3.1159 \(\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=551 \[ \frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {b \left (156 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^2 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {b \left (2064 a^4+512 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{7680 a^4 d}-\frac {b \left (2064 a^4+512 a^2 b^2-105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{7680 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (64 a^6+144 a^4 b^2-36 a^2 b^4+7 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{512 a^4 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (2544 a^4+176 a^2 b^2-35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{7680 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (240 a^4-168 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{3840 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d} \]

[Out]

7/96*(4*a^2-b^2)*cot(d*x+c)*csc(d*x+c)^3*(a+b*sin(d*x+c))^(3/2)/a^2/d+7/60*b*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(
d*x+c))^(5/2)/a^2/d-1/6*cot(d*x+c)*csc(d*x+c)^5*(a+b*sin(d*x+c))^(5/2)/a/d-1/7680*b*(2064*a^4+512*a^2*b^2-105*
b^4)*cot(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^4/d-1/3840*(240*a^4-168*a^2*b^2+35*b^4)*cot(d*x+c)*csc(d*x+c)*(a+b*si
n(d*x+c))^(1/2)/a^3/d+1/960*b*(156*a^2-35*b^2)*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+c))^(1/2)/a^2/d+1/7680*b*(
2064*a^4+512*a^2*b^2-105*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*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/a^4/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/7680*
b*(2544*a^4+176*a^2*b^2-35*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^3/d/(a+b*sin(d*x+c))^(1/2)-1/512
*(64*a^6+144*a^4*b^2-36*a^2*b^4+7*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)*EllipticP
i(cos(1/2*c+1/4*Pi+1/2*d*x),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/a^4/d/(a+b*sin(d*x+c))^(
1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.98, 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} \[ -\frac {b \left (512 a^2 b^2+2064 a^4-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{7680 a^4 d}+\frac {b \left (176 a^2 b^2+2544 a^4-35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{7680 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {b \left (512 a^2 b^2+2064 a^4-105 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{7680 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (144 a^4 b^2-36 a^2 b^4+64 a^6+7 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{512 a^4 d \sqrt {a+b \sin (c+d x)}}+\frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {b \left (156 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^2 d}-\frac {\left (-168 a^2 b^2+240 a^4+35 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{3840 a^3 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(2064*a^4 + 512*a^2*b^2 - 105*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(7680*a^4*d) - ((240*a^4 - 168*a
^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3840*a^3*d) + (b*(156*a^2 - 35*b^2)*Cot[
c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(960*a^2*d) + (7*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(
a + b*Sin[c + d*x])^(3/2))/(96*a^2*d) + (7*b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(60*a^2*d
) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2))/(6*a*d) - (b*(2064*a^4 + 512*a^2*b^2 - 105*b^4)*E
llipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(7680*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a
+ b)]) + (b*(2544*a^4 + 176*a^2*b^2 - 35*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c
+ d*x])/(a + b)])/(7680*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + ((64*a^6 + 144*a^4*b^2 - 36*a^2*b^4 + 7*b^6)*Ellipti
cPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(512*a^4*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))^{3/2} \, dx &=\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}-\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {35}{4} \left (4 a^2-b^2\right )+\frac {3}{2} a b \sin (c+d x)-\frac {3}{4} \left (40 a^2-7 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{30 a^2}\\ &=\frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}-\frac {\int \csc ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {3}{8} b \left (156 a^2-35 b^2\right )-\frac {3}{4} a \left (20 a^2-b^2\right ) \sin (c+d x)-\frac {9}{8} b \left (60 a^2-7 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2}\\ &=\frac {b \left (156 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^2 d}+\frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}-\frac {\int \frac {\csc ^3(c+d x) \left (-\frac {3}{16} \left (240 a^4-168 a^2 b^2+35 b^4\right )-\frac {3}{8} a b \left (348 a^2+b^2\right ) \sin (c+d x)-\frac {9}{16} b^2 \left (204 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{360 a^2}\\ &=-\frac {\left (240 a^4-168 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{3840 a^3 d}+\frac {b \left (156 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^2 d}+\frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}-\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {3}{32} b \left (2064 a^4+512 a^2 b^2-105 b^4\right )-\frac {3}{16} a \left (240 a^4+1056 a^2 b^2-7 b^4\right ) \sin (c+d x)-\frac {3}{32} b \left (240 a^4-168 a^2 b^2+35 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{720 a^3}\\ &=-\frac {b \left (2064 a^4+512 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{7680 a^4 d}-\frac {\left (240 a^4-168 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{3840 a^3 d}+\frac {b \left (156 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^2 d}+\frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {45}{64} \left (64 a^6+144 a^4 b^2-36 a^2 b^4+7 b^6\right )-\frac {3}{32} a b \left (240 a^4-168 a^2 b^2+35 b^4\right ) \sin (c+d x)+\frac {3}{64} b^2 \left (2064 a^4+512 a^2 b^2-105 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{720 a^4}\\ &=-\frac {b \left (2064 a^4+512 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{7680 a^4 d}-\frac {\left (240 a^4-168 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{3840 a^3 d}+\frac {b \left (156 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^2 d}+\frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}+\frac {\int \frac {\csc (c+d x) \left (\frac {45}{64} b \left (64 a^6+144 a^4 b^2-36 a^2 b^4+7 b^6\right )+\frac {3}{64} a b^2 \left (2544 a^4+176 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{720 a^4 b}-\frac {\left (b \left (2064 a^4+512 a^2 b^2-105 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{15360 a^4}\\ &=-\frac {b \left (2064 a^4+512 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{7680 a^4 d}-\frac {\left (240 a^4-168 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{3840 a^3 d}+\frac {b \left (156 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^2 d}+\frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}+\frac {\left (b \left (2544 a^4+176 a^2 b^2-35 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{15360 a^3}+\frac {\left (64 a^6+144 a^4 b^2-36 a^2 b^4+7 b^6\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{1024 a^4}-\frac {\left (b \left (2064 a^4+512 a^2 b^2-105 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}{15360 a^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=-\frac {b \left (2064 a^4+512 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{7680 a^4 d}-\frac {\left (240 a^4-168 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{3840 a^3 d}+\frac {b \left (156 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^2 d}+\frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}-\frac {b \left (2064 a^4+512 a^2 b^2-105 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)}}{7680 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (b \left (2544 a^4+176 a^2 b^2-35 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}{15360 a^3 \sqrt {a+b \sin (c+d x)}}+\frac {\left (\left (64 a^6+144 a^4 b^2-36 a^2 b^4+7 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^4 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {b \left (2064 a^4+512 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{7680 a^4 d}-\frac {\left (240 a^4-168 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{3840 a^3 d}+\frac {b \left (156 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^2 d}+\frac {7 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac {7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}-\frac {b \left (2064 a^4+512 a^2 b^2-105 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)}}{7680 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {b \left (2544 a^4+176 a^2 b^2-35 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}}}{7680 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (64 a^6+144 a^4 b^2-36 a^2 b^4+7 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^4 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}

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Mathematica [C]  time = 6.78, size = 771, normalized size = 1.40 \[ \frac {\sqrt {a+b \sin (c+d x)} \left (\frac {\csc ^3(c+d x) \left (436 a^2 b \cos (c+d x)+7 b^3 \cos (c+d x)\right )}{960 a^2}+\frac {\csc ^4(c+d x) \left (140 a^2 \cos (c+d x)-3 b^2 \cos (c+d x)\right )}{480 a}+\frac {\csc (c+d x) \left (-2064 a^4 b \cos (c+d x)-512 a^2 b^3 \cos (c+d x)+105 b^5 \cos (c+d x)\right )}{7680 a^4}+\frac {\csc ^2(c+d x) \left (-240 a^4 \cos (c+d x)+168 a^2 b^2 \cos (c+d x)-35 b^4 \cos (c+d x)\right )}{3840 a^3}-\frac {1}{6} a \cot (c+d x) \csc ^5(c+d x)-\frac {13}{60} b \cot (c+d x) \csc ^4(c+d x)\right )}{d}+\frac {-\frac {2 \left (960 a^5 b-672 a^3 b^3+140 a b^5\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (2064 a^4 b^2+512 a^2 b^4-105 b^6\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right )}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt {-\frac {a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac {2 \left (1920 a^6+2256 a^4 b^2-1592 a^2 b^4+315 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}}{30720 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((((-2064*a^4*b*Cos[c + d*x] - 512*a^2*b^3*Cos[c + d*x] + 105*b^5*Cos[c + d*x])*Csc[c + d*x])/(7680*a^4) + ((-
240*a^4*Cos[c + d*x] + 168*a^2*b^2*Cos[c + d*x] - 35*b^4*Cos[c + d*x])*Csc[c + d*x]^2)/(3840*a^3) + ((436*a^2*
b*Cos[c + d*x] + 7*b^3*Cos[c + d*x])*Csc[c + d*x]^3)/(960*a^2) + ((140*a^2*Cos[c + d*x] - 3*b^2*Cos[c + d*x])*
Csc[c + d*x]^4)/(480*a) - (13*b*Cot[c + d*x]*Csc[c + d*x]^4)/60 - (a*Cot[c + d*x]*Csc[c + d*x]^5)/6)*Sqrt[a +
b*Sin[c + d*x]])/d + ((-2*(960*a^5*b - 672*a^3*b^3 + 140*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*(1920*a^6 + 2256*a^4*b^2 - 1592*a^2*b^4 + 31
5*b^6)*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)*(2064*a^4*b^2 + 512*a^2*b^4 - 105*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*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)]))/(30720*a^4*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))^(3/2),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))^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 3.03, 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))^(3/2),x)

[Out]

-1/7680*(-960*((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)*E
llipticPi(((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)^6-540*((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+540*((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)*Ellipti
cPi(((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+105*((a+b*sin(d*x+c))/(a-
b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^
(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a*b^6*sin(d*x+c)^6+1552*((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+617*((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)^6-105*((a+b*sin(d*x+c))
/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b
))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6*sin(d*x+c)^6+2544*((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)^6-1728*((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)^6-105*a*b^6*sin(d*x+c)^6-105*
((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+
b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*b^7*sin(d*x+c)^6-35*a^2*b^5*sin(d*x+c)^5+14*a^3*b^4*si
n(d*x+c)^4-8*a^4*b^3*sin(d*x+c)^3+1712*a^5*b^2*sin(d*x+c)^2+1280*a^7-2064*((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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+
b))^(1/2))*a^7*sin(d*x+c)^6-480*((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+960*((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)^6-2064*a^5*b^2*sin(d*x+c)^8-512*a^3*b^4*sin(d*x+c
)^8+105*a*b^6*sin(d*x+c)^8-2544*a^6*b*sin(d*x+c)^7-176*a^4*b^3*sin(d*x+c)^7+184*a^4*b^3*sin(d*x+c)^5+498*a^3*b
^4*sin(d*x+c)^6+5888*a^5*b^2*sin(d*x+c)^6+35*a^2*b^5*sin(d*x+c)^7+8272*a^6*b*sin(d*x+c)^5-5536*a^5*b^2*sin(d*x
+c)^4-8672*a^6*b*sin(d*x+c)^3+2944*a^6*b*sin(d*x+c)-480*a^7*sin(d*x+c)^6+2720*a^7*sin(d*x+c)^4-3520*a^7*sin(d*
x+c)^2+176*((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
pticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^3*sin(d*x+c)^6-582*((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)^6-35*((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+105*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF
(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6*sin(d*x+c)^6)/a^5/sin(d*x+c)^6/cos(d*x+c)/(a+b*sin(
d*x+c))^(1/2)/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{3}\,{d x} \]

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))^(3/2),x, algorithm="maxima")

[Out]

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

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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))^(3/2))/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))**(3/2),x)

[Out]

Timed out

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