3.603 \(\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=194 \[ -\frac{a^2 \cot ^{11}(c+d x)}{11 d}-\frac{a^2 \cot ^9(c+d x)}{3 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]

[Out]

(3*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(3*d) - (a^2*Cot[c
+ d*x]^11)/(11*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a
^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a^2*Cot[c + d*x]^5*Csc[c
+ d*x]^5)/(5*d)

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Rubi [A]  time = 0.301631, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2607, 14, 2611, 3768, 3770, 270} \[ -\frac{a^2 \cot ^{11}(c+d x)}{11 d}-\frac{a^2 \cot ^9(c+d x)}{3 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(3*d) - (a^2*Cot[c
+ d*x]^11)/(11*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a
^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a^2*Cot[c + d*x]^5*Csc[c
+ d*x]^5)/(5*d)

Rule 2873

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 2607

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 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 2611

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 3768

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 3770

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

Rule 270

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]

Rubi steps

\begin{align*} \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x) \csc ^4(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^5(c+d x)+a^2 \cot ^6(c+d x) \csc ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac{a^2 \operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{a^2 \operatorname{Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{1}{8} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{a^2 \operatorname{Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{a^2 \operatorname{Subst}\left (\int \left (x^6+2 x^8+x^{10}\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)}{3 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 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{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac{1}{16} a^2 \int \csc ^5(c+d x) \, dx\\ &=-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^9(c+d x)}{3 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{64 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{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac{1}{64} \left (3 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)}{3 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{64 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{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac{1}{128} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^9(c+d x)}{3 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{64 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{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 3.03577, size = 187, normalized size = 0.96 \[ \frac{a^2 (\sin (c+d x)+1)^2 \left (887040 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (1073226 \sin (c+d x)+869484 \sin (3 (c+d x))+727188 \sin (5 (c+d x))+40425 \sin (7 (c+d x))-3465 \sin (9 (c+d x))+1798400 \cos (2 (c+d x))+440320 \cos (4 (c+d x))-81280 \cos (6 (c+d x))-38400 \cos (8 (c+d x))+3200 \cos (10 (c+d x))+1318400)\right )}{37847040 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(1 + Sin[c + d*x])^2*(887040*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^
10*(1318400 + 1798400*Cos[2*(c + d*x)] + 440320*Cos[4*(c + d*x)] - 81280*Cos[6*(c + d*x)] - 38400*Cos[8*(c + d
*x)] + 3200*Cos[10*(c + d*x)] + 1073226*Sin[c + d*x] + 869484*Sin[3*(c + d*x)] + 727188*Sin[5*(c + d*x)] + 404
25*Sin[7*(c + d*x)] - 3465*Sin[9*(c + d*x)])))/(37847040*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)

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Maple [A]  time = 0.087, size = 264, normalized size = 1.4 \begin{align*} -{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{33\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{10\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{231\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{40\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{320\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{640\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{640\,d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{128\,d}}-{\frac{3\,{a}^{2}\cos \left ( dx+c \right ) }{128\,d}}-{\frac{3\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{11\,d \left ( \sin \left ( dx+c \right ) \right ) ^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^2,x)

[Out]

-5/33/d*a^2/sin(d*x+c)^9*cos(d*x+c)^7-10/231/d*a^2/sin(d*x+c)^7*cos(d*x+c)^7-1/5/d*a^2/sin(d*x+c)^10*cos(d*x+c
)^7-3/40/d*a^2/sin(d*x+c)^8*cos(d*x+c)^7-1/80/d*a^2/sin(d*x+c)^6*cos(d*x+c)^7+1/320/d*a^2/sin(d*x+c)^4*cos(d*x
+c)^7-3/640/d*a^2/sin(d*x+c)^2*cos(d*x+c)^7-3/640*a^2*cos(d*x+c)^5/d-1/128*a^2*cos(d*x+c)^3/d-3/128*a^2*cos(d*
x+c)/d-3/128/d*a^2*ln(csc(d*x+c)-cot(d*x+c))-1/11/d*a^2/sin(d*x+c)^11*cos(d*x+c)^7

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Maxima [A]  time = 1.06697, size = 266, normalized size = 1.37 \begin{align*} -\frac{693 \, 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 )} + \frac{14080 \,{\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}} + \frac{1280 \,{\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{2}}{\tan \left (d x + c\right )^{11}}}{887040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/887040*(693*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)) + 14080*(9*tan(d*x + c)^2 + 7)*a^2/tan(d*x + c)^9 + 12
80*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^2/tan(d*x + c)^11)/d

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Fricas [B]  time = 1.28571, size = 940, normalized size = 4.85 \begin{align*} \frac{12800 \, a^{2} \cos \left (d x + c\right )^{11} - 70400 \, a^{2} \cos \left (d x + c\right )^{9} + 84480 \, a^{2} \cos \left (d x + c\right )^{7} + 3465 \,{\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 ) \sin \left (d x + c\right ) - 3465 \,{\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 ) \sin \left (d x + c\right ) - 462 \,{\left (15 \, a^{2} \cos \left (d x + c\right )^{9} - 70 \, a^{2} \cos \left (d x + c\right )^{7} - 128 \, a^{2} \cos \left (d x + c\right )^{5} + 70 \, 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 )}{295680 \,{\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/295680*(12800*a^2*cos(d*x + c)^11 - 70400*a^2*cos(d*x + c)^9 + 84480*a^2*cos(d*x + c)^7 + 3465*(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)*l
og(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3465*(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)*sin(d*x + c) - 462*(15
*a^2*cos(d*x + c)^9 - 70*a^2*cos(d*x + c)^7 - 128*a^2*cos(d*x + c)^5 + 70*a^2*cos(d*x + c)^3 - 15*a^2*cos(d*x
+ c))*sin(d*x + c))/((d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d
*cos(d*x + c)^2 - d)*sin(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**12*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.36363, size = 524, normalized size = 2.7 \begin{align*} \frac{105 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 462 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 385 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1155 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 2805 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 2310 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1155 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9240 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 16170 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4620 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 55440 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 39270 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{167422 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 39270 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 4620 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 16170 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 9240 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1155 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 2310 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2805 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1155 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 385 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 462 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 105 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11}}}{2365440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/2365440*(105*a^2*tan(1/2*d*x + 1/2*c)^11 + 462*a^2*tan(1/2*d*x + 1/2*c)^10 + 385*a^2*tan(1/2*d*x + 1/2*c)^9
- 1155*a^2*tan(1/2*d*x + 1/2*c)^8 - 2805*a^2*tan(1/2*d*x + 1/2*c)^7 - 2310*a^2*tan(1/2*d*x + 1/2*c)^6 + 1155*a
^2*tan(1/2*d*x + 1/2*c)^5 + 9240*a^2*tan(1/2*d*x + 1/2*c)^4 + 16170*a^2*tan(1/2*d*x + 1/2*c)^3 + 4620*a^2*tan(
1/2*d*x + 1/2*c)^2 - 55440*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 39270*a^2*tan(1/2*d*x + 1/2*c) + (167422*a^2*t
an(1/2*d*x + 1/2*c)^11 + 39270*a^2*tan(1/2*d*x + 1/2*c)^10 - 4620*a^2*tan(1/2*d*x + 1/2*c)^9 - 16170*a^2*tan(1
/2*d*x + 1/2*c)^8 - 9240*a^2*tan(1/2*d*x + 1/2*c)^7 - 1155*a^2*tan(1/2*d*x + 1/2*c)^6 + 2310*a^2*tan(1/2*d*x +
1/2*c)^5 + 2805*a^2*tan(1/2*d*x + 1/2*c)^4 + 1155*a^2*tan(1/2*d*x + 1/2*c)^3 - 385*a^2*tan(1/2*d*x + 1/2*c)^2
- 462*a^2*tan(1/2*d*x + 1/2*c) - 105*a^2)/tan(1/2*d*x + 1/2*c)^11)/d