3.463 \(\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=329 \[ -\frac {1587 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{16384 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a \sin (c+d x)+a}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a \sin (c+d x)+a}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a \sin (c+d x)+a}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a \sin (c+d x)+a}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a \sin (c+d x)+a}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{112 d} \]

[Out]

-1587/16384*a^(3/2)*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/d-1/8*cot(d*x+c)*csc(d*x+c)^7*(a+a*sin(
d*x+c))^(3/2)/d-1587/16384*a^2*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-529/8192*a^2*cot(d*x+c)*csc(d*x+c)/d/(a+a*s
in(d*x+c))^(1/2)-529/10240*a^2*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x+c))^(1/2)+8653/35840*a^2*cot(d*x+c)*csc(
d*x+c)^3/d/(a+a*sin(d*x+c))^(1/2)+1957/4480*a^2*cot(d*x+c)*csc(d*x+c)^4/d/(a+a*sin(d*x+c))^(1/2)+83/448*a^2*co
t(d*x+c)*csc(d*x+c)^5/d/(a+a*sin(d*x+c))^(1/2)-3/112*a*cot(d*x+c)*csc(d*x+c)^6*(a+a*sin(d*x+c))^(1/2)/d

________________________________________________________________________________________

Rubi [A]  time = 1.19, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2881, 2762, 21, 2772, 2773, 206, 3044, 2975, 2980} \[ -\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a \sin (c+d x)+a}}-\frac {1587 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{16384 d}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a \sin (c+d x)+a}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a \sin (c+d x)+a}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a \sin (c+d x)+a}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a \sin (c+d x)+a}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{112 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(-1587*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(16384*d) - (1587*a^2*Cot[c + d*x])/(
16384*d*Sqrt[a + a*Sin[c + d*x]]) - (529*a^2*Cot[c + d*x]*Csc[c + d*x])/(8192*d*Sqrt[a + a*Sin[c + d*x]]) - (5
29*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(10240*d*Sqrt[a + a*Sin[c + d*x]]) + (8653*a^2*Cot[c + d*x]*Csc[c + d*x]^3
)/(35840*d*Sqrt[a + a*Sin[c + d*x]]) + (1957*a^2*Cot[c + d*x]*Csc[c + d*x]^4)/(4480*d*Sqrt[a + a*Sin[c + d*x]]
) + (83*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(448*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Csc[c + d*x]^6*S
qrt[a + a*Sin[c + d*x]])/(112*d) - (Cot[c + d*x]*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2))/(8*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 206

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

Rule 2762

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(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 2772

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 2773

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 2881

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 2975

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*S
in[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 2980

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 3044

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*Sin[
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*} \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^9(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 ^3(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {\int \csc ^8(c+d x) \left (\frac {3 a}{2}-\frac {21}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{8 a}-\frac {1}{4} a \int \frac {\csc ^4(c+d x) \left (-\frac {15 a}{2}-\frac {15}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {\int \csc ^7(c+d x) \sqrt {a+a \sin (c+d x)} \left (-\frac {249 a^2}{4}-\frac {261}{4} a^2 \sin (c+d x)\right ) \, dx}{56 a}+\frac {1}{8} (15 a) \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {5 a^2 \cot (c+d x) \csc ^2(c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {1}{16} (25 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {1}{896} (1957 a) \int \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {25 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a^2 \cot (c+d x) \csc ^2(c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {1}{64} (75 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {(17613 a) \int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{8960}\\ &=-\frac {75 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {25 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a^2 \cot (c+d x) \csc ^2(c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {1}{128} (75 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {(17613 a) \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{10240}\\ &=-\frac {75 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {25 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}-\frac {(5871 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{4096}-\frac {\left (75 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}\\ &=-\frac {75 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {75 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}-\frac {(17613 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{16384}\\ &=-\frac {75 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}-\frac {(17613 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{32768}\\ &=-\frac {75 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {\left (17613 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{16384 d}\\ &=-\frac {1587 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{16384 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}\\ \end {align*}

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Mathematica [B]  time = 6.26, size = 2303, normalized size = 7.00 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(6053*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (6053*Cot[(c + d*x)/4
]*(a*(1 + Sin[c + d*x]))^(3/2))/(286720*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (179*Csc[(c + d*x)/4]^2*(
a*(1 + Sin[c + d*x]))^(3/2))/(131072*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (107*Cot[(c + d*x)/4]*Csc[(c
+ d*x)/4]^2*(a*(1 + Sin[c + d*x]))^(3/2))/(573440*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (113*Csc[(c +
d*x)/4]^4*(a*(1 + Sin[c + d*x]))^(3/2))/(262144*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (31*Cot[(c + d*x)
/4]*Csc[(c + d*x)/4]^4*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (Csc
[(c + d*x)/4]^6*(a*(1 + Sin[c + d*x]))^(3/2))/(131072*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (3*Cot[(c +
d*x)/4]*Csc[(c + d*x)/4]^6*(a*(1 + Sin[c + d*x]))^(3/2))/(229376*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) -
(Csc[(c + d*x)/4]^8*(a*(1 + Sin[c + d*x]))^(3/2))/(524288*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (1587*
Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*(1 + Sin[c + d*x]))^(3/2))/(32768*d*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^3) + (1587*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*(1 + Sin[c + d*x]))^(3/2))/(32768*d*(Co
s[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (179*Sec[(c + d*x)/4]^2*(a*(1 + Sin[c + d*x]))^(3/2))/(131072*d*(Cos[(
c + d*x)/2] + Sin[(c + d*x)/2])^3) - (113*Sec[(c + d*x)/4]^4*(a*(1 + Sin[c + d*x]))^(3/2))/(262144*d*(Cos[(c +
d*x)/2] + Sin[(c + d*x)/2])^3) - (Sec[(c + d*x)/4]^6*(a*(1 + Sin[c + d*x]))^(3/2))/(131072*d*(Cos[(c + d*x)/2
] + Sin[(c + d*x)/2])^3) + (Sec[(c + d*x)/4]^8*(a*(1 + Sin[c + d*x]))^(3/2))/(524288*d*(Cos[(c + d*x)/2] + Sin
[(c + d*x)/2])^3) + (a*(1 + Sin[c + d*x]))^(3/2)/(32768*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^8*(Cos[(c + d*
x)/2] + Sin[(c + d*x)/2])^3) + (5*(a*(1 + Sin[c + d*x]))^(3/2))/(114688*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]
)^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (5939*(a*(1 + Sin[c + d*x]))^(3/2))/(2293760*d*(Cos[(c + d*x)/4
] - Sin[(c + d*x)/4])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (5409*(a*(1 + Sin[c + d*x]))^(3/2))/(229376
0*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (3*Sin[(c + d*x)/4]*(a*
(1 + Sin[c + d*x]))^(3/2))/(14336*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^7*(Cos[(c + d*x)/2] + Sin[(c + d*x)/
2])^3) - (31*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(17920*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^5*(
Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (107*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(
c + d*x)/4] - Sin[(c + d*x)/4])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (6053*Sin[(c + d*x)/4]*(a*(1 + Si
n[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) -
(a*(1 + Sin[c + d*x]))^(3/2)/(32768*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^8*(Cos[(c + d*x)/2] + Sin[(c + d*
x)/2])^3) - (3*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(14336*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^7
*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (19*(a*(1 + Sin[c + d*x]))^(3/2))/(114688*d*(Cos[(c + d*x)/4] + Si
n[(c + d*x)/4])^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (31*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2)
)/(17920*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^5*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (1971*(a*(1 + Si
n[c + d*x]))^(3/2))/(2293760*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3
) + (107*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^3*(Cos
[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (7121*(a*(1 + Sin[c + d*x]))^(3/2))/(2293760*d*(Cos[(c + d*x)/4] + Sin[
(c + d*x)/4])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (6053*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2)
)/(143360*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (6053*(a*(1 + Sin
[c + d*x]))^(3/2)*Tan[(c + d*x)/4])/(286720*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (107*Sec[(c + d*x)/4]
^2*(a*(1 + Sin[c + d*x]))^(3/2)*Tan[(c + d*x)/4])/(573440*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (31*Sec
[(c + d*x)/4]^4*(a*(1 + Sin[c + d*x]))^(3/2)*Tan[(c + d*x)/4])/(143360*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])
^3) - (3*Sec[(c + d*x)/4]^6*(a*(1 + Sin[c + d*x]))^(3/2)*Tan[(c + d*x)/4])/(229376*d*(Cos[(c + d*x)/2] + Sin[(
c + d*x)/2])^3)

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fricas [B]  time = 0.51, size = 657, normalized size = 2.00 \[ \frac {55545 \, {\left (a \cos \left (d x + c\right )^{9} + a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{7} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{5} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{3} - 4 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + {\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} + 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 (55545 \, a \cos \left (d x + c\right )^{8} + 37030 \, a \cos \left (d x + c\right )^{7} - 214774 \, a \cos \left (d x + c\right )^{6} + 27358 \, a \cos \left (d x + c\right )^{5} + 199004 \, a \cos \left (d x + c\right )^{4} - 185006 \, a \cos \left (d x + c\right )^{3} - 153786 \, a \cos \left (d x + c\right )^{2} + 48938 \, a \cos \left (d x + c\right ) + {\left (55545 \, a \cos \left (d x + c\right )^{7} + 18515 \, a \cos \left (d x + c\right )^{6} - 196259 \, a \cos \left (d x + c\right )^{5} - 223617 \, a \cos \left (d x + c\right )^{4} - 24613 \, a \cos \left (d x + c\right )^{3} + 160393 \, a \cos \left (d x + c\right )^{2} + 6607 \, a \cos \left (d x + c\right ) - 42331 \, a\right )} \sin \left (d x + c\right ) + 42331 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2293760 \, {\left (d \cos \left (d x + c\right )^{9} + d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{7} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{5} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{3} - 4 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + {\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 ) + d\right )}} \]

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))^(3/2),x, algorithm="fricas")

[Out]

1/2293760*(55545*(a*cos(d*x + c)^9 + a*cos(d*x + c)^8 - 4*a*cos(d*x + c)^7 - 4*a*cos(d*x + c)^6 + 6*a*cos(d*x
+ c)^5 + 6*a*cos(d*x + c)^4 - 4*a*cos(d*x + c)^3 - 4*a*cos(d*x + c)^2 + a*cos(d*x + c) + (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)*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*(55545*a*cos(d*x + c)^
8 + 37030*a*cos(d*x + c)^7 - 214774*a*cos(d*x + c)^6 + 27358*a*cos(d*x + c)^5 + 199004*a*cos(d*x + c)^4 - 1850
06*a*cos(d*x + c)^3 - 153786*a*cos(d*x + c)^2 + 48938*a*cos(d*x + c) + (55545*a*cos(d*x + c)^7 + 18515*a*cos(d
*x + c)^6 - 196259*a*cos(d*x + c)^5 - 223617*a*cos(d*x + c)^4 - 24613*a*cos(d*x + c)^3 + 160393*a*cos(d*x + c)
^2 + 6607*a*cos(d*x + c) - 42331*a)*sin(d*x + c) + 42331*a)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^9 + d*co
s(d*x + c)^8 - 4*d*cos(d*x + c)^7 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^5 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x
+ c)^3 - 4*d*cos(d*x + c)^2 + d*cos(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) + d)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 1.88, size = 234, normalized size = 0.71 \[ \frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (55545 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {15}{2}} a^{\frac {7}{2}}-425845 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {13}{2}} a^{\frac {9}{2}}+1418249 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {11}{2}}-55545 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{11} \left (\sin ^{8}\left (d x +c \right )\right )-2509197 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {13}{2}}+2176627 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {15}{2}}-416759 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {17}{2}}-425845 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {19}{2}}+55545 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {21}{2}}\right )}{573440 a^{\frac {19}{2}} \sin \left (d x +c \right )^{8} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]

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))^(3/2),x)

[Out]

1/573440*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(19/2)*(55545*(-a*(sin(d*x+c)-1))^(15/2)*a^(7/2)-425845*(-
a*(sin(d*x+c)-1))^(13/2)*a^(9/2)+1418249*(-a*(sin(d*x+c)-1))^(11/2)*a^(11/2)-55545*arctanh((-a*(sin(d*x+c)-1))
^(1/2)/a^(1/2))*a^11*sin(d*x+c)^8-2509197*(-a*(sin(d*x+c)-1))^(9/2)*a^(13/2)+2176627*(-a*(sin(d*x+c)-1))^(7/2)
*a^(15/2)-416759*(-a*(sin(d*x+c)-1))^(5/2)*a^(17/2)-425845*(-a*(sin(d*x+c)-1))^(3/2)*a^(19/2)+55545*(-a*(sin(d
*x+c)-1))^(1/2)*a^(21/2))/sin(d*x+c)^8/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{9}\,{d x} \]

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))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^4*csc(d*x + c)^9, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^9} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^9,x)

[Out]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^9, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**(3/2),x)

[Out]

Timed out

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