\(\int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\) [618]

Optimal result

Integrand size = 29, antiderivative size = 200 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {55 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {25 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \]

[Out]

55/128*a^3*arctanh(cos(d*x+c))/d-4/7*a^3*cot(d*x+c)^7/d-1/9*a^3*cot(d*x+c)^9/d-25/128*a^3*cot(d*x+c)*csc(d*x+c
)/d+5/24*a^3*cot(d*x+c)^3*csc(d*x+c)/d-1/6*a^3*cot(d*x+c)^5*csc(d*x+c)/d-15/64*a^3*cot(d*x+c)*csc(d*x+c)^3/d+5
/16*a^3*cot(d*x+c)^3*csc(d*x+c)^3/d-3/8*a^3*cot(d*x+c)^5*csc(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2691, 3855, 2687, 30, 3853, 14} \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {55 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {25 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

(55*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/(9*d) - (25*a^3*C
ot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a^3*Cot[c + d*x]^5*Csc[c + d
*x])/(6*d) - (15*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(16*d) - (3*a
^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

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

Rule 2691

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

Rule 2952

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

Rule 3853

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

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \cot ^6(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^3(c+d x)+a^3 \cot ^6(c+d x) \csc ^4(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^6(c+d x) \csc (c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx \\ & = -\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{6} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx-\frac {1}{8} \left (15 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {3 a^3 \cot ^7(c+d x)}{7 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {1}{8} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\frac {1}{16} \left (15 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {5 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{64} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{16} \left (5 a^3\right ) \int \csc (c+d x) \, dx \\ & = \frac {5 a^3 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {25 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{128} \left (15 a^3\right ) \int \csc (c+d x) \, dx \\ & = \frac {55 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {25 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(459\) vs. \(2(200)=400\).

Time = 0.36 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.30 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 \left (\frac {29 \cot \left (\frac {1}{2} (c+d x)\right )}{126 d}-\frac {73 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {4163 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32256 d}-\frac {13 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {319 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{10752 d}+\frac {17 \csc ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}-\frac {53 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^6\left (\frac {1}{2} (c+d x)\right )}{32256 d}-\frac {3 \csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^8\left (\frac {1}{2} (c+d x)\right )}{4608 d}+\frac {55 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {55 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}+\frac {73 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {13 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {17 \sec ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}+\frac {3 \sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {29 \tan \left (\frac {1}{2} (c+d x)\right )}{126 d}+\frac {4163 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{32256 d}-\frac {319 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{10752 d}+\frac {53 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{32256 d}+\frac {\sec ^8\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{4608 d}\right ) \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

a^3*((29*Cot[(c + d*x)/2])/(126*d) - (73*Csc[(c + d*x)/2]^2)/(512*d) - (4163*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]
^2)/(32256*d) - (13*Csc[(c + d*x)/2]^4)/(1024*d) + (319*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(10752*d) + (17*C
sc[(c + d*x)/2]^6)/(1536*d) - (53*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^6)/(32256*d) - (3*Csc[(c + d*x)/2]^8)/(204
8*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^8)/(4608*d) + (55*Log[Cos[(c + d*x)/2]])/(128*d) - (55*Log[Sin[(c +
d*x)/2]])/(128*d) + (73*Sec[(c + d*x)/2]^2)/(512*d) + (13*Sec[(c + d*x)/2]^4)/(1024*d) - (17*Sec[(c + d*x)/2]^
6)/(1536*d) + (3*Sec[(c + d*x)/2]^8)/(2048*d) - (29*Tan[(c + d*x)/2])/(126*d) + (4163*Sec[(c + d*x)/2]^2*Tan[(
c + d*x)/2])/(32256*d) - (319*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(10752*d) + (53*Sec[(c + d*x)/2]^6*Tan[(c +
d*x)/2])/(32256*d) + (Sec[(c + d*x)/2]^8*Tan[(c + d*x)/2])/(4608*d))

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(-\frac {11 \left (2560 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\csc ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {14 \cos \left (3 d x +3 c \right )}{33}-\frac {2 \cos \left (5 d x +5 c \right )}{77}-\frac {13 \cos \left (7 d x +7 c \right )}{154}-\frac {29 \cos \left (9 d x +9 c \right )}{1386}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5975 \cos \left (d x +c \right )}{1056}+\frac {1781 \cos \left (3 d x +3 c \right )}{1056}+\frac {1679 \cos \left (5 d x +5 c \right )}{1056}-\frac {73 \cos \left (7 d x +7 c \right )}{352}\right ) \left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{3}}{65536 d}\)	\(157\)
risch	\(\frac {a^{3} \left (-64512 i {\mathrm e}^{14 i \left (d x +c \right )}+4599 \,{\mathrm e}^{17 i \left (d x +c \right )}+24192 i {\mathrm e}^{16 i \left (d x +c \right )}-39858 \,{\mathrm e}^{15 i \left (d x +c \right )}+145152 i {\mathrm e}^{8 i \left (d x +c \right )}-2142 \,{\mathrm e}^{13 i \left (d x +c \right )}-193536 i {\mathrm e}^{6 i \left (d x +c \right )}-88074 \,{\mathrm e}^{11 i \left (d x +c \right )}+118272 i {\mathrm e}^{12 i \left (d x +c \right )}-322560 i {\mathrm e}^{10 i \left (d x +c \right )}+88074 \,{\mathrm e}^{7 i \left (d x +c \right )}+69120 i {\mathrm e}^{4 i \left (d x +c \right )}+2142 \,{\mathrm e}^{5 i \left (d x +c \right )}-9216 i {\mathrm e}^{2 i \left (d x +c \right )}+39858 \,{\mathrm e}^{3 i \left (d x +c \right )}+3712 i-4599 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{4032 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}-\frac {55 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}+\frac {55 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}\)	\(250\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(297\)
default	\(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(297\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^10*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-11/65536*(2560*ln(tan(1/2*d*x+1/2*c))+csc(1/2*d*x+1/2*c)^8*(sec(1/2*d*x+1/2*c)*(cos(d*x+c)+14/33*cos(3*d*x+3*
c)-2/77*cos(5*d*x+5*c)-13/154*cos(7*d*x+7*c)-29/1386*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)+5975/1056*cos(d*x+c)+1
781/1056*cos(3*d*x+3*c)+1679/1056*cos(5*d*x+5*c)-73/352*cos(7*d*x+7*c))*sec(1/2*d*x+1/2*c)^8)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.46 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {7424 \, a^{3} \cos \left (d x + c\right )^{9} - 9216 \, a^{3} \cos \left (d x + c\right )^{7} + 3465 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3465 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 42 \, {\left (219 \, a^{3} \cos \left (d x + c\right )^{7} - 803 \, a^{3} \cos \left (d x + c\right )^{5} + 605 \, a^{3} \cos \left (d x + c\right )^{3} - 165 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16128 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^10*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/16128*(7424*a^3*cos(d*x + c)^9 - 9216*a^3*cos(d*x + c)^7 + 3465*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 +
6*a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3465*(a^3*cos(d
*x + c)^8 - 4*a^3*cos(d*x + c)^6 + 6*a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) +
1/2)*sin(d*x + c) + 42*(219*a^3*cos(d*x + c)^7 - 803*a^3*cos(d*x + c)^5 + 605*a^3*cos(d*x + c)^3 - 165*a^3*cos
(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d
)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**10*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.23 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {63 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 168 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {6912 \, a^{3}}{\tan \left (d x + c\right )^{7}} + \frac {256 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{16128 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^10*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/16128*(63*a^3*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15*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) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x
+ c) - 1)) - 168*a^3*(2*(33*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x
+ c)^4 + 3*cos(d*x + c)^2 - 1) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 6912*a^3/tan(d*x + c)^
7 + 256*(9*tan(d*x + c)^2 + 7)*a^3/tan(d*x + c)^9)/d

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.62 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {28 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 189 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 324 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 672 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3024 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1512 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 9744 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18144 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 55440 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 16632 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {156838 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 16632 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 18144 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9744 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1512 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3024 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 324 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 189 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 28 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{129024 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^10*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/129024*(28*a^3*tan(1/2*d*x + 1/2*c)^9 + 189*a^3*tan(1/2*d*x + 1/2*c)^8 + 324*a^3*tan(1/2*d*x + 1/2*c)^7 - 67
2*a^3*tan(1/2*d*x + 1/2*c)^6 - 3024*a^3*tan(1/2*d*x + 1/2*c)^5 - 1512*a^3*tan(1/2*d*x + 1/2*c)^4 + 9744*a^3*ta
n(1/2*d*x + 1/2*c)^3 + 18144*a^3*tan(1/2*d*x + 1/2*c)^2 - 55440*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 16632*a^3
*tan(1/2*d*x + 1/2*c) + (156838*a^3*tan(1/2*d*x + 1/2*c)^9 + 16632*a^3*tan(1/2*d*x + 1/2*c)^8 - 18144*a^3*tan(
1/2*d*x + 1/2*c)^7 - 9744*a^3*tan(1/2*d*x + 1/2*c)^6 + 1512*a^3*tan(1/2*d*x + 1/2*c)^5 + 3024*a^3*tan(1/2*d*x
+ 1/2*c)^4 + 672*a^3*tan(1/2*d*x + 1/2*c)^3 - 324*a^3*tan(1/2*d*x + 1/2*c)^2 - 189*a^3*tan(1/2*d*x + 1/2*c) -
28*a^3)/tan(1/2*d*x + 1/2*c)^9)/d

Mupad [B] (verification not implemented)

Time = 11.06 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.78 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {29\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {9\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {9\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {9\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {29\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {9\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {55\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}+\frac {33\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {33\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^10,x)

[Out]

(3*a^3*cot(c/2 + (d*x)/2)^4)/(256*d) - (29*a^3*cot(c/2 + (d*x)/2)^3)/(384*d) - (9*a^3*cot(c/2 + (d*x)/2)^2)/(6
4*d) + (3*a^3*cot(c/2 + (d*x)/2)^5)/(128*d) + (a^3*cot(c/2 + (d*x)/2)^6)/(192*d) - (9*a^3*cot(c/2 + (d*x)/2)^7
)/(3584*d) - (3*a^3*cot(c/2 + (d*x)/2)^8)/(2048*d) - (a^3*cot(c/2 + (d*x)/2)^9)/(4608*d) + (9*a^3*tan(c/2 + (d
*x)/2)^2)/(64*d) + (29*a^3*tan(c/2 + (d*x)/2)^3)/(384*d) - (3*a^3*tan(c/2 + (d*x)/2)^4)/(256*d) - (3*a^3*tan(c
/2 + (d*x)/2)^5)/(128*d) - (a^3*tan(c/2 + (d*x)/2)^6)/(192*d) + (9*a^3*tan(c/2 + (d*x)/2)^7)/(3584*d) + (3*a^3
*tan(c/2 + (d*x)/2)^8)/(2048*d) + (a^3*tan(c/2 + (d*x)/2)^9)/(4608*d) - (55*a^3*log(tan(c/2 + (d*x)/2)))/(128*
d) + (33*a^3*cot(c/2 + (d*x)/2))/(256*d) - (33*a^3*tan(c/2 + (d*x)/2))/(256*d)