\(\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) [1165]

Optimal result

Integrand size = 29, antiderivative size = 449 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}+\frac {b \left (492 a^2-5 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{64 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {b \left (148 a^2+169 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{64 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (48 a^4-360 a^2 b^2-5 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{64 a d \sqrt {a+b \sin (c+d x)}} \]

[Out]

5/192*b*(68*a^2+b^2)*cot(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a^2/d+1/96*(60*a^2+b^2)*cot(d*x+c)*csc(d*x+c)*(a+b*sin(
d*x+c))^(5/2)/a^2/d+1/24*b*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+c))^(7/2)/a^2/d-1/4*cot(d*x+c)*csc(d*x+c)^3*(a
+b*sin(d*x+c))^(7/2)/a/d-1/64*b^2*(196*a^2+5*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^2/d-1/64*b*(492*a^2-5*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/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+1/64*b*(148*a^2+169*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)*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)/d/(a+b*sin(d*x+c))^(1/2)-1/64*(48*a^4-360*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)*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/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2972, 3126, 3128, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac {b \left (148 a^2+169 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{64 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{64 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}+\frac {\left (48 a^4-360 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{64 a d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d} \]

[In]

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

[Out]

-1/64*(b^2*(196*a^2 + 5*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(a^2*d) + (5*b*(68*a^2 + b^2)*Cot[c + d*x]
*(a + b*Sin[c + d*x])^(3/2))/(192*a^2*d) + ((60*a^2 + b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2
))/(96*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(7/2))/(24*a^2*d) - (Cot[c + d*x]*Csc[c +
d*x]^3*(a + b*Sin[c + d*x])^(7/2))/(4*a*d) + (b*(492*a^2 - 5*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]
*Sqrt[a + b*Sin[c + d*x]])/(64*a*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (b*(148*a^2 + 169*b^2)*EllipticF[(c -
Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4
- 360*a^2*b^2 - 5*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(6
4*a*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 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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 ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac {1}{4} \left (60 a^2+b^2\right )+\frac {5}{2} a b \sin (c+d x)-\frac {3}{4} \left (16 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{12 a^2} \\ & = \frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac {\int \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {5}{8} b \left (68 a^2+b^2\right )-\frac {3}{4} a \left (12 a^2-5 b^2\right ) \sin (c+d x)-\frac {3}{8} b \left (124 a^2+5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2} \\ & = \frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac {\int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-\frac {3}{16} \left (48 a^4-360 a^2 b^2-5 b^4\right )-\frac {3}{8} a b \left (148 a^2-5 b^2\right ) \sin (c+d x)-\frac {9}{16} b^2 \left (196 a^2+5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2} \\ & = -\frac {b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {9}{32} a \left (48 a^4-360 a^2 b^2-5 b^4\right )-\frac {9}{16} a^2 b \left (172 a^2-87 b^2\right ) \sin (c+d x)-\frac {9}{32} a b^2 \left (492 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{36 a^2} \\ & = -\frac {b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}+\frac {\int \frac {\csc (c+d x) \left (\frac {9}{32} a b \left (48 a^4-360 a^2 b^2-5 b^4\right )-\frac {9}{32} a^2 b^2 \left (148 a^2+169 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{36 a^2 b}+\frac {\left (b \left (492 a^2-5 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{128 a} \\ & = -\frac {b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac {1}{128} \left (b \left (148 a^2+169 b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx+\frac {\left (48 a^4-360 a^2 b^2-5 b^4\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{128 a}+\frac {\left (b \left (492 a^2-5 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}{128 a \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \\ & = -\frac {b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}+\frac {b \left (492 a^2-5 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)}}{64 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (b \left (148 a^2+169 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}{128 \sqrt {a+b \sin (c+d x)}}+\frac {\left (\left (48 a^4-360 a^2 b^2-5 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}{128 a \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}+\frac {b \left (492 a^2-5 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)}}{64 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {b \left (148 a^2+169 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}}}{64 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (48 a^4-360 a^2 b^2-5 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}}}{64 a d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.33 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.46 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\left (-\frac {2}{3} b^2 \cos (c+d x)+\frac {5 \left (116 a^2 b \cos (c+d x)-3 b^3 \cos (c+d x)\right ) \csc (c+d x)}{192 a}+\frac {1}{96} \left (60 a^2 \cos (c+d x)-59 b^2 \cos (c+d x)\right ) \csc ^2(c+d x)-\frac {17}{24} a b \cot (c+d x) \csc ^2(c+d x)-\frac {1}{4} a^2 \cot (c+d x) \csc ^3(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}{d}+\frac {-\frac {2 \left (688 a^3 b-348 a b^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (96 a^4-228 a^2 b^2-15 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (-492 a^2 b^2+5 b^4\right ) \cos (c+d x) \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b+b \sin (c+d x)}{a-b}}}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+b^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2\right ) \sqrt {-\frac {a^2-b^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2}{b^2}}}}{256 a d} \]

[In]

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

[Out]

(((-2*b^2*Cos[c + d*x])/3 + (5*(116*a^2*b*Cos[c + d*x] - 3*b^3*Cos[c + d*x])*Csc[c + d*x])/(192*a) + ((60*a^2*
Cos[c + d*x] - 59*b^2*Cos[c + d*x])*Csc[c + d*x]^2)/96 - (17*a*b*Cot[c + d*x]*Csc[c + d*x]^2)/24 - (a^2*Cot[c
+ d*x]*Csc[c + d*x]^3)/4)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(688*a^3*b - 348*a*b^3)*EllipticF[(-c + Pi/2 - d*
x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(96*a^4 - 228*a^2*b^2 -
15*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[
c + d*x]] - ((2*I)*(-492*a^2*b^2 + 5*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(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)]*Sq
rt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*S
in[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a
*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x]
)^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(256*a*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1776\) vs. \(2(510)=1020\).

Time = 21.63 (sec) , antiderivative size = 1777, normalized size of antiderivative = 3.96

method	result	size
default	\(\text {Expression too large to display}\)	\(1777\)

[In]

int(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/192*(-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
pticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4*sin(d*x+c)^4+444*((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)^4-1998*((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^2*sin(d*x+c)^4+
507*((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^3*sin(d*x+c)^4+15*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(s
in(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)^4+144*((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*sin(d*x+c)^4+
1080*((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)^4-1080*((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)^4+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^4*sin(d*x+c)^4+1491*((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)^4-15*a*b^4*sin(d*
x+c)^4-133*a^2*b^3*sin(d*x+c)^3-254*a^3*b^2*sin(d*x+c)^2+1032*((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*sin(d*x+c)^4-144*((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*sin(d*x+c)^4-48*a^5-15*((a+b*s
in(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)^4-1476*((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)^4+128*a^2*b^3*sin(d*x+c)^7-452*a^3*b^2*sin(d*x+c)^6+15*a*b^4*sin(d*x+c)^6-700*a^4*b*sin
(d*x+c)^5-184*a^4*b*sin(d*x+c)+884*a^4*b*sin(d*x+c)^3+5*a^2*b^3*sin(d*x+c)^5+706*a^3*b^2*sin(d*x+c)^4-120*a^5*
sin(d*x+c)^4+168*a^5*sin(d*x+c)^2)/a^2/sin(d*x+c)^4/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

Fricas [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)**4*csc(d*x+c)*(a+b*sin(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right ) \,d x } \]

[In]

integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)^(5/2)*cot(d*x + c)^4*csc(d*x + c), x)

Giac [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Hanged} \]

[In]

int((cot(c + d*x)^4*(a + b*sin(c + d*x))^(5/2))/sin(c + d*x),x)

[Out]

\text{Hanged}