\(\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) [1158]

Optimal result

Integrand size = 31, antiderivative size = 484 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{640 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (128 a^4+692 a^2 b^2+5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{640 a^2 d \sqrt {a+b \sin (c+d x)}}+\frac {3 b \left (48 a^4+8 a^2 b^2-b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{128 a^3 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

1/80*(32*a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+c))^(3/2)/a^2/d+1/8*b*cot(d*x+c)*csc(d*x+c)^3*(a+b*si
n(d*x+c))^(5/2)/a^2/d-1/5*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^(5/2)/a/d-1/640*(128*a^4-116*a^2*b^2+15*b^4
)*cot(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^3/d+3/320*b*(36*a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(1/2)/
a^2/d+1/640*(128*a^4-116*a^2*b^2+15*b^4)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*Ellipti
cE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/a^3/d/((a+b*sin(d*x+c))/(a+b))^(1
/2)-1/640*(128*a^4+692*a^2*b^2+5*b^4)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(
cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/a^2/d/(a+b*sin(d*x+c))^(1/2)
-3/128*b*(48*a^4+8*a^2*b^2-b^4)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1
/2*c+1/4*Pi+1/2*d*x),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/a^3/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2972, 3126, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}+\frac {\left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{640 a^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{640 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {3 b \left (48 a^4+8 a^2 b^2-b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{128 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d} \]

[In]

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

[Out]

-1/640*((128*a^4 - 116*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(a^3*d) + (3*b*(36*a^2 - 5*b^2
)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(320*a^2*d) + ((32*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x
]^2*(a + b*Sin[c + d*x])^(3/2))/(80*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(8*a^2
*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(5*a*d) - ((128*a^4 - 116*a^2*b^2 + 15*b^4)*Ell
ipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(640*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a +
b)]) + ((128*a^4 + 692*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])
/(a + b)])/(640*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + (3*b*(48*a^4 + 8*a^2*b^2 - b^4)*EllipticPi[2, (c - Pi/2 + d*
x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(128*a^3*d*Sqrt[a + b*Sin[c + d*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

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

Rule 2884

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

Rule 2886

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

Rule 2972

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] +
(-Dist[1/(a^2*d^2*(n + 1)*(n + 2)), Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) -
b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e +
f*x]^2, x], x], x] - Simp[b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 2)/(
a^2*d^2*f*(n + 1)*(n + 2))), x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || Intege
rsQ[2*m, 2*n]) &&  !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])

Rule 3081

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

Rule 3126

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

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3138

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

Rubi steps \begin{align*} \text {integral}& = \frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {3}{4} \left (32 a^2-5 b^2\right )+\frac {3}{2} a b \sin (c+d x)-\frac {5}{4} \left (16 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{20 a^2} \\ & = \frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\int \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {9}{8} b \left (36 a^2-5 b^2\right )-\frac {3}{4} a \left (16 a^2-b^2\right ) \sin (c+d x)-\frac {3}{8} b \left (128 a^2-5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{60 a^2} \\ & = \frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {3}{16} \left (128 a^4-116 a^2 b^2+15 b^4\right )-\frac {3}{8} a b \left (212 a^2+b^2\right ) \sin (c+d x)-\frac {3}{16} b^2 \left (404 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^2} \\ & = -\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {45}{32} b \left (48 a^4+8 a^2 b^2-b^4\right )-\frac {3}{16} a b^2 \left (404 a^2-5 b^2\right ) \sin (c+d x)+\frac {3}{32} b \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^3} \\ & = -\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}+\frac {\int \frac {\csc (c+d x) \left (\frac {45}{32} b^2 \left (48 a^4+8 a^2 b^2-b^4\right )+\frac {3}{32} a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^3 b}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{1280 a^3} \\ & = -\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}+\frac {\left (3 b \left (48 a^4+8 a^2 b^2-b^4\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{256 a^3}+\frac {\left (128 a^4+692 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{1280 a^2}-\frac {\left (\left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{1280 a^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \\ & = -\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{640 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (3 b \left (48 a^4+8 a^2 b^2-b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{256 a^3 \sqrt {a+b \sin (c+d x)}}+\frac {\left (\left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{1280 a^2 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{640 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (128 a^4+692 a^2 b^2+5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{640 a^2 d \sqrt {a+b \sin (c+d x)}}+\frac {3 b \left (48 a^4+8 a^2 b^2-b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{128 a^3 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.21 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.12 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {-4 \cot (c+d x) \left (128 a^4-116 a^2 b^2+15 b^4-2 a b \left (236 a^2+5 b^2\right ) \csc (c+d x)+8 a^2 \left (-32 a^2+b^2\right ) \csc ^2(c+d x)+176 a^3 b \csc ^3(c+d x)+128 a^4 \csc ^4(c+d x)\right ) \sqrt {a+b \sin (c+d x)}+b \left (-\frac {2 i \left (128 a^4-116 a^2 b^2+15 b^4\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}} \left (-2+\csc ^2(c+d x)\right )}-\frac {8 a b \left (404 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (1312 a^4+356 a^2 b^2-45 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}\right )}{2560 a^3 d} \]

[In]

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

[Out]

(-4*Cot[c + d*x]*(128*a^4 - 116*a^2*b^2 + 15*b^4 - 2*a*b*(236*a^2 + 5*b^2)*Csc[c + d*x] + 8*a^2*(-32*a^2 + b^2
)*Csc[c + d*x]^2 + 176*a^3*b*Csc[c + d*x]^3 + 128*a^4*Csc[c + d*x]^4)*Sqrt[a + b*Sin[c + d*x]] + b*(((-2*I)*(1
28*a^4 - 116*a^2*b^2 + 15*b^4)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^
(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*
Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*
x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(
a - b))])/(a*b^2*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (8*a*b*(404*a^2 - 5*b^2)*EllipticF[(-2*c + Pi -
2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(1312*a^4 + 356*a^2
*b^2 - 45*b^4)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a
+ b*Sin[c + d*x]]))/(2560*a^3*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2074\) vs. \(2(545)=1090\).

Time = 2.39 (sec) , antiderivative size = 2075, normalized size of antiderivative = 4.29

method	result	size
default	\(\text {Expression too large to display}\)	\(2075\)

[In]

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

[Out]

-1/640*(15*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Elli
pticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*b^7*sin(d*x+c)^5-128*((a+b*sin(d*x+c))/(a-b
))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1
/2),((a-b)/(a+b))^(1/2))*a^7*sin(d*x+c)^5+720*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(
-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^5*b^2*
sin(d*x+c)^5-720*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2
)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^4*b^3*sin(d*x+c)^5+120*((a+b*sin(d*
x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c)
)/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^3*b^4*sin(d*x+c)^5-120*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x
+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)
/(a+b))^(1/2))*a^2*b^5*sin(d*x+c)^5-15*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin
(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a*b^6*sin(d*x+c
)^5+244*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Ellipti
cE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2*sin(d*x+c)^5-131*((a+b*sin(d*x+c))/(a-b))^(1/2)
*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-
b)/(a+b))^(1/2))*a^3*b^4*sin(d*x+c)^5+15*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+s
in(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6*sin(d*x+c)^5+128
*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+
b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b*sin(d*x+c)^5-936*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d
*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))
^(1/2))*a^5*b^2*sin(d*x+c)^5+692*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c
))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^3*sin(d*x+c)^5+126*((a+b
*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(
d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^4*sin(d*x+c)^5+5*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-
1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2)
)*a^2*b^5*sin(d*x+c)^5-15*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a
-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6*sin(d*x+c)^5+128*a^6*b+856*a^5*
b^2*sin(d*x+c)^5+15*a*b^6*sin(d*x+c)^5-114*a^3*b^4*sin(d*x+c)^5-128*a^5*b^2*sin(d*x+c)^7+116*a^3*b^4*sin(d*x+c
)^7-15*a*b^6*sin(d*x+c)^7-128*a^6*b*sin(d*x+c)^6+588*a^4*b^3*sin(d*x+c)^6-5*a^2*b^5*sin(d*x+c)^6+384*a^6*b*sin
(d*x+c)^4-772*a^4*b^3*sin(d*x+c)^4+5*a^2*b^5*sin(d*x+c)^4-1032*a^5*b^2*sin(d*x+c)^3-2*a^3*b^4*sin(d*x+c)^3-384
*a^6*b*sin(d*x+c)^2+184*a^4*b^3*sin(d*x+c)^2+304*a^5*b^2*sin(d*x+c))/a^4/b/sin(d*x+c)^5/cos(d*x+c)/(a+b*sin(d*
x+c))^(1/2)/d

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}