3.1166 \(\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=482 \[ -\frac{\left (-580 a^2 b^2+128 a^4+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{\left (492 a^2 b^2+128 a^4-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{640 a d \sqrt{a+b \sin (c+d x)}}-\frac{\left (-2476 a^2 b^2+128 a^4-15 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{640 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{3 b \left (-40 a^2 b^2+80 a^4+b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{128 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d} \]

[Out]

-((128*a^4 - 580*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(640*a^2*d) + (b*(36*a^2 - b^2)*Cot[
c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(64*a^2*d) + ((32*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a
+ b*Sin[c + d*x])^(5/2))/(80*a^2*d) + (3*b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(7/2))/(40*a^2*d)
- (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(7/2))/(5*a*d) - ((128*a^4 - 2476*a^2*b^2 - 15*b^4)*Ellipt
icE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(640*a^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]
) + ((128*a^4 + 492*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*d*Sqrt[a + b*Sin[c + d*x]]) + (3*b*(80*a^4 - 40*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^2*d*Sqrt[a + b*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.62341, antiderivative size = 482, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {2893, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{\left (-580 a^2 b^2+128 a^4+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{\left (492 a^2 b^2+128 a^4-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{640 a d \sqrt{a+b \sin (c+d x)}}-\frac{\left (-2476 a^2 b^2+128 a^4-15 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{640 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{3 b \left (-40 a^2 b^2+80 a^4+b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{128 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((128*a^4 - 580*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(640*a^2*d) + (b*(36*a^2 - b^2)*Cot[
c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(64*a^2*d) + ((32*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a
+ b*Sin[c + d*x])^(5/2))/(80*a^2*d) + (3*b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(7/2))/(40*a^2*d)
- (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(7/2))/(5*a*d) - ((128*a^4 - 2476*a^2*b^2 - 15*b^4)*Ellipt
icE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(640*a^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]
) + ((128*a^4 + 492*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*d*Sqrt[a + b*Sin[c + d*x]]) + (3*b*(80*a^4 - 40*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^2*d*Sqrt[a + b*Sin[c + d*x]])

Rule 2893

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 3047

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 3059

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]

Rule 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

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

Rule 3002

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 2663

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

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

Rule 2807

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*
Sin[e + f*x])/(c + d)]), 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 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[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]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac{\int \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac{3}{4} \left (32 a^2-b^2\right )+\frac{5}{2} a b \sin (c+d x)-\frac{1}{4} \left (80 a^2+3 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{20 a^2}\\ &=\frac{\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac{\int \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac{15}{8} b \left (36 a^2-b^2\right )-\frac{3}{4} a \left (16 a^2-5 b^2\right ) \sin (c+d x)-\frac{3}{8} b \left (192 a^2+5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{60 a^2}\\ &=\frac{b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac{\int \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (-\frac{3}{16} \left (128 a^4-580 a^2 b^2+15 b^4\right )-\frac{3}{8} a b \left (268 a^2-5 b^2\right ) \sin (c+d x)-\frac{9}{16} b^2 \left (316 a^2+5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2}\\ &=-\frac{\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac{\int \frac{\csc (c+d x) \left (-\frac{45}{32} b \left (80 a^4-40 a^2 b^2+b^4\right )-\frac{3}{16} a b^2 \left (1484 a^2+5 b^2\right ) \sin (c+d x)+\frac{3}{32} b \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{120 a^2}\\ &=-\frac{\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}+\frac{\int \frac{\csc (c+d x) \left (\frac{45}{32} b^2 \left (80 a^4-40 a^2 b^2+b^4\right )+\frac{3}{32} a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{120 a^2 b}-\frac{\left (128 a^4-2476 a^2 b^2-15 b^4\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{1280 a^2}\\ &=-\frac{\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}+\frac{\left (128 a^4+492 a^2 b^2-5 b^4\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{1280 a}+\frac{\left (3 b \left (80 a^4-40 a^2 b^2+b^4\right )\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{256 a^2}-\frac{\left (\left (128 a^4-2476 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^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=-\frac{\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac{\left (128 a^4-2476 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^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (\left (128 a^4+492 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 \sqrt{a+b \sin (c+d x)}}+\frac{\left (3 b \left (80 a^4-40 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^2 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac{\left (128 a^4-2476 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^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (128 a^4+492 a^2 b^2-5 b^4\right ) F\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 d \sqrt{a+b \sin (c+d x)}}+\frac{3 b \left (80 a^4-40 a^2 b^2+b^4\right ) \Pi \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^2 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 6.73159, size = 700, normalized size = 1.45 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (\frac{1}{80} \csc ^3(c+d x) \left (32 a^2 \cos (c+d x)-31 b^2 \cos (c+d x)\right )+\frac{\csc ^2(c+d x) \left (436 a^2 b \cos (c+d x)-5 b^3 \cos (c+d x)\right )}{320 a}+\frac{\csc (c+d x) \left (1196 a^2 b^2 \cos (c+d x)-128 a^4 \cos (c+d x)+15 b^4 \cos (c+d x)\right )}{640 a^2}-\frac{1}{5} a^2 \cot (c+d x) \csc ^4(c+d x)-\frac{21}{40} a b \cot (c+d x) \csc ^3(c+d x)\right )}{d}+\frac{b \left (-\frac{2 \left (5936 a^3 b+20 a b^3\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left (1276 a^2 b^2+2272 a^4+45 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left (-2476 a^2 b^2+128 a^4-15 b^4\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right )}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}\right )}{2560 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((((-128*a^4*Cos[c + d*x] + 1196*a^2*b^2*Cos[c + d*x] + 15*b^4*Cos[c + d*x])*Csc[c + d*x])/(640*a^2) + ((436*a
^2*b*Cos[c + d*x] - 5*b^3*Cos[c + d*x])*Csc[c + d*x]^2)/(320*a) + ((32*a^2*Cos[c + d*x] - 31*b^2*Cos[c + d*x])
*Csc[c + d*x]^3)/80 - (21*a*b*Cot[c + d*x]*Csc[c + d*x]^3)/40 - (a^2*Cot[c + d*x]*Csc[c + d*x]^4)/5)*Sqrt[a +
b*Sin[c + d*x]])/d + (b*((-2*(5936*a^3*b + 20*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*(2272*a^4 + 1276*a^2*b^2 + 45*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)*(128*a^4 -
2476*a^2*b^2 - 15*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqr
t[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)]))*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)])))/(2560*a^2*d)

________________________________________________________________________________________

Maple [B]  time = 2.108, size = 2075, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/640*(-2466*((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))*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-
1200*((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+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))*b^7*sin(d*x+c)^5-128*a^6*b+128*a^5*b^2*sin(d*x+c)^7-1196*a^3*b^4*sin(d*x+c)^7-1
5*a*b^6*sin(d*x+c)^7+128*a^6*b*sin(d*x+c)^6-2068*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+2652*a^4*b^3*sin(d*x+c)^4+5*a^2*b^5*sin(d*x+c)^4+1592*a^5*b^2*sin(d*x+c)^3-258*a^3*b^4*sin(d*x+c)^3+384*
a^6*b*sin(d*x+c)^2-584*a^4*b^3*sin(d*x+c)^2-464*a^5*b^2*sin(d*x+c)-1256*a^5*b^2*sin(d*x+c)^5+1454*a^3*b^4*sin(
d*x+c)^5+15*a*b^6*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)^5-2
604*((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*b^2*sin(d*x+c)^5+2461*((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+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^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*s
in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b*sin(d*x+c)^5+3096*((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)^5-492*((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+1200*((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+600*((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)*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)^5-600*((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+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)/a^3/b/sin(d*x+c)^5/cos(d*x+c
)/(a+b*sin(d*x+c))^(1/2)/d

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out