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

Optimal. Leaf size=200 \[ -\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{55 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{25 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.358502, antiderivative size = 200, 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, 2611, 3770, 2607, 30, 3768, 14} \[ -\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{55 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{25 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]

Antiderivative was successfully verified.

[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 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 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 3770

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

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 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 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 ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\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 \operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{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 \operatorname{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 \tanh ^{-1}(\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 \tanh ^{-1}(\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]  time = 0.141289, size = 459, normalized size = 2.3 \[ a^3 \left (-\frac{29 \tan \left (\frac{1}{2} (c+d x)\right )}{126 d}+\frac{29 \cot \left (\frac{1}{2} (c+d x)\right )}{126 d}-\frac{3 \csc ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}+\frac{17 \csc ^6\left (\frac{1}{2} (c+d x)\right )}{1536 d}-\frac{13 \csc ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}-\frac{73 \csc ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}+\frac{3 \sec ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}-\frac{17 \sec ^6\left (\frac{1}{2} (c+d x)\right )}{1536 d}+\frac{13 \sec ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}+\frac{73 \sec ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}-\frac{55 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{128 d}+\frac{55 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{128 d}-\frac{\cot \left (\frac{1}{2} (c+d x)\right ) \csc ^8\left (\frac{1}{2} (c+d x)\right )}{4608 d}-\frac{53 \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^6\left (\frac{1}{2} (c+d x)\right )}{32256 d}+\frac{319 \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^4\left (\frac{1}{2} (c+d x)\right )}{10752 d}-\frac{4163 \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{32256 d}+\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^8\left (\frac{1}{2} (c+d x)\right )}{4608 d}+\frac{53 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6\left (\frac{1}{2} (c+d x)\right )}{32256 d}-\frac{319 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )}{10752 d}+\frac{4163 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{32256 d}\right ) \]

Antiderivative was successfully verified.

[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]  time = 0.096, size = 216, normalized size = 1.1 \begin{align*} -{\frac{11\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{11\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{192\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{11\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{11\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d}}-{\frac{55\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{384\,d}}-{\frac{55\,{a}^{3}\cos \left ( dx+c \right ) }{128\,d}}-{\frac{55\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{29\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/48/d*a^3/sin(d*x+c)^6*cos(d*x+c)^7+11/192/d*a^3/sin(d*x+c)^4*cos(d*x+c)^7-11/128/d*a^3/sin(d*x+c)^2*cos(d*
x+c)^7-11/128*a^3*cos(d*x+c)^5/d-55/384*a^3*cos(d*x+c)^3/d-55/128*a^3*cos(d*x+c)/d-55/128/d*a^3*ln(csc(d*x+c)-
cot(d*x+c))-29/63/d*a^3/sin(d*x+c)^7*cos(d*x+c)^7-3/8/d*a^3/sin(d*x+c)^8*cos(d*x+c)^7-1/9/d*a^3/sin(d*x+c)^9*c
os(d*x+c)^7

________________________________________________________________________________________

Maxima [A]  time = 1.08448, size = 332, normalized size = 1.66 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 1.22804, size = 765, normalized size = 3.82 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  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)**10*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.41675, size = 437, normalized size = 2.18 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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