\(\int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [1182]

Optimal result

Integrand size = 23, antiderivative size = 416 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {5 \left (16 a^2-21 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{24 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^2-35 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{24 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (36 a^2-35 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 a^4 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

1/3*(6*a^2-7*b^2)*cot(d*x+c)*csc(d*x+c)/a^2/b/d/(a+b*sin(d*x+c))^(1/2)-1/3*cot(d*x+c)*csc(d*x+c)^2/a/d/(a+b*si
n(d*x+c))^(1/2)+5/24*(16*a^2-21*b^2)*cot(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^4/d-1/12*(24*a^2-35*b^2)*cot(d*x+c)*c
sc(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^3/b/d-5/24*(16*a^2-21*b^2)*(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)*EllipticE(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^4/d/((a+b*
sin(d*x+c))/(a+b))^(1/2)+1/24*(32*a^2-35*b^2)*(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)*El
lipticF(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^3/d/(a+b*sin(d*x+c
))^(1/2)-1/8*b*(36*a^2-35*b^2)*(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^4/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2803, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}+\frac {5 \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{24 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {b \left (36 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (32 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{24 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}} \]

[In]

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

[Out]

((6*a^2 - 7*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^
2)/(3*a*d*Sqrt[a + b*Sin[c + d*x]]) + (5*(16*a^2 - 21*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d) -
((24*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^3*b*d) + (5*(16*a^2 - 21*b^2)*El
lipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a +
b)]) - ((32*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(24
*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + (b*(36*a^2 - 35*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[
(a + b*Sin[c + d*x])/(a + b)])/(8*a^4*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 2803

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[(-Cos[e + f*x])*(
(a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[e + f*x]^3)), x] + (-Dist[1/(3*a^2*b*(m + 1)), Int[((a + b*Sin[e + f*x
])^(m + 1)/Sin[e + f*x]^3)*Simp[6*a^2 - b^2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2
))*Sin[e + f*x]^2, x], x], x] - Simp[(3*a^2 + b^2*(m - 2))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(3*a^2*b
*f*(m + 1)*Sin[e + f*x]^2)), x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

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 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 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 {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {2 \int \frac {\csc ^3(c+d x) \left (\frac {1}{4} \left (24 a^2-35 b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {3}{4} \left (4 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^2 b} \\ & = \frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {5}{8} b \left (16 a^2-21 b^2\right )+\frac {7}{4} a b^2 \sin (c+d x)+\frac {1}{8} b \left (24 a^2-35 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^3 b} \\ & = \frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {\int \frac {\csc (c+d x) \left (\frac {3}{16} b^2 \left (36 a^2-35 b^2\right )+\frac {1}{8} a b \left (24 a^2-35 b^2\right ) \sin (c+d x)+\frac {5}{16} b^2 \left (16 a^2-21 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^4 b} \\ & = \frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {3}{16} b^3 \left (36 a^2-35 b^2\right )+\frac {1}{16} a b^2 \left (32 a^2-35 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^4 b^2}+\frac {\left (5 \left (16 a^2-21 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{48 a^4} \\ & = \frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}-\frac {\left (32 a^2-35 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{48 a^3}+\frac {\left (b \left (36 a^2-35 b^2\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{16 a^4}+\frac {\left (5 \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{48 a^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \\ & = \frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {5 \left (16 a^2-21 b^2\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)}}{24 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (\left (32 a^2-35 b^2\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}{48 a^3 \sqrt {a+b \sin (c+d x)}}+\frac {\left (b \left (36 a^2-35 b^2\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}{16 a^4 \sqrt {a+b \sin (c+d x)}} \\ & = \frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {5 \left (16 a^2-21 b^2\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)}}{24 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^2-35 b^2\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}}}{24 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (36 a^2-35 b^2\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}}}{8 a^4 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.83 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {-\frac {4 \left (\left (-80 a^2 b+105 b^3\right ) \cos (c+d x)+a \cot (c+d x) \left (-32 a^2+35 b^2-14 a b \csc (c+d x)+8 a^2 \csc ^2(c+d x)\right )\right )}{a^4 \sqrt {a+b \sin (c+d x)}}+\frac {\frac {10 i \left (-16 a^2+21 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b \sqrt {-\frac {1}{a+b}}}-\frac {8 a \left (24 a^2-35 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 b \left (-296 a^2+315 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}}{a^4}}{96 d} \]

[In]

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

[Out]

((-4*((-80*a^2*b + 105*b^3)*Cos[c + d*x] + a*Cot[c + d*x]*(-32*a^2 + 35*b^2 - 14*a*b*Csc[c + d*x] + 8*a^2*Csc[
c + d*x]^2)))/(a^4*Sqrt[a + b*Sin[c + d*x]]) + (((10*I)*(-16*a^2 + 21*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[S
qrt[-(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 + Si
n[c + d*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]) - (8*a*(24*a^2 - 35*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*b*(-296*a^2 + 315*b^2)*Elliptic
Pi[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]])/a^4)
/(96*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1495\) vs. \(2(481)=962\).

Time = 1.70 (sec) , antiderivative size = 1496, normalized size of antiderivative = 3.60

method	result	size
default	\(\text {Expression too large to display}\)	\(1496\)

[In]

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

[Out]

1/24*(48*a^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)*El
lipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*sin(d*x+c)^3+32*((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*sin(d*x+c)^3-150*b^2*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+si
n(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*sin(d*x+c)^3-35*b^3
*((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*sin(d*x+c)^3+105*((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^4*sin(d*x+c)^3-80*((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^5*sin(d*x+c)^3+185*((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^2*sin(d*x+c)^3-105*((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*b^4
*sin(d*x+c)^3-108*((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^2*sin(d*x+c)^3+108*((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^3*sin(d*x+c)^3+105*((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^4*sin(d*x+c)^3-105*((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))*b^5*sin(d*x+c)^
3-80*a^3*b^2*sin(d*x+c)^5+105*a*b^4*sin(d*x+c)^5-32*a^4*b*sin(d*x+c)^4+35*a^2*b^3*sin(d*x+c)^4+66*b^2*a^3*sin(
d*x+c)^3-105*a*b^4*sin(d*x+c)^3+40*a^4*b*sin(d*x+c)^2-35*a^2*b^3*sin(d*x+c)^2+14*a^3*b^2*sin(d*x+c)-8*a^4*b)/b
/a^5/sin(d*x+c)^3/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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