3.1164 \(\int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=429 \[ -\frac{b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}-\frac{b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{40 a d}+\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{a \left (96 a^2+179 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{40 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (176 a^2-167 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{40 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{5 b \left (12 a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 d \sqrt{a+b \sin (c+d x)}}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d} \]

[Out]

-(b*(96*a^2 - 25*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(40*a*d) - (b*(208*a^2 - 25*b^2)*Cos[c + d*x]*(a
+ b*Sin[c + d*x])^(3/2))/(120*a^2*d) + ((32*a^2 - 3*b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(24*a^2*d) -
(b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin
[c + d*x])^(7/2))/(3*a*d) + ((176*a^2 - 167*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c
+ d*x]])/(40*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (a*(96*a^2 + 179*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b
)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(40*d*Sqrt[a + b*Sin[c + d*x]]) - (5*b*(12*a^2 - b^2)*EllipticP
i[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*d*Sqrt[a + b*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.35532, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {2725, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}-\frac{b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{40 a d}+\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{a \left (96 a^2+179 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{40 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (176 a^2-167 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{40 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{5 b \left (12 a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 d \sqrt{a+b \sin (c+d x)}}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(96*a^2 - 25*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(40*a*d) - (b*(208*a^2 - 25*b^2)*Cos[c + d*x]*(a
+ b*Sin[c + d*x])^(3/2))/(120*a^2*d) + ((32*a^2 - 3*b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(24*a^2*d) -
(b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin
[c + d*x])^(7/2))/(3*a*d) + ((176*a^2 - 167*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c
+ d*x]])/(40*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (a*(96*a^2 + 179*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b
)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(40*d*Sqrt[a + b*Sin[c + d*x]]) - (5*b*(12*a^2 - b^2)*EllipticP
i[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*d*Sqrt[a + b*Sin[c + d*x]])

Rule 2725

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/(6*a^2), Int[((a + b*Sin[e + f*x])^m*Simp[8*
a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x])/Sin[e + f*x]^2, x
], x] - Simp[(b*(m - 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(6*a^2*f*Sin[e + f*x]^2), x]) /; FreeQ[{a,
b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1] && IntegerQ[2*m]

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 3049

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 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) (a+b \sin (c+d x))^{5/2} \, dx &=-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac{\int \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac{1}{4} \left (32 a^2-3 b^2\right )+\frac{5}{2} a b \sin (c+d x)-\frac{1}{4} \left (24 a^2-5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6 a^2}\\ &=\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac{\int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac{15}{8} b \left (12 a^2-b^2\right )-\frac{3}{4} a \left (8 a^2-5 b^2\right ) \sin (c+d x)-\frac{1}{8} b \left (208 a^2-25 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6 a^2}\\ &=-\frac{b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac{\int \csc (c+d x) \sqrt{a+b \sin (c+d x)} \left (\frac{75}{16} a b \left (12 a^2-b^2\right )-\frac{3}{8} a^2 \left (40 a^2-71 b^2\right ) \sin (c+d x)-\frac{9}{16} a b \left (96 a^2-25 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{15 a^2}\\ &=-\frac{b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{40 a d}-\frac{b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac{2 \int \frac{\csc (c+d x) \left (\frac{225}{32} a^2 b \left (12 a^2-b^2\right )-\frac{9}{16} a^3 \left (40 a^2-173 b^2\right ) \sin (c+d x)-\frac{9}{32} a^2 b \left (176 a^2-167 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{45 a^2}\\ &=-\frac{b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{40 a d}-\frac{b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}+\frac{2 \int \frac{\csc (c+d x) \left (-\frac{225}{32} a^2 b^2 \left (12 a^2-b^2\right )-\frac{9}{32} a^3 b \left (96 a^2+179 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{45 a^2 b}-\frac{1}{80} \left (-176 a^2+167 b^2\right ) \int \sqrt{a+b \sin (c+d x)} \, dx\\ &=-\frac{b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{40 a d}-\frac{b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac{1}{16} \left (5 b \left (12 a^2-b^2\right )\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx-\frac{1}{80} \left (a \left (96 a^2+179 b^2\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx-\frac{\left (\left (-176 a^2+167 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}{80 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=-\frac{b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{40 a d}-\frac{b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}+\frac{\left (176 a^2-167 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)}}{40 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (5 b \left (12 a^2-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 \sqrt{a+b \sin (c+d x)}}-\frac{\left (a \left (96 a^2+179 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}{80 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{40 a d}-\frac{b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}+\frac{\left (176 a^2-167 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)}}{40 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{a \left (96 a^2+179 b^2\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}}}{40 d \sqrt{a+b \sin (c+d x)}}-\frac{5 b \left (12 a^2-b^2\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}}}{8 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 3.4295, size = 466, normalized size = 1.09 \[ \frac{-\frac{8 a \left (40 a^2-173 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}+\frac{2 b \left (424 a^2+117 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{4}{3} \sqrt{a+b \sin (c+d x)} \left (5 \cot (c+d x) \left (8 a^2 \csc ^2(c+d x)-32 a^2+26 a b \csc (c+d x)+33 b^2\right )+176 a b \cos (c+d x)+24 b^2 \sin (2 (c+d x))\right )+\frac{2 i \left (167 b^2-176 a^2\right ) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left (b \left (b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )}{a b \sqrt{-\frac{1}{a+b}}}}{160 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((2*I)*(-176*a^2 + 167*b^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*Sqrt[-(a + b)^(-1
)]) - (8*a*(40*a^2 - 173*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*(424*a^2 + 117*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sq
rt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (4*Sqrt[a + b*Sin[c + d*x]]*(176*a*b*Cos[c + d*x]
+ 5*Cot[c + d*x]*(-32*a^2 + 33*b^2 + 26*a*b*Csc[c + d*x] + 8*a^2*Csc[c + d*x]^2) + 24*b^2*Sin[2*(c + d*x)]))/
3)/(160*d)

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Maple [B]  time = 1.86, size = 1526, normalized size = 3.6 \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*(a+b*sin(d*x+c))^(5/2),x)

[Out]

1/120*(240*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)*
EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*sin(d*x+c)^3+288*((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)^3-1566*b^2*((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*sin(d*x+c)^3+53
7*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+501*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(si
n(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)^3-528*((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+1029*((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)*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)^3-501*((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+900*((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)^3-900*((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*si
n(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^2*b^3*sin(d*x+c)^3-75*((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+75*((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+48*a*b^4*sin(d*x+c)^7+224*a^2*b^3*sin(d*x+c)^6+16*a^3*b^2*sin(d*x+c)^5+117*a*b^4*sin(d*x+c)^5-160*a^4*b
*sin(d*x+c)^4+71*a^2*b^3*sin(d*x+c)^4+154*a^3*b^2*sin(d*x+c)^3-165*a*b^4*sin(d*x+c)^3+200*a^4*b*sin(d*x+c)^2-2
95*a^2*b^3*sin(d*x+c)^2-170*a^3*b^2*sin(d*x+c)-40*a^4*b)/a/b/sin(d*x+c)^3/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

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

[Out]

Timed out

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

[Out]

Timed out

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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*(a+b*sin(d*x+c))**(5/2),x)

[Out]

Timed out

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

[Out]

Timed out