\(\int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx\) [406]

Optimal result

Integrand size = 29, antiderivative size = 216 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {21 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d} \]

[Out]

-21/256*a^3*arctanh(cos(d*x+c))/d-4/5*a^3*cot(d*x+c)^5/d-a^3*cot(d*x+c)^7/d-1/3*a^3*cot(d*x+c)^9/d-21/256*a^3*
cot(d*x+c)*csc(d*x+c)/d-7/128*a^3*cot(d*x+c)*csc(d*x+c)^3/d+29/160*a^3*cot(d*x+c)*csc(d*x+c)^5/d-3/8*a^3*cot(d
*x+c)^3*csc(d*x+c)^5/d+3/80*a^3*cot(d*x+c)*csc(d*x+c)^7/d-1/10*a^3*cot(d*x+c)^3*csc(d*x+c)^7/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2687, 14, 2691, 3853, 3855, 276} \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {21 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

(-21*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/d - (a^3*Cot[c +
d*x]^9)/(3*d) - (21*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (7*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (2
9*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^3*Cot[c + d*x]
*Csc[c + d*x]^7)/(80*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 276

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

Rule 2687

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

Rule 2691

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

Rule 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \cot ^4(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^6(c+d x)+a^3 \cot ^4(c+d x) \csc ^7(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx \\ & = -\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {1}{10} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx-\frac {1}{8} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{80} \left (3 a^3\right ) \int \csc ^7(c+d x) \, dx+\frac {1}{16} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{32} a^3 \int \csc ^5(c+d x) \, dx+\frac {1}{64} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {9 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{128} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac {1}{128} \left (9 a^3\right ) \int \csc (c+d x) \, dx \\ & = -\frac {9 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{256} \left (3 a^3\right ) \int \csc (c+d x) \, dx \\ & = -\frac {21 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.52 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.69 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (-4096 \cot \left (\frac {1}{2} (c+d x)\right )-1260 \csc ^2\left (\frac {1}{2} (c+d x)\right )-5040 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+5040 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1260 \sec ^2\left (\frac {1}{2} (c+d x)\right )-180 \sec ^4\left (\frac {1}{2} (c+d x)\right )-390 \sec ^6\left (\frac {1}{2} (c+d x)\right )+75 \sec ^8\left (\frac {1}{2} (c+d x)\right )+6 \sec ^{10}\left (\frac {1}{2} (c+d x)\right )+64 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-4 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (-45+\sin (c+d x))+5 \csc ^8\left (\frac {1}{2} (c+d x)\right ) (-15+4 \sin (c+d x))-2 \csc ^{10}\left (\frac {1}{2} (c+d x)\right ) (3+10 \sin (c+d x))+6 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (65+42 \sin (c+d x))+4096 \tan \left (\frac {1}{2} (c+d x)\right )-504 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )-40 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+40 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{61440 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(1 + Sin[c + d*x])^3*(-4096*Cot[(c + d*x)/2] - 1260*Csc[(c + d*x)/2]^2 - 5040*Log[Cos[(c + d*x)/2]] + 504
0*Log[Sin[(c + d*x)/2]] + 1260*Sec[(c + d*x)/2]^2 - 180*Sec[(c + d*x)/2]^4 - 390*Sec[(c + d*x)/2]^6 + 75*Sec[(
c + d*x)/2]^8 + 6*Sec[(c + d*x)/2]^10 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[(c + d*x)/2]^4*(-45 + Sin
[c + d*x]) + 5*Csc[(c + d*x)/2]^8*(-15 + 4*Sin[c + d*x]) - 2*Csc[(c + d*x)/2]^10*(3 + 10*Sin[c + d*x]) + 6*Csc
[(c + d*x)/2]^6*(65 + 42*Sin[c + d*x]) + 4096*Tan[(c + d*x)/2] - 504*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] - 40*
Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2] + 40*Sec[(c + d*x)/2]^8*Tan[(c + d*x)/2]))/(61440*d*(Cos[(c + d*x)/2] + Si
n[(c + d*x)/2])^6)

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(-\frac {3231 \left (-\frac {917504 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1077}+\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {926 \cos \left (3 d x +3 c \right )}{3231}-\frac {3946 \cos \left (5 d x +5 c \right )}{16155}-\frac {203 \cos \left (7 d x +7 c \right )}{6462}+\frac {7 \cos \left (9 d x +9 c \right )}{2154}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {77824 \cos \left (d x +c \right )}{16155}+\frac {31744 \cos \left (3 d x +3 c \right )}{16155}+\frac {1024 \cos \left (5 d x +5 c \right )}{16155}-\frac {512 \cos \left (7 d x +7 c \right )}{5385}+\frac {512 \cos \left (9 d x +9 c \right )}{48465}\right ) \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{3}}{33554432 d}\)	\(168\)
risch	\(\frac {a^{3} \left (315 \,{\mathrm e}^{19 i \left (d x +c \right )}-3045 \,{\mathrm e}^{17 i \left (d x +c \right )}-23676 \,{\mathrm e}^{15 i \left (d x +c \right )}-76800 i {\mathrm e}^{14 i \left (d x +c \right )}+27780 \,{\mathrm e}^{13 i \left (d x +c \right )}+7680 i {\mathrm e}^{16 i \left (d x +c \right )}+96930 \,{\mathrm e}^{11 i \left (d x +c \right )}+122880 i {\mathrm e}^{8 i \left (d x +c \right )}+96930 \,{\mathrm e}^{9 i \left (d x +c \right )}+15360 i {\mathrm e}^{6 i \left (d x +c \right )}+27780 \,{\mathrm e}^{7 i \left (d x +c \right )}-15360 i {\mathrm e}^{12 i \left (d x +c \right )}-23676 \,{\mathrm e}^{5 i \left (d x +c \right )}-64512 i {\mathrm e}^{10 i \left (d x +c \right )}-3045 \,{\mathrm e}^{3 i \left (d x +c \right )}+15360 i {\mathrm e}^{4 i \left (d x +c \right )}+315 \,{\mathrm e}^{i \left (d x +c \right )}-5120 i {\mathrm e}^{2 i \left (d x +c \right )}+512 i\right )}{1920 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}+\frac {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}-\frac {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\)	\(272\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315 \sin \left (d x +c \right )^{5}}\right )+a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{256 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(352\)
default	\(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315 \sin \left (d x +c \right )^{5}}\right )+a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{256 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(352\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-3231/33554432*(-917504/1077*ln(tan(1/2*d*x+1/2*c))+(sec(1/2*d*x+1/2*c)*(cos(d*x+c)+926/3231*cos(3*d*x+3*c)-39
46/16155*cos(5*d*x+5*c)-203/6462*cos(7*d*x+7*c)+7/2154*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)+77824/16155*cos(d*x+
c)+31744/16155*cos(3*d*x+3*c)+1024/16155*cos(5*d*x+5*c)-512/5385*cos(7*d*x+7*c)+512/48465*cos(9*d*x+9*c))*sec(
1/2*d*x+1/2*c)^9*csc(1/2*d*x+1/2*c)^9)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.57 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {630 \, a^{3} \cos \left (d x + c\right )^{9} - 2940 \, a^{3} \cos \left (d x + c\right )^{7} + 768 \, a^{3} \cos \left (d x + c\right )^{5} + 2940 \, a^{3} \cos \left (d x + c\right )^{3} - 630 \, a^{3} \cos \left (d x + c\right ) - 315 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 512 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{9} - 9 \, a^{3} \cos \left (d x + c\right )^{7} + 12 \, a^{3} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{7680 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/7680*(630*a^3*cos(d*x + c)^9 - 2940*a^3*cos(d*x + c)^7 + 768*a^3*cos(d*x + c)^5 + 2940*a^3*cos(d*x + c)^3 -
630*a^3*cos(d*x + c) - 315*(a^3*cos(d*x + c)^10 - 5*a^3*cos(d*x + c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a^3*cos(d*
x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) + 315*(a^3*cos(d*x + c)^10 - 5*a^3*cos(d*x
+ c)^8 + 10*a^3*cos(d*x + c)^6 - 10*a^3*cos(d*x + c)^4 + 5*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1
/2) + 512*(2*a^3*cos(d*x + c)^9 - 9*a^3*cos(d*x + c)^7 + 12*a^3*cos(d*x + c)^5)*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)

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**11*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.43 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {21 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 630 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1536 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}} - \frac {512 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{53760 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/53760*(21*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)) + 630*a^3*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*c
os(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)) - 1536*(7*tan(d*x + c)^2 + 5)*a^3/tan(d*x + c)^7 - 512*(6
3*tan(d*x + c)^4 + 90*tan(d*x + c)^2 + 35)*a^3/tan(d*x + c)^9)/d

Giac [A] (verification not implemented)

none

Time = 0.63 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.65 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5040 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 3600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {14762 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 3600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{61440 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^11*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/61440*(6*a^3*tan(1/2*d*x + 1/2*c)^10 + 40*a^3*tan(1/2*d*x + 1/2*c)^9 + 105*a^3*tan(1/2*d*x + 1/2*c)^8 + 120*
a^3*tan(1/2*d*x + 1/2*c)^7 - 30*a^3*tan(1/2*d*x + 1/2*c)^6 - 384*a^3*tan(1/2*d*x + 1/2*c)^5 - 840*a^3*tan(1/2*
d*x + 1/2*c)^4 - 960*a^3*tan(1/2*d*x + 1/2*c)^3 + 60*a^3*tan(1/2*d*x + 1/2*c)^2 + 5040*a^3*log(abs(tan(1/2*d*x
+ 1/2*c))) + 3600*a^3*tan(1/2*d*x + 1/2*c) - (14762*a^3*tan(1/2*d*x + 1/2*c)^10 + 3600*a^3*tan(1/2*d*x + 1/2*
c)^9 + 60*a^3*tan(1/2*d*x + 1/2*c)^8 - 960*a^3*tan(1/2*d*x + 1/2*c)^7 - 840*a^3*tan(1/2*d*x + 1/2*c)^6 - 384*a
^3*tan(1/2*d*x + 1/2*c)^5 - 30*a^3*tan(1/2*d*x + 1/2*c)^4 + 120*a^3*tan(1/2*d*x + 1/2*c)^3 + 105*a^3*tan(1/2*d
*x + 1/2*c)^2 + 40*a^3*tan(1/2*d*x + 1/2*c) + 6*a^3)/tan(1/2*d*x + 1/2*c)^10)/d

Mupad [B] (verification not implemented)

Time = 10.74 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.83 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512\,d}-\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512\,d}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {21\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}-\frac {15\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}+\frac {15\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^3)/sin(c + d*x)^11,x)

[Out]

(a^3*cot(c/2 + (d*x)/2)^3)/(64*d) - (a^3*cot(c/2 + (d*x)/2)^2)/(1024*d) + (7*a^3*cot(c/2 + (d*x)/2)^4)/(512*d)
+ (a^3*cot(c/2 + (d*x)/2)^5)/(160*d) + (a^3*cot(c/2 + (d*x)/2)^6)/(2048*d) - (a^3*cot(c/2 + (d*x)/2)^7)/(512*
d) - (7*a^3*cot(c/2 + (d*x)/2)^8)/(4096*d) - (a^3*cot(c/2 + (d*x)/2)^9)/(1536*d) - (a^3*cot(c/2 + (d*x)/2)^10)
/(10240*d) + (a^3*tan(c/2 + (d*x)/2)^2)/(1024*d) - (a^3*tan(c/2 + (d*x)/2)^3)/(64*d) - (7*a^3*tan(c/2 + (d*x)/
2)^4)/(512*d) - (a^3*tan(c/2 + (d*x)/2)^5)/(160*d) - (a^3*tan(c/2 + (d*x)/2)^6)/(2048*d) + (a^3*tan(c/2 + (d*x
)/2)^7)/(512*d) + (7*a^3*tan(c/2 + (d*x)/2)^8)/(4096*d) + (a^3*tan(c/2 + (d*x)/2)^9)/(1536*d) + (a^3*tan(c/2 +
(d*x)/2)^10)/(10240*d) + (21*a^3*log(tan(c/2 + (d*x)/2)))/(256*d) - (15*a^3*cot(c/2 + (d*x)/2))/(256*d) + (15
*a^3*tan(c/2 + (d*x)/2))/(256*d)