\(\int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx\) [391]

Optimal result

Integrand size = 29, antiderivative size = 218 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {9 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {9 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d} \]

[Out]

-9/256*a^2*arctanh(cos(d*x+c))/d-2/5*a^2*cot(d*x+c)^5/d-4/7*a^2*cot(d*x+c)^7/d-2/9*a^2*cot(d*x+c)^9/d-9/256*a^
2*cot(d*x+c)*csc(d*x+c)/d-3/128*a^2*cot(d*x+c)*csc(d*x+c)^3/d+9/160*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+3/80*a^2*cot(d*x+c)*csc(d*x+c)^7/d-1/10*a^2*cot(d*x+c)^3*csc(d*x+c)^7/d

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2691, 3853, 3855, 2687, 276} \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {9 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac {9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {9 a^2 \cot (c+d x) \csc (c+d x)}{256 d} \]

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]

[Out]

(-9*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (4*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*
Cot[c + d*x]^9)/(9*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d
) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^2*Cot[c + d
*x]*Csc[c + d*x]^7)/(80*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^7)/(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 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 ^4(c+d x) \csc ^5(c+d x)+2 a^2 \cot ^4(c+d x) \csc ^6(c+d x)+a^2 \cot ^4(c+d x) \csc ^7(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+a^2 \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx \\ & = -\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {1}{10} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx-\frac {1}{8} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {\left (2 a^2\right ) \text {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{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 {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{80} \left (3 a^2\right ) \int \csc ^7(c+d x) \, dx+\frac {1}{16} a^2 \int \csc ^5(c+d x) \, dx+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{32} a^2 \int \csc ^5(c+d x) \, dx+\frac {1}{64} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}-\frac {3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{128} \left (3 a^2\right ) \int \csc (c+d x) \, dx+\frac {1}{128} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {9 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{256} \left (3 a^2\right ) \int \csc (c+d x) \, dx \\ & = -\frac {9 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {9 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.78 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.62 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^{10}(c+d x) \left (3219300 \cos (c+d x)+1237320 \cos (3 (c+d x))-278712 \cos (5 (c+d x))-54810 \cos (7 (c+d x))+5670 \cos (9 (c+d x))+357210 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-595350 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+340200 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-127575 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+28350 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2835 \cos (10 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-357210 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+595350 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-340200 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+127575 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-28350 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2835 \cos (10 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1720320 \sin (2 (c+d x))+1228800 \sin (4 (c+d x))+184320 \sin (6 (c+d x))-40960 \sin (8 (c+d x))+4096 \sin (10 (c+d x))\right )}{41287680 d} \]

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]

[Out]

-1/41287680*(a^2*Csc[c + d*x]^10*(3219300*Cos[c + d*x] + 1237320*Cos[3*(c + d*x)] - 278712*Cos[5*(c + d*x)] -
54810*Cos[7*(c + d*x)] + 5670*Cos[9*(c + d*x)] + 357210*Log[Cos[(c + d*x)/2]] - 595350*Cos[2*(c + d*x)]*Log[Co
s[(c + d*x)/2]] + 340200*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 127575*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]
] + 28350*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 2835*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 357210*Log[S
in[(c + d*x)/2]] + 595350*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 340200*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2
]] + 127575*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 28350*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 2835*Cos[1
0*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 1720320*Sin[2*(c + d*x)] + 1228800*Sin[4*(c + d*x)] + 184320*Sin[6*(c + d
*x)] - 40960*Sin[8*(c + d*x)] + 4096*Sin[10*(c + d*x)]))/d

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.77

method	result	size
parallelrisch	\(-\frac {2555 \left (-\frac {1179648 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2555}+\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {982 \cos \left (3 d x +3 c \right )}{2555}-\frac {158 \cos \left (5 d x +5 c \right )}{1825}-\frac {87 \cos \left (7 d x +7 c \right )}{5110}+\frac {9 \cos \left (9 d x +9 c \right )}{5110}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {49152 \cos \left (d x +c \right )}{12775}+\frac {65536 \cos \left (3 d x +3 c \right )}{38325}+\frac {16384 \cos \left (5 d x +5 c \right )}{89425}-\frac {4096 \cos \left (7 d x +7 c \right )}{89425}+\frac {4096 \cos \left (9 d x +9 c \right )}{804825}\right ) \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{2}}{33554432 d}\)	\(168\)
risch	\(\frac {a^{2} \left (2835 \,{\mathrm e}^{19 i \left (d x +c \right )}-27405 \,{\mathrm e}^{17 i \left (d x +c \right )}-139356 \,{\mathrm e}^{15 i \left (d x +c \right )}+618660 \,{\mathrm e}^{13 i \left (d x +c \right )}-860160 i {\mathrm e}^{14 i \left (d x +c \right )}+1609650 \,{\mathrm e}^{11 i \left (d x +c \right )}+1290240 i {\mathrm e}^{8 i \left (d x +c \right )}+1609650 \,{\mathrm e}^{9 i \left (d x +c \right )}+368640 i {\mathrm e}^{6 i \left (d x +c \right )}+618660 \,{\mathrm e}^{7 i \left (d x +c \right )}-430080 i {\mathrm e}^{12 i \left (d x +c \right )}-139356 \,{\mathrm e}^{5 i \left (d x +c \right )}-516096 i {\mathrm e}^{10 i \left (d x +c \right )}-27405 \,{\mathrm e}^{3 i \left (d x +c \right )}+184320 i {\mathrm e}^{4 i \left (d x +c \right )}+2835 \,{\mathrm e}^{i \left (d x +c \right )}-40960 i {\mathrm e}^{2 i \left (d x +c \right )}+4096 i\right )}{40320 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}-\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}+\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}\)	\(260\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315 \sin \left (d x +c \right )^{5}}\right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{256 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(310\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315 \sin \left (d x +c \right )^{5}}\right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{256 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(310\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-2555/33554432*(-1179648/2555*ln(tan(1/2*d*x+1/2*c))+(sec(1/2*d*x+1/2*c)*(cos(d*x+c)+982/2555*cos(3*d*x+3*c)-1
58/1825*cos(5*d*x+5*c)-87/5110*cos(7*d*x+7*c)+9/5110*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)+49152/12775*cos(d*x+c)
+65536/38325*cos(3*d*x+3*c)+16384/89425*cos(5*d*x+5*c)-4096/89425*cos(7*d*x+7*c)+4096/804825*cos(9*d*x+9*c))*s
ec(1/2*d*x+1/2*c)^9*csc(1/2*d*x+1/2*c)^9)*a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.56 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5670 \, a^{2} \cos \left (d x + c\right )^{9} - 26460 \, a^{2} \cos \left (d x + c\right )^{7} + 16128 \, a^{2} \cos \left (d x + c\right )^{5} + 26460 \, a^{2} \cos \left (d x + c\right )^{3} - 5670 \, a^{2} \cos \left (d x + c\right ) - 2835 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2835 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1024 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{9} - 36 \, a^{2} \cos \left (d x + c\right )^{7} + 63 \, a^{2} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/161280*(5670*a^2*cos(d*x + c)^9 - 26460*a^2*cos(d*x + c)^7 + 16128*a^2*cos(d*x + c)^5 + 26460*a^2*cos(d*x +
c)^3 - 5670*a^2*cos(d*x + c) - 2835*(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) + 2835*(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) + 1024*(8*a^2*cos(d*x + c)^9 - 36*a^2*cos(d*x + c)^7 + 63*a^2*cos(d*x + c)^5)*sin(d*x + c))/(d*c
os(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**11*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.30 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {63 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 630 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1024 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/161280*(63*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)) + 630*a^2*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*
cos(d*x + c)^3 + 3*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)
- 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 1024*(63*tan(d*x + c)^4 + 90*tan(d*x + c)^2 + 35)*a^2/
tan(d*x + c)^9)/d

Giac [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.64 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {126 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 945 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4032 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45360 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {132858 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1260 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 6720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4032 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 945 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 126 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1290240*(126*a^2*tan(1/2*d*x + 1/2*c)^10 + 560*a^2*tan(1/2*d*x + 1/2*c)^9 + 945*a^2*tan(1/2*d*x + 1/2*c)^8 +
720*a^2*tan(1/2*d*x + 1/2*c)^7 - 630*a^2*tan(1/2*d*x + 1/2*c)^6 - 4032*a^2*tan(1/2*d*x + 1/2*c)^5 - 7560*a^2*
tan(1/2*d*x + 1/2*c)^4 - 6720*a^2*tan(1/2*d*x + 1/2*c)^3 + 1260*a^2*tan(1/2*d*x + 1/2*c)^2 + 45360*a^2*log(abs
(tan(1/2*d*x + 1/2*c))) + 30240*a^2*tan(1/2*d*x + 1/2*c) - (132858*a^2*tan(1/2*d*x + 1/2*c)^10 + 30240*a^2*tan
(1/2*d*x + 1/2*c)^9 + 1260*a^2*tan(1/2*d*x + 1/2*c)^8 - 6720*a^2*tan(1/2*d*x + 1/2*c)^7 - 7560*a^2*tan(1/2*d*x
+ 1/2*c)^6 - 4032*a^2*tan(1/2*d*x + 1/2*c)^5 - 630*a^2*tan(1/2*d*x + 1/2*c)^4 + 720*a^2*tan(1/2*d*x + 1/2*c)^
3 + 945*a^2*tan(1/2*d*x + 1/2*c)^2 + 560*a^2*tan(1/2*d*x + 1/2*c) + 126*a^2)/tan(1/2*d*x + 1/2*c)^10)/d

Mupad [B] (verification not implemented)

Time = 10.86 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.81 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}-\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}+\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {9\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}-\frac {3\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^2)/sin(c + d*x)^11,x)

[Out]

(a^2*cot(c/2 + (d*x)/2)^3)/(192*d) - (a^2*cot(c/2 + (d*x)/2)^2)/(1024*d) + (3*a^2*cot(c/2 + (d*x)/2)^4)/(512*d
) + (a^2*cot(c/2 + (d*x)/2)^5)/(320*d) + (a^2*cot(c/2 + (d*x)/2)^6)/(2048*d) - (a^2*cot(c/2 + (d*x)/2)^7)/(179
2*d) - (3*a^2*cot(c/2 + (d*x)/2)^8)/(4096*d) - (a^2*cot(c/2 + (d*x)/2)^9)/(2304*d) - (a^2*cot(c/2 + (d*x)/2)^1
0)/(10240*d) + (a^2*tan(c/2 + (d*x)/2)^2)/(1024*d) - (a^2*tan(c/2 + (d*x)/2)^3)/(192*d) - (3*a^2*tan(c/2 + (d*
x)/2)^4)/(512*d) - (a^2*tan(c/2 + (d*x)/2)^5)/(320*d) - (a^2*tan(c/2 + (d*x)/2)^6)/(2048*d) + (a^2*tan(c/2 + (
d*x)/2)^7)/(1792*d) + (3*a^2*tan(c/2 + (d*x)/2)^8)/(4096*d) + (a^2*tan(c/2 + (d*x)/2)^9)/(2304*d) + (a^2*tan(c
/2 + (d*x)/2)^10)/(10240*d) + (9*a^2*log(tan(c/2 + (d*x)/2)))/(256*d) - (3*a^2*cot(c/2 + (d*x)/2))/(128*d) + (
3*a^2*tan(c/2 + (d*x)/2))/(128*d)