\(\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\) [603]

Optimal result

Integrand size = 29, antiderivative size = 194 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{3 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d} \]

[Out]

3/128*a^2*arctanh(cos(d*x+c))/d-2/7*a^2*cot(d*x+c)^7/d-1/3*a^2*cot(d*x+c)^9/d-1/11*a^2*cot(d*x+c)^11/d+3/128*a
^2*cot(d*x+c)*csc(d*x+c)/d+1/64*a^2*cot(d*x+c)*csc(d*x+c)^3/d-1/16*a^2*cot(d*x+c)*csc(d*x+c)^5/d+1/8*a^2*cot(d
*x+c)^3*csc(d*x+c)^5/d-1/5*a^2*cot(d*x+c)^5*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2687, 14, 2691, 3853, 3855, 276} \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}-\frac {a^2 \cot ^9(c+d x)}{3 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]

[Out]

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

Mathematica [A] (verified)

Time = 7.84 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.96 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (1+\sin (c+d x))^2 \left (887040 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (1318400+1798400 \cos (2 (c+d x))+440320 \cos (4 (c+d x))-81280 \cos (6 (c+d x))-38400 \cos (8 (c+d x))+3200 \cos (10 (c+d x))+1073226 \sin (c+d x)+869484 \sin (3 (c+d x))+727188 \sin (5 (c+d x))+40425 \sin (7 (c+d x))-3465 \sin (9 (c+d x)))\right )}{37847040 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]

[Out]

(a^2*(1 + Sin[c + d*x])^2*(887040*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^
10*(1318400 + 1798400*Cos[2*(c + d*x)] + 440320*Cos[4*(c + d*x)] - 81280*Cos[6*(c + d*x)] - 38400*Cos[8*(c + d
*x)] + 3200*Cos[10*(c + d*x)] + 1073226*Sin[c + d*x] + 869484*Sin[3*(c + d*x)] + 727188*Sin[5*(c + d*x)] + 404
25*Sin[7*(c + d*x)] - 3465*Sin[9*(c + d*x)])))/(37847040*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.92

method	result	size
parallelrisch	\(-\frac {5 \left (\frac {5677056 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\left (\sec ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (11 d x +11 c \right )+1386 \cos \left (d x +c \right )+\frac {3498 \cos \left (3 d x +3 c \right )}{5}+\frac {561 \cos \left (5 d x +5 c \right )}{5}-\frac {187 \cos \left (7 d x +7 c \right )}{5}-11 \cos \left (9 d x +9 c \right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {434049 \cos \left (d x +c \right )}{80}+\frac {119889 \cos \left (3 d x +3 c \right )}{40}+\frac {200277 \cos \left (5 d x +5 c \right )}{200}+\frac {6699 \cos \left (7 d x +7 c \right )}{160}-\frac {693 \cos \left (9 d x +9 c \right )}{160}\right ) \left (\csc ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{2}}{242221056 d}\)	\(179\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(248\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(248\)
risch	\(-\frac {a^{2} \left (3465 \,{\mathrm e}^{21 i \left (d x +c \right )}-1182720 i {\mathrm e}^{14 i \left (d x +c \right )}-36960 \,{\mathrm e}^{19 i \left (d x +c \right )}-887040 i {\mathrm e}^{16 i \left (d x +c \right )}-767613 \,{\mathrm e}^{17 i \left (d x +c \right )}-3294720 i {\mathrm e}^{8 i \left (d x +c \right )}-1596672 \,{\mathrm e}^{15 i \left (d x +c \right )}+295680 i {\mathrm e}^{18 i \left (d x +c \right )}-1942710 \,{\mathrm e}^{13 i \left (d x +c \right )}+168960 i {\mathrm e}^{6 i \left (d x +c \right )}-5913600 i {\mathrm e}^{12 i \left (d x +c \right )}+1942710 \,{\mathrm e}^{9 i \left (d x +c \right )}-2956800 i {\mathrm e}^{10 i \left (d x +c \right )}+1596672 \,{\mathrm e}^{7 i \left (d x +c \right )}-56320 i {\mathrm e}^{4 i \left (d x +c \right )}+767613 \,{\mathrm e}^{5 i \left (d x +c \right )}+70400 i {\mathrm e}^{2 i \left (d x +c \right )}+36960 \,{\mathrm e}^{3 i \left (d x +c \right )}-6400 i-3465 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{73920 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}\)	\(284\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-5/242221056*(5677056/5*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)+138
6*cos(d*x+c)+3498/5*cos(3*d*x+3*c)+561/5*cos(5*d*x+5*c)-187/5*cos(7*d*x+7*c)-11*cos(9*d*x+9*c))*csc(1/2*d*x+1/
2*c)+434049/80*cos(d*x+c)+119889/40*cos(3*d*x+3*c)+200277/200*cos(5*d*x+5*c)+6699/160*cos(7*d*x+7*c)-693/160*c
os(9*d*x+9*c))*csc(1/2*d*x+1/2*c)^10)*a^2/d

Fricas [B] (verification not implemented)

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

Time = 0.30 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.86 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {12800 \, a^{2} \cos \left (d x + c\right )^{11} - 70400 \, a^{2} \cos \left (d x + c\right )^{9} + 84480 \, a^{2} \cos \left (d x + c\right )^{7} + 3465 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3465 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 462 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{9} - 70 \, a^{2} \cos \left (d x + c\right )^{7} - 128 \, a^{2} \cos \left (d x + c\right )^{5} + 70 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{295680 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/295680*(12800*a^2*cos(d*x + c)^11 - 70400*a^2*cos(d*x + c)^9 + 84480*a^2*cos(d*x + c)^7 + 3465*(a^2*cos(d*x
+ c)^10 - 5*a^2*cos(d*x + c)^8 + 10*a^2*cos(d*x + c)^6 - 10*a^2*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*l
og(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3465*(a^2*cos(d*x + c)^10 - 5*a^2*cos(d*x + c)^8 + 10*a^2*cos(d*x +
c)^6 - 10*a^2*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 462*(15
*a^2*cos(d*x + c)^9 - 70*a^2*cos(d*x + c)^7 - 128*a^2*cos(d*x + c)^5 + 70*a^2*cos(d*x + c)^3 - 15*a^2*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. \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**12*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.02 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {693 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {14080 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}} + \frac {1280 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{2}}{\tan \left (d x + c\right )^{11}}}{887040 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/887040*(693*a^2*(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)) + 14080*(9*tan(d*x + c)^2 + 7)*a^2/tan(d*x + c)^9 + 12
80*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^2/tan(d*x + c)^11)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (176) = 352\).

Time = 0.44 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.00 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {105 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 462 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 385 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2805 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2310 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16170 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4620 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 55440 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 39270 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {167422 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 39270 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 4620 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 16170 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 9240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2310 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2805 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 462 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{2365440 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/2365440*(105*a^2*tan(1/2*d*x + 1/2*c)^11 + 462*a^2*tan(1/2*d*x + 1/2*c)^10 + 385*a^2*tan(1/2*d*x + 1/2*c)^9
- 1155*a^2*tan(1/2*d*x + 1/2*c)^8 - 2805*a^2*tan(1/2*d*x + 1/2*c)^7 - 2310*a^2*tan(1/2*d*x + 1/2*c)^6 + 1155*a
^2*tan(1/2*d*x + 1/2*c)^5 + 9240*a^2*tan(1/2*d*x + 1/2*c)^4 + 16170*a^2*tan(1/2*d*x + 1/2*c)^3 + 4620*a^2*tan(
1/2*d*x + 1/2*c)^2 - 55440*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 39270*a^2*tan(1/2*d*x + 1/2*c) + (167422*a^2*t
an(1/2*d*x + 1/2*c)^11 + 39270*a^2*tan(1/2*d*x + 1/2*c)^10 - 4620*a^2*tan(1/2*d*x + 1/2*c)^9 - 16170*a^2*tan(1
/2*d*x + 1/2*c)^8 - 9240*a^2*tan(1/2*d*x + 1/2*c)^7 - 1155*a^2*tan(1/2*d*x + 1/2*c)^6 + 2310*a^2*tan(1/2*d*x +
1/2*c)^5 + 2805*a^2*tan(1/2*d*x + 1/2*c)^4 + 1155*a^2*tan(1/2*d*x + 1/2*c)^3 - 385*a^2*tan(1/2*d*x + 1/2*c)^2
- 462*a^2*tan(1/2*d*x + 1/2*c) - 105*a^2)/tan(1/2*d*x + 1/2*c)^11)/d

Mupad [B] (verification not implemented)

Time = 12.23 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.23 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1024\,d}-\frac {7\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1024\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{512\,d}+\frac {17\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6144\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5120\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{512\,d}+\frac {7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1024\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1024\,d}-\frac {17\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6144\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5120\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}-\frac {3\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}+\frac {17\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d}-\frac {17\,a^2\,\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))^2)/sin(c + d*x)^12,x)

[Out]

(a^2*cot(c/2 + (d*x)/2)^6)/(1024*d) - (7*a^2*cot(c/2 + (d*x)/2)^3)/(1024*d) - (a^2*cot(c/2 + (d*x)/2)^4)/(256*
d) - (a^2*cot(c/2 + (d*x)/2)^5)/(2048*d) - (a^2*cot(c/2 + (d*x)/2)^2)/(512*d) + (17*a^2*cot(c/2 + (d*x)/2)^7)/
(14336*d) + (a^2*cot(c/2 + (d*x)/2)^8)/(2048*d) - (a^2*cot(c/2 + (d*x)/2)^9)/(6144*d) - (a^2*cot(c/2 + (d*x)/2
)^10)/(5120*d) - (a^2*cot(c/2 + (d*x)/2)^11)/(22528*d) + (a^2*tan(c/2 + (d*x)/2)^2)/(512*d) + (7*a^2*tan(c/2 +
(d*x)/2)^3)/(1024*d) + (a^2*tan(c/2 + (d*x)/2)^4)/(256*d) + (a^2*tan(c/2 + (d*x)/2)^5)/(2048*d) - (a^2*tan(c/
2 + (d*x)/2)^6)/(1024*d) - (17*a^2*tan(c/2 + (d*x)/2)^7)/(14336*d) - (a^2*tan(c/2 + (d*x)/2)^8)/(2048*d) + (a^
2*tan(c/2 + (d*x)/2)^9)/(6144*d) + (a^2*tan(c/2 + (d*x)/2)^10)/(5120*d) + (a^2*tan(c/2 + (d*x)/2)^11)/(22528*d
) - (3*a^2*log(tan(c/2 + (d*x)/2)))/(128*d) + (17*a^2*cot(c/2 + (d*x)/2))/(1024*d) - (17*a^2*tan(c/2 + (d*x)/2
))/(1024*d)