\(\int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\) [601]

Optimal result

Integrand size = 29, antiderivative size = 152 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d} \]

[Out]

5/64*a^2*arctanh(cos(d*x+c))/d-2/7*a^2*cot(d*x+c)^7/d-1/9*a^2*cot(d*x+c)^9/d+5/64*a^2*cot(d*x+c)*csc(d*x+c)/d-
5/32*a^2*cot(d*x+c)*csc(d*x+c)^3/d+5/24*a^2*cot(d*x+c)^3*csc(d*x+c)^3/d-1/4*a^2*cot(d*x+c)^5*csc(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2687, 30, 2691, 3853, 3855, 14} \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

(5*a^2*ArcTanh[Cos[c + d*x]])/(64*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(9*d) + (5*a^2*Cot[
c + d*x]*Csc[c + d*x])/(64*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x
]^3)/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(4*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^2 \cot ^6(c+d x) \csc ^2(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^3(c+d x)+a^2 \cot ^6(c+d x) \csc ^4(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx \\ & = -\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{4} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {a^2 \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}+\frac {a^2 \text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a^2 \cot ^7(c+d x)}{7 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {a^2 \text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{32} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{64} \left (5 a^2\right ) \int \csc (c+d x) \, dx \\ & = \frac {5 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(313\) vs. \(2(152)=304\).

Time = 6.77 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.06 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^9(c+d x) \left (72576 \cos (c+d x)+37632 \cos (3 (c+d x))+6912 \cos (5 (c+d x))-1728 \cos (7 (c+d x))-704 \cos (9 (c+d x))-39690 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+39690 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+36540 \sin (2 (c+d x))+26460 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-26460 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+20916 \sin (4 (c+d x))-11340 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+11340 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+16044 \sin (6 (c+d x))+2835 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-2835 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+630 \sin (8 (c+d x))-315 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))+315 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))\right )}{1032192 d} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

-1/1032192*(a^2*Csc[c + d*x]^9*(72576*Cos[c + d*x] + 37632*Cos[3*(c + d*x)] + 6912*Cos[5*(c + d*x)] - 1728*Cos
[7*(c + d*x)] - 704*Cos[9*(c + d*x)] - 39690*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 39690*Log[Sin[(c + d*x)/2]]*
Sin[c + d*x] + 36540*Sin[2*(c + d*x)] + 26460*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 26460*Log[Sin[(c + d*x)
/2]]*Sin[3*(c + d*x)] + 20916*Sin[4*(c + d*x)] - 11340*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 11340*Log[Sin[
(c + d*x)/2]]*Sin[5*(c + d*x)] + 16044*Sin[6*(c + d*x)] + 2835*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] - 2835*L
og[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] + 630*Sin[8*(c + d*x)] - 315*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] + 31
5*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/d

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03

method	result	size
parallelrisch	\(-\frac {9 \left (\frac {5120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\left (\csc ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {14 \cos \left (3 d x +3 c \right )}{27}+\frac {2 \cos \left (5 d x +5 c \right )}{21}-\frac {\cos \left (7 d x +7 c \right )}{42}-\frac {11 \cos \left (9 d x +9 c \right )}{1134}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1765 \cos \left (d x +c \right )}{432}+\frac {895 \cos \left (3 d x +3 c \right )}{432}+\frac {397 \cos \left (5 d x +5 c \right )}{432}+\frac {5 \cos \left (7 d x +7 c \right )}{144}\right ) \left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{2}}{65536 d}\)	\(157\)
derivativedivides	\(\frac {-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(192\)
default	\(\frac {-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(192\)
risch	\(-\frac {a^{2} \left (16128 i {\mathrm e}^{14 i \left (d x +c \right )}+315 \,{\mathrm e}^{17 i \left (d x +c \right )}-4032 i {\mathrm e}^{16 i \left (d x +c \right )}+8022 \,{\mathrm e}^{15 i \left (d x +c \right )}-8064 i {\mathrm e}^{8 i \left (d x +c \right )}+10458 \,{\mathrm e}^{13 i \left (d x +c \right )}+48384 i {\mathrm e}^{6 i \left (d x +c \right )}+18270 \,{\mathrm e}^{11 i \left (d x +c \right )}-10752 i {\mathrm e}^{12 i \left (d x +c \right )}+80640 i {\mathrm e}^{10 i \left (d x +c \right )}-18270 \,{\mathrm e}^{7 i \left (d x +c \right )}-9216 i {\mathrm e}^{4 i \left (d x +c \right )}-10458 \,{\mathrm e}^{5 i \left (d x +c \right )}+2304 i {\mathrm e}^{2 i \left (d x +c \right )}-8022 \,{\mathrm e}^{3 i \left (d x +c \right )}-704 i-315 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2016 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 d}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 d}\)	\(250\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^10*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-9/65536*(5120/9*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)+14/27*cos(3*d*x+3
*c)+2/21*cos(5*d*x+5*c)-1/42*cos(7*d*x+7*c)-11/1134*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)+1765/432*cos(d*x+c)+895
/432*cos(3*d*x+3*c)+397/432*cos(5*d*x+5*c)+5/144*cos(7*d*x+7*c))*sec(1/2*d*x+1/2*c)^8)*a^2/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (138) = 276\).

Time = 0.28 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.91 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {1408 \, a^{2} \cos \left (d x + c\right )^{9} - 2304 \, a^{2} \cos \left (d x + c\right )^{7} + 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 42 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{7} + 73 \, a^{2} \cos \left (d x + c\right )^{5} - 55 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^10*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/8064*(1408*a^2*cos(d*x + c)^9 - 2304*a^2*cos(d*x + c)^7 + 315*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6
*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 315*(a^2*cos(d*x
+ c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(-1/2*cos(d*x + c) + 1/2
)*sin(d*x + c) - 42*(15*a^2*cos(d*x + c)^7 + 73*a^2*cos(d*x + c)^5 - 55*a^2*cos(d*x + c)^3 + 15*a^2*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. \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**10*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.02 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {21 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {1152 \, a^{2}}{\tan \left (d x + c\right )^{7}} + \frac {128 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{8064 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^10*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/8064*(21*a^2*(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)) + 1152*a^2/tan(d*x + c)^7 + 128*(9*tan(d*x + c)^2 + 7)*a^2/tan(d*x + c)^9)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (138) = 276\).

Time = 0.42 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.13 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {14 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1848 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 3276 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {14258 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3276 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1848 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{64512 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^10*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/64512*(14*a^2*tan(1/2*d*x + 1/2*c)^9 + 63*a^2*tan(1/2*d*x + 1/2*c)^8 + 18*a^2*tan(1/2*d*x + 1/2*c)^7 - 336*a
^2*tan(1/2*d*x + 1/2*c)^6 - 504*a^2*tan(1/2*d*x + 1/2*c)^5 + 504*a^2*tan(1/2*d*x + 1/2*c)^4 + 1848*a^2*tan(1/2
*d*x + 1/2*c)^3 + 1008*a^2*tan(1/2*d*x + 1/2*c)^2 - 5040*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 3276*a^2*tan(1/2
*d*x + 1/2*c) + (14258*a^2*tan(1/2*d*x + 1/2*c)^9 + 3276*a^2*tan(1/2*d*x + 1/2*c)^8 - 1008*a^2*tan(1/2*d*x + 1
/2*c)^7 - 1848*a^2*tan(1/2*d*x + 1/2*c)^6 - 504*a^2*tan(1/2*d*x + 1/2*c)^5 + 504*a^2*tan(1/2*d*x + 1/2*c)^4 +
336*a^2*tan(1/2*d*x + 1/2*c)^3 - 18*a^2*tan(1/2*d*x + 1/2*c)^2 - 63*a^2*tan(1/2*d*x + 1/2*c) - 14*a^2)/tan(1/2
*d*x + 1/2*c)^9)/d

Mupad [B] (verification not implemented)

Time = 10.78 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.35 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {11\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{1024\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{1024\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {5\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d}+\frac {13\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {13\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^10,x)

[Out]

(a^2*cot(c/2 + (d*x)/2)^5)/(128*d) - (11*a^2*cot(c/2 + (d*x)/2)^3)/(384*d) - (a^2*cot(c/2 + (d*x)/2)^4)/(128*d
) - (a^2*cot(c/2 + (d*x)/2)^2)/(64*d) + (a^2*cot(c/2 + (d*x)/2)^6)/(192*d) - (a^2*cot(c/2 + (d*x)/2)^7)/(3584*
d) - (a^2*cot(c/2 + (d*x)/2)^8)/(1024*d) - (a^2*cot(c/2 + (d*x)/2)^9)/(4608*d) + (a^2*tan(c/2 + (d*x)/2)^2)/(6
4*d) + (11*a^2*tan(c/2 + (d*x)/2)^3)/(384*d) + (a^2*tan(c/2 + (d*x)/2)^4)/(128*d) - (a^2*tan(c/2 + (d*x)/2)^5)
/(128*d) - (a^2*tan(c/2 + (d*x)/2)^6)/(192*d) + (a^2*tan(c/2 + (d*x)/2)^7)/(3584*d) + (a^2*tan(c/2 + (d*x)/2)^
8)/(1024*d) + (a^2*tan(c/2 + (d*x)/2)^9)/(4608*d) - (5*a^2*log(tan(c/2 + (d*x)/2)))/(64*d) + (13*a^2*cot(c/2 +
(d*x)/2))/(256*d) - (13*a^2*tan(c/2 + (d*x)/2))/(256*d)