3.611 \(\int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=181 \[ \frac{3 a^3 \cos ^5(c+d x)}{5 d}+\frac{2 a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac{5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{85 a^3 x}{16} \]

[Out]

(-85*a^3*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (a^3*Cos[c + d*x])/d + (2*a^3*Cos[c + d*x]^3)/(3*d) + (3*
a^3*Cos[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (43*a^3*Cos[c + d
*x]*Sin[c + d*x])/(16*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d
)

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Rubi [A]  time = 0.252971, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2872, 3767, 8, 3768, 3770, 2638, 2635, 2633} \[ \frac{3 a^3 \cos ^5(c+d x)}{5 d}+\frac{2 a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac{5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{85 a^3 x}{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-85*a^3*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (a^3*Cos[c + d*x])/d + (2*a^3*Cos[c + d*x]^3)/(3*d) + (3*
a^3*Cos[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (43*a^3*Cos[c + d
*x]*Sin[c + d*x])/(16*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d
)

Rule 2872

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2638

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

Rule 2635

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

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\int \left (-8 a^9+3 a^9 \csc ^2(c+d x)+a^9 \csc ^3(c+d x)-6 a^9 \sin (c+d x)+6 a^9 \sin ^2(c+d x)+8 a^9 \sin ^3(c+d x)-3 a^9 \sin ^5(c+d x)-a^9 \sin ^6(c+d x)\right ) \, dx}{a^6}\\ &=-8 a^3 x+a^3 \int \csc ^3(c+d x) \, dx-a^3 \int \sin ^6(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^5(c+d x) \, dx-\left (6 a^3\right ) \int \sin (c+d x) \, dx+\left (6 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (8 a^3\right ) \int \sin ^3(c+d x) \, dx\\ &=-8 a^3 x+\frac{6 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{d}+\frac{a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac{1}{2} a^3 \int \csc (c+d x) \, dx-\frac{1}{6} \left (5 a^3\right ) \int \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int 1 \, dx-\frac{\left (3 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (8 a^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-5 a^3 x-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{a^3 \cos (c+d x)}{d}+\frac{2 a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{d}+\frac{5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac{a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac{1}{8} \left (5 a^3\right ) \int \sin ^2(c+d x) \, dx\\ &=-5 a^3 x-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{a^3 \cos (c+d x)}{d}+\frac{2 a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac{a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac{1}{16} \left (5 a^3\right ) \int 1 \, dx\\ &=-\frac{85 a^3 x}{16}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{a^3 \cos (c+d x)}{d}+\frac{2 a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac{a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}\\ \end{align*}

Mathematica [B]  time = 6.37225, size = 664, normalized size = 3.67 \[ -\frac{81 \sin (2 (c+d x)) (a \sin (c+d x)+a)^3}{64 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}-\frac{3 \sin (4 (c+d x)) (a \sin (c+d x)+a)^3}{64 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}+\frac{\sin (6 (c+d x)) (a \sin (c+d x)+a)^3}{192 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}-\frac{85 (c+d x) (a \sin (c+d x)+a)^3}{16 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}+\frac{15 \cos (c+d x) (a \sin (c+d x)+a)^3}{8 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}+\frac{17 \cos (3 (c+d x)) (a \sin (c+d x)+a)^3}{48 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}+\frac{3 \cos (5 (c+d x)) (a \sin (c+d x)+a)^3}{80 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}-\frac{(a \sin (c+d x)+a)^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}+\frac{(a \sin (c+d x)+a)^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}+\frac{3 \tan \left (\frac{1}{2} (c+d x)\right ) (a \sin (c+d x)+a)^3}{2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}-\frac{3 \cot \left (\frac{1}{2} (c+d x)\right ) (a \sin (c+d x)+a)^3}{2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \sin (c+d x)+a)^3}{8 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}+\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sin (c+d x)+a)^3}{8 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[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]

[Out]

(-85*(c + d*x)*(a + a*Sin[c + d*x])^3)/(16*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (15*Cos[c + d*x]*(a +
a*Sin[c + d*x])^3)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (17*Cos[3*(c + d*x)]*(a + a*Sin[c + d*x])^3
)/(48*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (3*Cos[5*(c + d*x)]*(a + a*Sin[c + d*x])^3)/(80*d*(Cos[(c +
d*x)/2] + Sin[(c + d*x)/2])^6) - (3*Cot[(c + d*x)/2]*(a + a*Sin[c + d*x])^3)/(2*d*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^6) - (Csc[(c + d*x)/2]^2*(a + a*Sin[c + d*x])^3)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (L
og[Cos[(c + d*x)/2]]*(a + a*Sin[c + d*x])^3)/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (Log[Sin[(c + d*x
)/2]]*(a + a*Sin[c + d*x])^3)/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (Sec[(c + d*x)/2]^2*(a + a*Sin[c
+ d*x])^3)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (81*(a + a*Sin[c + d*x])^3*Sin[2*(c + d*x)])/(64*d
*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (3*(a + a*Sin[c + d*x])^3*Sin[4*(c + d*x)])/(64*d*(Cos[(c + d*x)/2
] + Sin[(c + d*x)/2])^6) + ((a + a*Sin[c + d*x])^3*Sin[6*(c + d*x)])/(192*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/
2])^6) + (3*(a + a*Sin[c + d*x])^3*Tan[(c + d*x)/2])/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

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Maple [A]  time = 0.086, size = 199, normalized size = 1.1 \begin{align*} -{\frac{17\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{6\,d}}-{\frac{85\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{24\,d}}-{\frac{85\,{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{16\,d}}-{\frac{85\,{a}^{3}x}{16}}-{\frac{85\,{a}^{3}c}{16\,d}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{10\,d}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{6\,d}}+{\frac{{a}^{3}\cos \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}-3\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d\sin \left ( dx+c \right ) }}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-17/6*a^3*cos(d*x+c)^5*sin(d*x+c)/d-85/24*a^3*cos(d*x+c)^3*sin(d*x+c)/d-85/16*a^3*cos(d*x+c)*sin(d*x+c)/d-85/1
6*a^3*x-85/16/d*a^3*c+1/10*a^3*cos(d*x+c)^5/d+1/6*a^3*cos(d*x+c)^3/d+1/2*a^3*cos(d*x+c)/d+1/2/d*a^3*ln(csc(d*x
+c)-cot(d*x+c))-3/d*a^3/sin(d*x+c)*cos(d*x+c)^7-1/2/d*a^3/sin(d*x+c)^2*cos(d*x+c)^7

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Maxima [A]  time = 1.71735, size = 323, normalized size = 1.78 \begin{align*} \frac{96 \,{\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 80 \,{\left (4 \, \cos \left (d x + c\right )^{3} - \frac{6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 5 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 360 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3}}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/960*(96*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x
+ c) - 1))*a^3 - 80*(4*cos(d*x + c)^3 - 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*cos(d*x + c) - 15*log(cos(d*x
+ c) + 1) + 15*log(cos(d*x + c) - 1))*a^3 - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48
*sin(2*d*x + 2*c))*a^3 - 360*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(tan(d*x + c)^5 + 2*
tan(d*x + c)^3 + tan(d*x + c)))*a^3)/d

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Fricas [A]  time = 1.23528, size = 544, normalized size = 3.01 \begin{align*} \frac{144 \, a^{3} \cos \left (d x + c\right )^{7} + 16 \, a^{3} \cos \left (d x + c\right )^{5} - 1275 \, a^{3} d x \cos \left (d x + c\right )^{2} + 80 \, a^{3} \cos \left (d x + c\right )^{3} + 1275 \, a^{3} d x - 120 \, a^{3} \cos \left (d x + c\right ) - 60 \,{\left (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 ) + 60 \,{\left (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 ) + 5 \,{\left (8 \, a^{3} \cos \left (d x + c\right )^{7} - 34 \, a^{3} \cos \left (d x + c\right )^{5} - 85 \, a^{3} \cos \left (d x + c\right )^{3} + 255 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \,{\left (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)^6*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/240*(144*a^3*cos(d*x + c)^7 + 16*a^3*cos(d*x + c)^5 - 1275*a^3*d*x*cos(d*x + c)^2 + 80*a^3*cos(d*x + c)^3 +
1275*a^3*d*x - 120*a^3*cos(d*x + c) - 60*(a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) + 60*(a^3*cos(
d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1/2) + 5*(8*a^3*cos(d*x + c)^7 - 34*a^3*cos(d*x + c)^5 - 85*a^3*cos(
d*x + c)^3 + 255*a^3*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d)

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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)**3*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.37932, size = 413, normalized size = 2.28 \begin{align*} \frac{30 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1275 \,{\left (d x + c\right )} a^{3} + 120 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 360 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{30 \,{\left (6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} + \frac{2 \,{\left (645 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1440 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 1735 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 3360 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 450 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 5440 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 450 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4800 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1735 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1824 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 645 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 544 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(30*a^3*tan(1/2*d*x + 1/2*c)^2 - 1275*(d*x + c)*a^3 + 120*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 360*a^3*t
an(1/2*d*x + 1/2*c) - 30*(6*a^3*tan(1/2*d*x + 1/2*c)^2 + 12*a^3*tan(1/2*d*x + 1/2*c) + a^3)/tan(1/2*d*x + 1/2*
c)^2 + 2*(645*a^3*tan(1/2*d*x + 1/2*c)^11 + 1440*a^3*tan(1/2*d*x + 1/2*c)^10 + 1735*a^3*tan(1/2*d*x + 1/2*c)^9
+ 3360*a^3*tan(1/2*d*x + 1/2*c)^8 + 450*a^3*tan(1/2*d*x + 1/2*c)^7 + 5440*a^3*tan(1/2*d*x + 1/2*c)^6 - 450*a^
3*tan(1/2*d*x + 1/2*c)^5 + 4800*a^3*tan(1/2*d*x + 1/2*c)^4 - 1735*a^3*tan(1/2*d*x + 1/2*c)^3 + 1824*a^3*tan(1/
2*d*x + 1/2*c)^2 - 645*a^3*tan(1/2*d*x + 1/2*c) + 544*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d