3.404 \(\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=176 \[ -\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]

[Out]

(-27*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (27*a^
3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (23*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^3*Cot[c + d*x]^3*Csc[c
+ d*x]^3)/(2*d) + (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)

________________________________________________________________________________________

Rubi [A]  time = 0.338176, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2607, 30, 2611, 3768, 3770, 14} \[ -\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(-27*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (27*a^
3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (23*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^3*Cot[c + d*x]^3*Csc[c
+ d*x]^3)/(2*d) + (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*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 30

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

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 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]]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^4(c+d x)+a^3 \cot ^4(c+d x) \csc ^5(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{1}{8} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx-\frac{1}{2} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{16} a^3 \int \csc ^5(c+d x) \, dx+\frac{1}{8} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{64} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac{1}{16} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{128} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 5.16025, size = 313, normalized size = 1.78 \[ -\frac{a^3 \sin (c+d x) (\sin (c+d x)+1)^3 \left (10 (7 \csc (c+d x)+24) \csc ^8\left (\frac{1}{2} (c+d x)\right )+8 (105 \csc (c+d x)-76) \csc ^6\left (\frac{1}{2} (c+d x)\right )-4 (1715 \csc (c+d x)+856) \csc ^4\left (\frac{1}{2} (c+d x)\right )+8 (945 \csc (c+d x)+1664) \csc ^2\left (\frac{1}{2} (c+d x)\right )-4 \left ((1056 \cos (c+d x)+517 \cos (2 (c+d x))+104 \cos (3 (c+d x))+703) \sec ^8\left (\frac{1}{2} (c+d x)\right )+4480 \sin ^8\left (\frac{1}{2} (c+d x)\right ) \csc ^9(c+d x)+13440 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^7(c+d x)-27440 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)+7560 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-7560 \csc (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )\right )}{143360 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

-(a^3*(10*Csc[(c + d*x)/2]^8*(24 + 7*Csc[c + d*x]) + 8*Csc[(c + d*x)/2]^6*(-76 + 105*Csc[c + d*x]) + 8*Csc[(c
+ d*x)/2]^2*(1664 + 945*Csc[c + d*x]) - 4*Csc[(c + d*x)/2]^4*(856 + 1715*Csc[c + d*x]) - 4*(-7560*Csc[c + d*x]
*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + (703 + 1056*Cos[c + d*x] + 517*Cos[2*(c + d*x)] + 104*Cos[3
*(c + d*x)])*Sec[(c + d*x)/2]^8 + 7560*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 27440*Csc[c + d*x]^5*Sin[(c + d*x)/
2]^4 + 13440*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6 + 4480*Csc[c + d*x]^9*Sin[(c + d*x)/2]^8))*Sin[c + d*x]*(1 + Si
n[c + d*x])^3)/(143360*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

________________________________________________________________________________________

Maple [A]  time = 0.101, size = 200, normalized size = 1.1 \begin{align*} -{\frac{13\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{64\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{128\,d}}+{\frac{27\,{a}^{3}\cos \left ( dx+c \right ) }{128\,d}}+{\frac{27\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x)

[Out]

-13/35/d*a^3/sin(d*x+c)^5*cos(d*x+c)^5-9/16/d*a^3/sin(d*x+c)^6*cos(d*x+c)^5-9/64/d*a^3/sin(d*x+c)^4*cos(d*x+c)
^5+9/128/d*a^3/sin(d*x+c)^2*cos(d*x+c)^5+9/128*a^3*cos(d*x+c)^3/d+27/128*a^3*cos(d*x+c)/d+27/128/d*a^3*ln(csc(
d*x+c)-cot(d*x+c))-3/7/d*a^3/sin(d*x+c)^7*cos(d*x+c)^5-1/8/d*a^3/sin(d*x+c)^8*cos(d*x+c)^5

________________________________________________________________________________________

Maxima [A]  time = 1.11625, size = 332, normalized size = 1.89 \begin{align*} \frac{35 \, 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 )} + 280 \, a^{3}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \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} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac{1792 \, a^{3}}{\tan \left (d x + c\right )^{5}} - \frac{768 \,{\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{8960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8960*(35*a^3*(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)) + 280*a^3*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 +
3*cos(d*x + c)^2 - 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 1792*a^3/tan(d*x + c)^5 - 768*(7*
tan(d*x + c)^2 + 5)*a^3/tan(d*x + c)^7)/d

________________________________________________________________________________________

Fricas [A]  time = 1.59157, size = 707, normalized size = 4.02 \begin{align*} \frac{1890 \, a^{3} \cos \left (d x + c\right )^{7} + 2030 \, a^{3} \cos \left (d x + c\right )^{5} - 6930 \, a^{3} \cos \left (d x + c\right )^{3} + 1890 \, a^{3} \cos \left (d x + c\right ) - 945 \,{\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 ) + 945 \,{\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 ) + 256 \,{\left (13 \, a^{3} \cos \left (d x + c\right )^{7} - 28 \, a^{3} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{8960 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/8960*(1890*a^3*cos(d*x + c)^7 + 2030*a^3*cos(d*x + c)^5 - 6930*a^3*cos(d*x + c)^3 + 1890*a^3*cos(d*x + c) -
945*(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*co
s(d*x + c) + 1/2) + 945*(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) + 256*(13*a^3*cos(d*x + c)^7 - 28*a^3*cos(d*x + c)^5)*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)

________________________________________________________________________________________

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)**4*csc(d*x+c)**9*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.61677, size = 396, normalized size = 2.25 \begin{align*} \frac{35 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 240 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 560 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 112 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1960 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3920 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1680 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15120 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 9520 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{41094 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 9520 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1680 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 3920 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1960 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 112 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 560 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 240 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 35 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8}}}{71680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/71680*(35*a^3*tan(1/2*d*x + 1/2*c)^8 + 240*a^3*tan(1/2*d*x + 1/2*c)^7 + 560*a^3*tan(1/2*d*x + 1/2*c)^6 + 112
*a^3*tan(1/2*d*x + 1/2*c)^5 - 1960*a^3*tan(1/2*d*x + 1/2*c)^4 - 3920*a^3*tan(1/2*d*x + 1/2*c)^3 - 1680*a^3*tan
(1/2*d*x + 1/2*c)^2 + 15120*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 9520*a^3*tan(1/2*d*x + 1/2*c) - (41094*a^3*ta
n(1/2*d*x + 1/2*c)^8 + 9520*a^3*tan(1/2*d*x + 1/2*c)^7 - 1680*a^3*tan(1/2*d*x + 1/2*c)^6 - 3920*a^3*tan(1/2*d*
x + 1/2*c)^5 - 1960*a^3*tan(1/2*d*x + 1/2*c)^4 + 112*a^3*tan(1/2*d*x + 1/2*c)^3 + 560*a^3*tan(1/2*d*x + 1/2*c)
^2 + 240*a^3*tan(1/2*d*x + 1/2*c) + 35*a^3)/tan(1/2*d*x + 1/2*c)^8)/d