\(\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx\) [587]

Optimal result

Integrand size = 27, antiderivative size = 176 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {2 a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^{11}(c+d x)}{11 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2687, 276, 2691, 3853, 3855} \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a \cot ^{11}(c+d x)}{11 d}-\frac {2 a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d} \]

[In]

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

[Out]

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

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2917

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

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}& = a \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx \\ & = -\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{2} a \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac {a \text {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {1}{16} (3 a) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a \text {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \cot ^7(c+d x)}{7 d}-\frac {2 a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^{11}(c+d x)}{11 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{32} a \int \csc ^5(c+d x) \, dx \\ & = -\frac {a \cot ^7(c+d x)}{7 d}-\frac {2 a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^{11}(c+d x)}{11 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{128} (3 a) \int \csc ^3(c+d x) \, dx \\ & = -\frac {a \cot ^7(c+d x)}{7 d}-\frac {2 a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^{11}(c+d x)}{11 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{256} (3 a) \int \csc (c+d x) \, dx \\ & = \frac {3 a \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {2 a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^{11}(c+d x)}{11 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(363\) vs. \(2(176)=352\).

Time = 0.19 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.06 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {8 a \cot (c+d x)}{693 d}+\frac {3 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {3 a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {3 a \csc ^8\left (\frac {1}{2} (c+d x)\right )}{4096 d}-\frac {a \csc ^{10}\left (\frac {1}{2} (c+d x)\right )}{10240 d}+\frac {4 a \cot (c+d x) \csc ^2(c+d x)}{693 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{231 d}-\frac {113 a \cot (c+d x) \csc ^6(c+d x)}{693 d}+\frac {23 a \cot (c+d x) \csc ^8(c+d x)}{99 d}-\frac {a \cot (c+d x) \csc ^{10}(c+d x)}{11 d}+\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}-\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}-\frac {3 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {3 a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {3 a \sec ^8\left (\frac {1}{2} (c+d x)\right )}{4096 d}+\frac {a \sec ^{10}\left (\frac {1}{2} (c+d x)\right )}{10240 d} \]

[In]

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

[Out]

(8*a*Cot[c + d*x])/(693*d) + (3*a*Csc[(c + d*x)/2]^2)/(1024*d) - (a*Csc[(c + d*x)/2]^4)/(1024*d) - (3*a*Csc[(c
+ d*x)/2]^6)/(2048*d) + (3*a*Csc[(c + d*x)/2]^8)/(4096*d) - (a*Csc[(c + d*x)/2]^10)/(10240*d) + (4*a*Cot[c +
d*x]*Csc[c + d*x]^2)/(693*d) + (a*Cot[c + d*x]*Csc[c + d*x]^4)/(231*d) - (113*a*Cot[c + d*x]*Csc[c + d*x]^6)/(
693*d) + (23*a*Cot[c + d*x]*Csc[c + d*x]^8)/(99*d) - (a*Cot[c + d*x]*Csc[c + d*x]^10)/(11*d) + (3*a*Log[Cos[(c
+ d*x)/2]])/(256*d) - (3*a*Log[Sin[(c + d*x)/2]])/(256*d) - (3*a*Sec[(c + d*x)/2]^2)/(1024*d) + (a*Sec[(c + d
*x)/2]^4)/(1024*d) + (3*a*Sec[(c + d*x)/2]^6)/(2048*d) - (3*a*Sec[(c + d*x)/2]^8)/(4096*d) + (a*Sec[(c + d*x)/
2]^10)/(10240*d)

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.01

method	result	size
parallelrisch	\(-\frac {\left (2128896 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\sec ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (11 d x +11 c \right )+4620 \cos \left (d x +c \right )+2640 \cos \left (3 d x +3 c \right )+759 \cos \left (5 d x +5 c \right )+55 \cos \left (7 d x +7 c \right )-11 \cos \left (9 d x +9 c \right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1302147 \cos \left (d x +c \right )}{128}+\frac {359667 \cos \left (3 d x +3 c \right )}{64}+\frac {600831 \cos \left (5 d x +5 c \right )}{320}+\frac {20097 \cos \left (7 d x +7 c \right )}{256}-\frac {2079 \cos \left (9 d x +9 c \right )}{256}\right ) \left (\csc ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a}{181665792 d}\)	\(177\)
derivativedivides	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(202\)
default	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(202\)
risch	\(-\frac {a \left (10395 \,{\mathrm e}^{21 i \left (d x +c \right )}-110880 \,{\mathrm e}^{19 i \left (d x +c \right )}-11827200 i {\mathrm e}^{14 i \left (d x +c \right )}-2302839 \,{\mathrm e}^{17 i \left (d x +c \right )}-4730880 i {\mathrm e}^{16 i \left (d x +c \right )}-4790016 \,{\mathrm e}^{15 i \left (d x +c \right )}-15206400 i {\mathrm e}^{8 i \left (d x +c \right )}-5828130 \,{\mathrm e}^{13 i \left (d x +c \right )}-3041280 i {\mathrm e}^{6 i \left (d x +c \right )}-26019840 i {\mathrm e}^{12 i \left (d x +c \right )}+5828130 \,{\mathrm e}^{9 i \left (d x +c \right )}-21288960 i {\mathrm e}^{10 i \left (d x +c \right )}+4790016 \,{\mathrm e}^{7 i \left (d x +c \right )}-563200 i {\mathrm e}^{4 i \left (d x +c \right )}+2302839 \,{\mathrm e}^{5 i \left (d x +c \right )}+112640 i {\mathrm e}^{2 i \left (d x +c \right )}+110880 \,{\mathrm e}^{3 i \left (d x +c \right )}-10240 i-10395 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{443520 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\)	\(266\)

[In]

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

[Out]

-1/181665792*(2128896*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)+4620*
cos(d*x+c)+2640*cos(3*d*x+3*c)+759*cos(5*d*x+5*c)+55*cos(7*d*x+7*c)-11*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)+1302
147/128*cos(d*x+c)+359667/64*cos(3*d*x+3*c)+600831/320*cos(5*d*x+5*c)+20097/256*cos(7*d*x+7*c)-2079/256*cos(9*
d*x+9*c))*csc(1/2*d*x+1/2*c)^10)*a/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (158) = 316\).

Time = 0.29 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.82 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {20480 \, a \cos \left (d x + c\right )^{11} - 112640 \, a \cos \left (d x + c\right )^{9} + 253440 \, a \cos \left (d x + c\right )^{7} + 10395 \, {\left (a \cos \left (d x + c\right )^{10} - 5 \, a \cos \left (d x + c\right )^{8} + 10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 10395 \, {\left (a \cos \left (d x + c\right )^{10} - 5 \, a \cos \left (d x + c\right )^{8} + 10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 1386 \, {\left (15 \, a \cos \left (d x + c\right )^{9} - 70 \, a \cos \left (d x + c\right )^{7} - 128 \, a \cos \left (d x + c\right )^{5} + 70 \, a \cos \left (d x + c\right )^{3} - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1774080 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

1/1774080*(20480*a*cos(d*x + c)^11 - 112640*a*cos(d*x + c)^9 + 253440*a*cos(d*x + c)^7 + 10395*(a*cos(d*x + c)
^10 - 5*a*cos(d*x + c)^8 + 10*a*cos(d*x + c)^6 - 10*a*cos(d*x + c)^4 + 5*a*cos(d*x + c)^2 - a)*log(1/2*cos(d*x
+ c) + 1/2)*sin(d*x + c) - 10395*(a*cos(d*x + c)^10 - 5*a*cos(d*x + c)^8 + 10*a*cos(d*x + c)^6 - 10*a*cos(d*x
+ c)^4 + 5*a*cos(d*x + c)^2 - a)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 1386*(15*a*cos(d*x + c)^9 - 70*a
*cos(d*x + c)^7 - 128*a*cos(d*x + c)^5 + 70*a*cos(d*x + c)^3 - 15*a*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. \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.95 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {693 \, a {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {2560 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a}{\tan \left (d x + c\right )^{11}}}{1774080 \, d} \]

[In]

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

[Out]

-1/1774080*(693*a*(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)) + 2560*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a/
tan(d*x + c)^11)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (158) = 316\).

Time = 0.39 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.93 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {630 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1386 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 770 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3465 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 4950 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 6930 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 23100 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13860 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 166320 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 69300 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {502266 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 69300 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 13860 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 23100 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 27720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 6930 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4950 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 770 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1386 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 630 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{14192640 \, d} \]

[In]

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

[Out]

1/14192640*(630*a*tan(1/2*d*x + 1/2*c)^11 + 1386*a*tan(1/2*d*x + 1/2*c)^10 - 770*a*tan(1/2*d*x + 1/2*c)^9 - 34
65*a*tan(1/2*d*x + 1/2*c)^8 - 4950*a*tan(1/2*d*x + 1/2*c)^7 - 6930*a*tan(1/2*d*x + 1/2*c)^6 + 6930*a*tan(1/2*d
*x + 1/2*c)^5 + 27720*a*tan(1/2*d*x + 1/2*c)^4 + 23100*a*tan(1/2*d*x + 1/2*c)^3 + 13860*a*tan(1/2*d*x + 1/2*c)
^2 - 166320*a*log(abs(tan(1/2*d*x + 1/2*c))) - 69300*a*tan(1/2*d*x + 1/2*c) + (502266*a*tan(1/2*d*x + 1/2*c)^1
1 + 69300*a*tan(1/2*d*x + 1/2*c)^10 - 13860*a*tan(1/2*d*x + 1/2*c)^9 - 23100*a*tan(1/2*d*x + 1/2*c)^8 - 27720*
a*tan(1/2*d*x + 1/2*c)^7 - 6930*a*tan(1/2*d*x + 1/2*c)^6 + 6930*a*tan(1/2*d*x + 1/2*c)^5 + 4950*a*tan(1/2*d*x
+ 1/2*c)^4 + 3465*a*tan(1/2*d*x + 1/2*c)^3 + 770*a*tan(1/2*d*x + 1/2*c)^2 - 1386*a*tan(1/2*d*x + 1/2*c) - 630*
a)/tan(1/2*d*x + 1/2*c)^11)/d

Mupad [B] (verification not implemented)

Time = 11.09 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.20 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d}-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}-\frac {5\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {5\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{18432\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{18432\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}-\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d} \]

[In]

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

[Out]

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