\(\int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\) [462]

Optimal result

Integrand size = 31, antiderivative size = 291 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {171 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{1024 d}-\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d} \]

[Out]

-171/1024*a^(3/2)*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/d-1/7*cot(d*x+c)*csc(d*x+c)^6*(a+a*sin(d*
x+c))^(3/2)/d-171/1024*a^2*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-57/512*a^2*cot(d*x+c)*csc(d*x+c)/d/(a+a*sin(d*x
+c))^(1/2)+199/640*a^2*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x+c))^(1/2)+1237/2240*a^2*cot(d*x+c)*csc(d*x+c)^3/
d/(a+a*sin(d*x+c))^(1/2)+9/40*a^2*cot(d*x+c)*csc(d*x+c)^4/d/(a+a*sin(d*x+c))^(1/2)-1/28*a*cot(d*x+c)*csc(d*x+c
)^5*(a+a*sin(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2960, 2841, 21, 2851, 2852, 212, 3123, 3054, 3059} \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {171 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{1024 d}-\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a \sin (c+d x)+a}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a \sin (c+d x)+a}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a \sin (c+d x)+a}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a \sin (c+d x)+a}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{28 d} \]

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(-171*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(1024*d) - (171*a^2*Cot[c + d*x])/(102
4*d*Sqrt[a + a*Sin[c + d*x]]) - (57*a^2*Cot[c + d*x]*Csc[c + d*x])/(512*d*Sqrt[a + a*Sin[c + d*x]]) + (199*a^2
*Cot[c + d*x]*Csc[c + d*x]^2)/(640*d*Sqrt[a + a*Sin[c + d*x]]) + (1237*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*
d*Sqrt[a + a*Sin[c + d*x]]) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^4)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c +
d*x]*Csc[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(28*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2
))/(7*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2841

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c
+ a*d))), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1
)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{
a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1
] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2851

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x]
+ Dist[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]

Rule 2852

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[-2*(
b/f), Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2960

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Dist[1/d^4, Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^
n*(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&
!IGtQ[m, 0]

Rule 3054

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d
*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x
])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n
+ 1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a
*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*
n] || EqQ[c, 0])

Rule 3059

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n +
1)*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rule 3123

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Si
n[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x]
)^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n
+ 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ
[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2,
0])

Rubi steps \begin{align*} \text {integral}& = \int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^8(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {\int \csc ^7(c+d x) \left (\frac {3 a}{2}-\frac {19}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{7 a}-\frac {1}{3} a \int \frac {\csc ^3(c+d x) \left (-\frac {11 a}{2}-\frac {11}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {\int \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)} \left (-\frac {189 a^2}{4}-\frac {201}{4} a^2 \sin (c+d x)\right ) \, dx}{42 a}+\frac {1}{6} (11 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {1}{8} (11 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {1}{560} (1237 a) \int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {1}{16} (11 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {1}{640} (1237 a) \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {1}{768} (1237 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {\left (11 a^2\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d} \\ & = -\frac {11 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {(1237 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{1024} \\ & = -\frac {11 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {(1237 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{2048} \\ & = -\frac {11 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac {\left (1237 a^2\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{1024 d} \\ & = -\frac {171 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{1024 d}-\frac {171 a^2 \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt {a+a \sin (c+d x)}}+\frac {1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{28 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.58 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.79 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {a \csc ^{22}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-306488 \cos \left (\frac {1}{2} (c+d x)\right )-177170 \cos \left (\frac {3}{2} (c+d x)\right )+6566 \cos \left (\frac {5}{2} (c+d x)\right )-219540 \cos \left (\frac {7}{2} (c+d x)\right )+33292 \cos \left (\frac {9}{2} (c+d x)\right )-3990 \cos \left (\frac {11}{2} (c+d x)\right )+11970 \cos \left (\frac {13}{2} (c+d x)\right )+306488 \sin \left (\frac {1}{2} (c+d x)\right )-209475 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+209475 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-177170 \sin \left (\frac {3}{2} (c+d x)\right )-6566 \sin \left (\frac {5}{2} (c+d x)\right )+125685 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-125685 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-219540 \sin \left (\frac {7}{2} (c+d x)\right )-33292 \sin \left (\frac {9}{2} (c+d x)\right )-41895 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+41895 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-3990 \sin \left (\frac {11}{2} (c+d x)\right )-11970 \sin \left (\frac {13}{2} (c+d x)\right )+5985 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-5985 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{35840 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^7} \]

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(a*Csc[(c + d*x)/2]^22*Sqrt[a*(1 + Sin[c + d*x])]*(-306488*Cos[(c + d*x)/2] - 177170*Cos[(3*(c + d*x))/2] + 65
66*Cos[(5*(c + d*x))/2] - 219540*Cos[(7*(c + d*x))/2] + 33292*Cos[(9*(c + d*x))/2] - 3990*Cos[(11*(c + d*x))/2
] + 11970*Cos[(13*(c + d*x))/2] + 306488*Sin[(c + d*x)/2] - 209475*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]
]*Sin[c + d*x] + 209475*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 177170*Sin[(3*(c + d*x))/2
] - 6566*Sin[(5*(c + d*x))/2] + 125685*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 125685*
Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 219540*Sin[(7*(c + d*x))/2] - 33292*Sin[(9*(c
+ d*x))/2] - 41895*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 41895*Log[1 - Cos[(c + d*x)
/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 3990*Sin[(11*(c + d*x))/2] - 11970*Sin[(13*(c + d*x))/2] + 5985*Log
[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 5985*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]
*Sin[7*(c + d*x)]))/(35840*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^7)

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.74

method	result	size
default	\(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (5985 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {13}{2}} a^{\frac {5}{2}}-39900 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {7}{2}}+5985 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{9} \left (\sin ^{7}\left (d x +c \right )\right )+98581 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {9}{2}}-95232 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {11}{2}}+1771 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {13}{2}}+39900 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {15}{2}}-5985 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {17}{2}}\right )}{35840 a^{\frac {15}{2}} \sin \left (d x +c \right )^{7} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(216\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/35840*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(15/2)*(5985*(-a*(sin(d*x+c)-1))^(13/2)*a^(5/2)-39900*(-a*
(sin(d*x+c)-1))^(11/2)*a^(7/2)+5985*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^9*sin(d*x+c)^7+98581*(-a*(sin
(d*x+c)-1))^(9/2)*a^(9/2)-95232*(-a*(sin(d*x+c)-1))^(7/2)*a^(11/2)+1771*(-a*(sin(d*x+c)-1))^(5/2)*a^(13/2)+399
00*(-a*(sin(d*x+c)-1))^(3/2)*a^(15/2)-5985*(-a*(sin(d*x+c)-1))^(1/2)*a^(17/2))/sin(d*x+c)^7/cos(d*x+c)/(a+a*si
n(d*x+c))^(1/2)/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (255) = 510\).

Time = 0.32 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.06 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {5985 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{7} + a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{5} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (5985 \, a \cos \left (d x + c\right )^{7} + 1995 \, a \cos \left (d x + c\right )^{6} - 6811 \, a \cos \left (d x + c\right )^{5} - 14633 \, a \cos \left (d x + c\right )^{4} - 5997 \, a \cos \left (d x + c\right )^{3} + 10097 \, a \cos \left (d x + c\right )^{2} + 1703 \, a \cos \left (d x + c\right ) - {\left (5985 \, a \cos \left (d x + c\right )^{6} + 3990 \, a \cos \left (d x + c\right )^{5} - 2821 \, a \cos \left (d x + c\right )^{4} + 11812 \, a \cos \left (d x + c\right )^{3} + 5815 \, a \cos \left (d x + c\right )^{2} - 4282 \, a \cos \left (d x + c\right ) - 2579 \, a\right )} \sin \left (d x + c\right ) - 2579 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{143360 \, {\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} - {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} + 3 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/143360*(5985*(a*cos(d*x + c)^8 - 4*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 - 4*a*cos(d*x + c)^2 - (a*cos(d*x +
c)^7 + a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^5 - 3*a*cos(d*x + c)^4 + 3*a*cos(d*x + c)^3 + 3*a*cos(d*x + c)^2 -
a*cos(d*x + c) - a)*sin(d*x + c) + a)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2
+ (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) +
(a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2
- 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(5985*a*cos(d*x + c)^7 + 1995*a*cos(d*x + c)^6 - 6811*a*cos(d*x +
c)^5 - 14633*a*cos(d*x + c)^4 - 5997*a*cos(d*x + c)^3 + 10097*a*cos(d*x + c)^2 + 1703*a*cos(d*x + c) - (5985*a
*cos(d*x + c)^6 + 3990*a*cos(d*x + c)^5 - 2821*a*cos(d*x + c)^4 + 11812*a*cos(d*x + c)^3 + 5815*a*cos(d*x + c)
^2 - 4282*a*cos(d*x + c) - 2579*a)*sin(d*x + c) - 2579*a)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^8 - 4*d*co
s(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 - (d*cos(d*x + c)^7 + d*cos(d*x + c)^6 - 3*d*cos(d*x +
c)^5 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^3 + 3*d*cos(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d*x + c) + d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**8*(a+a*sin(d*x+c))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{8} \,d x } \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^4*csc(d*x + c)^8, x)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.06 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (5985 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (383040 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 1276800 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1577296 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 761856 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7084 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 79800 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5985 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}\right )} \sqrt {a}}{143360 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

-1/143360*sqrt(2)*(5985*sqrt(2)*a*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c))/abs(2*sqrt(2) + 4*sin
(-1/4*pi + 1/2*d*x + 1/2*c)))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) + 4*(383040*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/
2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^13 - 1276800*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x
+ 1/2*c)^11 + 1577296*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 - 761856*a*sgn(co
s(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 + 7084*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*si
n(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 79800*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3
- 5985*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c))/(2*sin(-1/4*pi + 1/2*d*x + 1/2*c)
^2 - 1)^7)*sqrt(a)/d

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^8} \,d x \]

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^8,x)

[Out]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^8, x)