[2*(c + d*x)] + 22551144*Cos[4*(c + d*x)] - 23426403*Cos[6*(c + d*x)] - 1799490*C
os[8*(c + d*x)] + 142065*Cos[10*(c + d*x)] + 49776640*Sin[c + d*x] + 84039680*Sin[3*(c + d*x)] + 38118400*Sin[
5*(c + d*x)] + 2206720*Sin[7*(c + d*x)] - 1530880*Sin[9*(c + d*x)] + 117760*Sin[11*(c + d*x)])))/(1816657920*d
*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.70

method	result	size
parallelrisch	\(-\frac {41 \left (4194304 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\sec ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (11 d x +11 c \right )+\frac {70774 \cos \left (d x +c \right )}{41}+\frac {368662 \cos \left (3 d x +3 c \right )}{615}-\frac {1263 \cos \left (5 d x +5 c \right )}{205}-\frac {36401 \cos \left (7 d x +7 c \right )}{205}-\frac {35 \cos \left (9 d x +9 c \right )}{3}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {94208 \cos \left (11 d x +11 c \right )}{28413}+\frac {1024000 \cos \left (d x +c \right )}{123}+\frac {3923968 \cos \left (3 d x +3 c \right )}{861}+\frac {962560 \cos \left (5 d x +5 c \right )}{861}-\frac {45056 \cos \left (7 d x +7 c \right )}{2583}-\frac {94208 \cos \left (9 d x +9 c \right )}{2583}\right ) \left (\csc ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{3}}{4294967296 d}\)	\(190\)
risch	\(-\frac {a^{3} \left (142065 \,{\mathrm e}^{23 i \left (d x +c \right )}-1657425 \,{\mathrm e}^{21 i \left (d x +c \right )}-25225893 \,{\mathrm e}^{19 i \left (d x +c \right )}-227082240 i {\mathrm e}^{14 i \left (d x +c \right )}-875259 \,{\mathrm e}^{17 i \left (d x +c \right )}+7096320 i {\mathrm e}^{20 i \left (d x +c \right )}[phone] \,{\mathrm e}^{15 i \left (d x +c \right )}-63866880 i {\mathrm e}^{16 i \left (d x +c \right )}[phone] \,{\mathrm e}^{13 i \left (d x +c \right )}[phone] i {\mathrm e}^{8 i \left (d x +c \right )}[phone] \,{\mathrm e}^{11 i \left (d x +c \right )}-66232320 i {\mathrm e}^{18 i \left (d x +c \right )}[phone] \,{\mathrm e}^{9 i \left (d x +c \right )}[phone] i {\mathrm e}^{6 i \left (d x +c \right )}-875259 \,{\mathrm e}^{7 i \left (d x +c \right )}[phone] i {\mathrm e}^{12 i \left (d x +c \right )}-25225893 \,{\mathrm e}^{5 i \left (d x +c \right )}[phone] i {\mathrm e}^{10 i \left (d x +c \right )}-1657425 \,{\mathrm e}^{3 i \left (d x +c \right )}+8448000 i {\mathrm e}^{4 i \left (d x +c \right )}+142065 \,{\mathrm e}^{i \left (d x +c \right )}-2826240 i {\mathrm e}^{2 i \left (d x +c \right )}+235520 i\right )}{1774080 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}+\frac {41 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{1024 d}-\frac {41 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{1024 d}\)	\(318\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{7}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{1024 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{1024}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )}{d}\)	\(408\)
default	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{7}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{1024 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{1024}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )}{d}\)	\(408\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^13*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-41/4294967296*(4194304*ln(tan(1/2*d*x+1/2*c))+sec(1/2*d*x+1/2*c)^11*(sec(1/2*d*x+1/2*c)*(cos(11*d*x+11*c)+707
74/41*cos(d*x+c)+368662/615*cos(3*d*x+3*c)-1263/205*cos(5*d*x+5*c)-36401/205*cos(7*d*x+7*c)-35/3*cos(9*d*x+9*c
))*csc(1/2*d*x+1/2*c)+94208/28413*cos(11*d*x+11*c)+1024000/123*cos(d*x+c)+3923968/861*cos(3*d*x+3*c)+962560/86
1*cos(5*d*x+5*c)-45056/2583*cos(7*d*x+7*c)-94208/2583*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)^11)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.42 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {284130 \, a^{3} \cos \left (d x + c\right )^{11} - 1610070 \, a^{3} \cos \left (d x + c\right )^{9} - 507276 \, a^{3} \cos \left (d x + c\right )^{7} + 3750516 \, a^{3} \cos \left (d x + c\right )^{5} - 1610070 \, a^{3} \cos \left (d x + c\right )^{3} + 284130 \, a^{3} \cos \left (d x + c\right ) - 142065 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 142065 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 10240 \, {\left (46 \, a^{3} \cos \left (d x + c\right )^{11} - 253 \, a^{3} \cos \left (d x + c\right )^{9} + 396 \, a^{3} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{7096320 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^13*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/7096320*(284130*a^3*cos(d*x + c)^11 - 1610070*a^3*cos(d*x + c)^9 - 507276*a^3*cos(d*x + c)^7 + 3750516*a^3*
cos(d*x + c)^5 - 1610070*a^3*cos(d*x + c)^3 + 284130*a^3*cos(d*x + c) - 142065*(a^3*cos(d*x + c)^12 - 6*a^3*co
s(d*x + c)^10 + 15*a^3*cos(d*x + c)^8 - 20*a^3*cos(d*x + c)^6 + 15*a^3*cos(d*x + c)^4 - 6*a^3*cos(d*x + c)^2 +
a^3)*log(1/2*cos(d*x + c) + 1/2) + 142065*(a^3*cos(d*x + c)^12 - 6*a^3*cos(d*x + c)^10 + 15*a^3*cos(d*x + c)^
8 - 20*a^3*cos(d*x + c)^6 + 15*a^3*cos(d*x + c)^4 - 6*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2) +
10240*(46*a^3*cos(d*x + c)^11 - 253*a^3*cos(d*x + c)^9 + 396*a^3*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c
)^12 - 6*d*cos(d*x + c)^10 + 15*d*cos(d*x + c)^8 - 20*d*cos(d*x + c)^6 + 15*d*cos(d*x + c)^4 - 6*d*cos(d*x + c
)^2 + d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**13*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.29 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {1155 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 8316 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {112640 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}} + \frac {30720 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}}}{7096320 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^13*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/7096320*(1155*a^3*(2*(15*cos(d*x + c)^11 - 85*cos(d*x + c)^9 + 198*cos(d*x + c)^7 + 198*cos(d*x + c)^5 - 85
*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^12 - 6*cos(d*x + c)^10 + 15*cos(d*x + c)^8 - 20*cos(d*x + c)^
6 + 15*cos(d*x + c)^4 - 6*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 8316*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)) + 112640*(9*tan(d*x + c)^2 + 7)*a^3/tan(d*x + c)^9 + 30720*(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 = 0.54 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.56 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {1155 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 7560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 16632 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 3080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 51975 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 106920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 83160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 83160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 384615 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 572880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 166320 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2273040 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1496880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {7053722 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 1496880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 166320 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 572880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 384615 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 83160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 83160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 106920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 51975 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16632 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1155 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12}}}{56770560 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^13*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/56770560*(1155*a^3*tan(1/2*d*x + 1/2*c)^12 + 7560*a^3*tan(1/2*d*x + 1/2*c)^11 + 16632*a^3*tan(1/2*d*x + 1/2*
c)^10 + 3080*a^3*tan(1/2*d*x + 1/2*c)^9 - 51975*a^3*tan(1/2*d*x + 1/2*c)^8 - 106920*a^3*tan(1/2*d*x + 1/2*c)^7
- 83160*a^3*tan(1/2*d*x + 1/2*c)^6 + 83160*a^3*tan(1/2*d*x + 1/2*c)^5 + 384615*a^3*tan(1/2*d*x + 1/2*c)^4 + 5
72880*a^3*tan(1/2*d*x + 1/2*c)^3 + 166320*a^3*tan(1/2*d*x + 1/2*c)^2 - 2273040*a^3*log(abs(tan(1/2*d*x + 1/2*c
))) - 1496880*a^3*tan(1/2*d*x + 1/2*c) + (7053722*a^3*tan(1/2*d*x + 1/2*c)^12 + 1496880*a^3*tan(1/2*d*x + 1/2*
c)^11 - 166320*a^3*tan(1/2*d*x + 1/2*c)^10 - 572880*a^3*tan(1/2*d*x + 1/2*c)^9 - 384615*a^3*tan(1/2*d*x + 1/2*
c)^8 - 83160*a^3*tan(1/2*d*x + 1/2*c)^7 + 83160*a^3*tan(1/2*d*x + 1/2*c)^6 + 106920*a^3*tan(1/2*d*x + 1/2*c)^5
+ 51975*a^3*tan(1/2*d*x + 1/2*c)^4 - 3080*a^3*tan(1/2*d*x + 1/2*c)^3 - 16632*a^3*tan(1/2*d*x + 1/2*c)^2 - 756
0*a^3*tan(1/2*d*x + 1/2*c) - 1155*a^3)/tan(1/2*d*x + 1/2*c)^12)/d

Mupad [B] (verification not implemented)

Time = 12.11 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.74 \[ \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {31\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}-\frac {111\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {27\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}+\frac {15\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}-\frac {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 {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{49152\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {31\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}+\frac {111\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {27\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}-\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}+\frac {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 {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{49152\,d}-\frac {41\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{1024\,d}+\frac {27\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d}-\frac {27\,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)^13,x)

[Out]

(3*a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) - (31*a^3*cot(c/2 + (d*x)/2)^3)/(3072*d) - (111*a^3*cot(c/2 + (d*x)/2)^4
)/(16384*d) - (3*a^3*cot(c/2 + (d*x)/2)^5)/(2048*d) - (3*a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (27*a^3*cot(c/2
+ (d*x)/2)^7)/(14336*d) + (15*a^3*cot(c/2 + (d*x)/2)^8)/(16384*d) - (a^3*cot(c/2 + (d*x)/2)^9)/(18432*d) - (3*
a^3*cot(c/2 + (d*x)/2)^10)/(10240*d) - (3*a^3*cot(c/2 + (d*x)/2)^11)/(22528*d) - (a^3*cot(c/2 + (d*x)/2)^12)/(
49152*d) + (3*a^3*tan(c/2 + (d*x)/2)^2)/(1024*d) + (31*a^3*tan(c/2 + (d*x)/2)^3)/(3072*d) + (111*a^3*tan(c/2 +
(d*x)/2)^4)/(16384*d) + (3*a^3*tan(c/2 + (d*x)/2)^5)/(2048*d) - (3*a^3*tan(c/2 + (d*x)/2)^6)/(2048*d) - (27*a
^3*tan(c/2 + (d*x)/2)^7)/(14336*d) - (15*a^3*tan(c/2 + (d*x)/2)^8)/(16384*d) + (a^3*tan(c/2 + (d*x)/2)^9)/(184
32*d) + (3*a^3*tan(c/2 + (d*x)/2)^10)/(10240*d) + (3*a^3*tan(c/2 + (d*x)/2)^11)/(22528*d) + (a^3*tan(c/2 + (d*
x)/2)^12)/(49152*d) - (41*a^3*log(tan(c/2 + (d*x)/2)))/(1024*d) + (27*a^3*cot(c/2 + (d*x)/2))/(1024*d) - (27*a
^3*tan(c/2 + (d*x)/2))/(1024*d)