3.1167 \(\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=551 \[ \frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}+\frac {b \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{1536 a^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{256 a^2 d}-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1536 a^3 d}-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{1536 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{512 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d} \]

[Out]

1/192*b*(52*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+1/96*(28*a^2-3*b^2)*cot(d*x+c)*csc
(d*x+c)^3*(a+b*sin(d*x+c))^(5/2)/a^2/d+1/12*b*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^(7/2)/a^2/d-1/6*cot(d*x
+c)*csc(d*x+c)^5*(a+b*sin(d*x+c))^(7/2)/a/d-1/1536*b*(720*a^4-176*a^2*b^2+15*b^4)*cot(d*x+c)*(a+b*sin(d*x+c))^
(1/2)/a^3/d-1/256*(16*a^4-56*a^2*b^2+5*b^4)*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^2/d+1/1536*b*(720*a
^4-176*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)*EllipticE(cos(1/2*c+1/4*P
i+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)-1/1536*b*(816*
a^4+1696*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)-1/512*(64*a^6
+720*a^4*b^2+60*a^2*b^4-5*b^6)*(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]  time = 1.97, antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2893, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {b \left (-176 a^2 b^2+720 a^4+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1536 a^3 d}+\frac {b \left (1696 a^2 b^2+816 a^4+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{1536 a^2 d \sqrt {a+b \sin (c+d x)}}-\frac {b \left (-176 a^2 b^2+720 a^4+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{1536 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (720 a^4 b^2+60 a^2 b^4+64 a^6-5 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{512 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac {\left (-56 a^2 b^2+16 a^4+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{256 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

-(b*(720*a^4 - 176*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1536*a^3*d) - ((16*a^4 - 56*a^2*b
^2 + 5*b^4)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(256*a^2*d) + (b*(52*a^2 - 5*b^2)*Cot[c + d*x]
*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(192*a^2*d) + ((28*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b
*Sin[c + d*x])^(5/2))/(96*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(7/2))/(12*a^2*d) - (Co
t[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^(7/2))/(6*a*d) - (b*(720*a^4 - 176*a^2*b^2 + 15*b^4)*EllipticE[
(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1536*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) +
(b*(816*a^4 + 1696*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)])/(1536*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + ((64*a^6 + 720*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*EllipticPi[2, (c
- Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(512*a^3*d*Sqrt[a + b*Sin[c + d*x]])

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 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 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 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 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]

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 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 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 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 3055

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] + Dis
t[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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 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]

Rubi steps

\begin {align*} \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac {5}{4} \left (28 a^2-3 b^2\right )+\frac {5}{2} a b \sin (c+d x)-\frac {5}{4} \left (24 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{30 a^2}\\ &=\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {15}{8} b \left (52 a^2-5 b^2\right )-\frac {15}{4} a \left (4 a^2-b^2\right ) \sin (c+d x)-\frac {5}{8} b \left (164 a^2-5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2}\\ &=\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac {\int \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (-\frac {45}{16} \left (16 a^4-56 a^2 b^2+5 b^4\right )-\frac {15}{8} a b \left (84 a^2-b^2\right ) \sin (c+d x)-\frac {15}{16} b^2 \left (276 a^2-5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{360 a^2}\\ &=-\frac {\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{256 a^2 d}+\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {15}{32} b \left (720 a^4-176 a^2 b^2+15 b^4\right )-\frac {15}{16} a \left (48 a^4+720 a^2 b^2+b^4\right ) \sin (c+d x)-\frac {15}{32} b \left (48 a^4+936 a^2 b^2-5 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{720 a^2}\\ &=-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1536 a^3 d}-\frac {\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{256 a^2 d}+\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {45}{64} \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right )-\frac {15}{32} a b \left (48 a^4+936 a^2 b^2-5 b^4\right ) \sin (c+d x)+\frac {15}{64} b^2 \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{720 a^3}\\ &=-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1536 a^3 d}-\frac {\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{256 a^2 d}+\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}+\frac {\int \frac {\csc (c+d x) \left (\frac {45}{64} b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right )+\frac {15}{64} a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{720 a^3 b}-\frac {\left (b \left (720 a^4-176 a^2 b^2+15 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{3072 a^3}\\ &=-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1536 a^3 d}-\frac {\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{256 a^2 d}+\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}+\frac {\left (b \left (816 a^4+1696 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{3072 a^2}+\frac {\left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{1024 a^3}-\frac {\left (b \left (720 a^4-176 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}{3072 a^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1536 a^3 d}-\frac {\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{256 a^2 d}+\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac {b \left (720 a^4-176 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)}}{1536 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (b \left (816 a^4+1696 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}{3072 a^2 \sqrt {a+b \sin (c+d x)}}+\frac {\left (\left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\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}{1024 a^3 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1536 a^3 d}-\frac {\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{256 a^2 d}+\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac {b \left (720 a^4-176 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)}}{1536 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {b \left (816 a^4+1696 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}}}{1536 a^2 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\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}}}{512 a^3 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}

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Mathematica [C]  time = 6.81, size = 771, normalized size = 1.40 \[ \frac {\sqrt {a+b \sin (c+d x)} \left (\frac {\csc ^3(c+d x) \left (164 a^2 b \cos (c+d x)-b^3 \cos (c+d x)\right )}{192 a}+\frac {1}{96} \csc ^4(c+d x) \left (28 a^2 \cos (c+d x)-27 b^2 \cos (c+d x)\right )-\frac {1}{6} a^2 \cot (c+d x) \csc ^5(c+d x)+\frac {\csc ^2(c+d x) \left (-48 a^4 \cos (c+d x)+600 a^2 b^2 \cos (c+d x)+5 b^4 \cos (c+d x)\right )}{768 a^2}+\frac {\csc (c+d x) \left (-720 a^4 b \cos (c+d x)+176 a^2 b^3 \cos (c+d x)-15 b^5 \cos (c+d x)\right )}{1536 a^3}-\frac {5}{12} a b \cot (c+d x) \csc ^4(c+d x)\right )}{d}+\frac {-\frac {2 \left (192 a^5 b+3744 a^3 b^3-20 a b^5\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (720 a^4 b^2-176 a^2 b^4+15 b^6\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right )}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt {-\frac {a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac {2 \left (384 a^6+3600 a^4 b^2+536 a^2 b^4-45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}}{6144 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

((((-720*a^4*b*Cos[c + d*x] + 176*a^2*b^3*Cos[c + d*x] - 15*b^5*Cos[c + d*x])*Csc[c + d*x])/(1536*a^3) + ((-48
*a^4*Cos[c + d*x] + 600*a^2*b^2*Cos[c + d*x] + 5*b^4*Cos[c + d*x])*Csc[c + d*x]^2)/(768*a^2) + ((164*a^2*b*Cos
[c + d*x] - b^3*Cos[c + d*x])*Csc[c + d*x]^3)/(192*a) + ((28*a^2*Cos[c + d*x] - 27*b^2*Cos[c + d*x])*Csc[c + d
*x]^4)/96 - (5*a*b*Cot[c + d*x]*Csc[c + d*x]^4)/12 - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/6)*Sqrt[a + b*Sin[c + d
*x]])/d + ((-2*(192*a^5*b + 3744*a^3*b^3 - 20*a*b^5)*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*(384*a^6 + 3600*a^4*b^2 + 536*a^2*b^4 - 45*b^6)*Ellipti
cPi[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)*(720*a^4*b^2 - 176*a^2*b^4 + 15*b^6)*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)]*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*S
qrt[-(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)]))/(6144*a^3*d)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 2.85, size = 2458, normalized size = 4.46 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x)

[Out]

1/1536*(192*((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)*Ell
ipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^6*b*sin(d*x+c)^6-2160*((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)^6+2160*((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)^6-180*((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)^
6+180*((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)*EllipticP
i(((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)^6+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)^6-896*((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)^6+191*((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)^6-15*((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)^6-816*((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
)^6+2592*((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)*Ellipt
icF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2*sin(d*x+c)^6-15*a*b^6*sin(d*x+c)^6-15*((a+b*si
n(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)^6-5*a^2*b^5*sin(d*x+c)^5+2*a^3*b^4*sin(d*x+c)^4
-440*a^4*b^3*sin(d*x+c)^3-1072*a^5*b^2*sin(d*x+c)^2-256*a^7+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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))
*a^7*sin(d*x+c)^6+96*((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^7*sin(d*x+c)^6-192*((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^7*sin(d*x+c)^6+720*a^5*b^2*sin(d*x+c)^8-176*a^3*b^4*sin(d*x+c)^8+15*a*b^6
*sin(d*x+c)^8+816*a^6*b*sin(d*x+c)^7-1376*a^4*b^3*sin(d*x+c)^7+1816*a^4*b^3*sin(d*x+c)^5+174*a^3*b^4*sin(d*x+c
)^6-3232*a^5*b^2*sin(d*x+c)^6+5*a^2*b^5*sin(d*x+c)^7-2576*a^6*b*sin(d*x+c)^5+3584*a^5*b^2*sin(d*x+c)^4+2656*a^
6*b*sin(d*x+c)^3-896*a^6*b*sin(d*x+c)+96*a^7*sin(d*x+c)^6-544*a^7*sin(d*x+c)^4+704*a^7*sin(d*x+c)^2-1696*((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)^6-186*((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)^6-5*((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)^6+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)^6)/a^4/sin(d*x+c)^6/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(c + d*x)^4*(a + b*sin(c + d*x))^(5/2))/sin(c + d*x)^3,x)

[Out]

\text{Hanged}

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*csc(d*x+c)**3*(a+b*sin(d*x+c))**(5/2),x)

[Out]

Timed out

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