\(\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) [1163]

Optimal result

Integrand size = 29, antiderivative size = 430 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}+\frac {a \left (8 a^2-247 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{28 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (8 a^4+3 a^2 b^2-32 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{28 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 a \left (4 a^2-5 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

-1/28*(8*a^2-35*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a/d-1/28*(8*a^2-21*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(5/
2)/a^2/d-3/4*b*cot(d*x+c)*(a+b*sin(d*x+c))^(7/2)/a^2/d-1/2*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(7/2)/a/d-1/
28*(8*a^2-73*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/d-1/28*a*(8*a^2-247*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/
2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/
2)/b/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+1/28*(8*a^4+3*a^2*b^2-32*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
)/b/d/(a+b*sin(d*x+c))^(1/2)+3/4*a*(4*a^2-5*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)
*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)/d/(a+b*sin(d*x
+c))^(1/2)

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2972, 3128, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}+\frac {a \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{28 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {3 a \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{4 d \sqrt {a+b \sin (c+d x)}}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\left (8 a^4+3 a^2 b^2-32 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{28 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d} \]

[In]

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

[Out]

-1/28*((8*a^2 - 73*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/d - ((8*a^2 - 35*b^2)*Cos[c + d*x]*(a + b*Sin[c
+ d*x])^(3/2))/(28*a*d) - ((8*a^2 - 21*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(28*a^2*d) - (3*b*Cot[c
+ d*x]*(a + b*Sin[c + d*x])^(7/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(2*a*d)
+ (a*(8*a^2 - 247*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(28*b*d*Sqrt[(a
+ b*Sin[c + d*x])/(a + b)]) - ((8*a^4 + 3*a^2*b^2 - 32*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[
(a + b*Sin[c + d*x])/(a + b)])/(28*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*a*(4*a^2 - 5*b^2)*EllipticPi[2, (c - Pi/
2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*d*Sqrt[a + b*Sin[c + d*x]])

Rule 2732

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

Rule 2734

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

Rule 2740

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

Rule 2742

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

Rule 2884

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

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 2972

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] +
(-Dist[1/(a^2*d^2*(n + 1)*(n + 2)), Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) -
b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e +
f*x]^2, x], x], x] - Simp[b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 2)/(
a^2*d^2*f*(n + 1)*(n + 2))), x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || Intege
rsQ[2*m, 2*n]) &&  !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])

Rule 3081

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3138

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac {3}{4} \left (4 a^2-5 b^2\right )+\frac {5}{2} a b \sin (c+d x)-\frac {1}{4} \left (8 a^2-21 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2 a^2} \\ & = -\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {21}{8} a \left (4 a^2-5 b^2\right )+\frac {57}{4} a^2 b \sin (c+d x)-\frac {5}{8} a \left (8 a^2-35 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{7 a^2} \\ & = -\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {2 \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {105}{16} a^2 \left (4 a^2-5 b^2\right )+\frac {435}{8} a^3 b \sin (c+d x)-\frac {15}{16} a^2 \left (8 a^2-73 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{35 a^2} \\ & = -\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {4 \int \frac {\csc (c+d x) \left (\frac {315}{32} a^3 \left (4 a^2-5 b^2\right )+\frac {15}{16} a^2 b \left (125 a^2-16 b^2\right ) \sin (c+d x)-\frac {15}{32} a^3 \left (8 a^2-247 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 a^2} \\ & = -\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}+\frac {4 \int \frac {\csc (c+d x) \left (-\frac {315}{32} a^3 b \left (4 a^2-5 b^2\right )-\frac {15}{32} a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 a^2 b}+\frac {\left (a \left (8 a^2-247 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{56 b} \\ & = -\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {1}{8} \left (3 a \left (4 a^2-5 b^2\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {\left (8 a^4+3 a^2 b^2-32 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{56 b}+\frac {\left (a \left (8 a^2-247 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}{56 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \\ & = -\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}+\frac {a \left (8 a^2-247 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)}}{28 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (3 a \left (4 a^2-5 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}{8 \sqrt {a+b \sin (c+d x)}}-\frac {\left (\left (8 a^4+3 a^2 b^2-32 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}{56 b \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}+\frac {a \left (8 a^2-247 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)}}{28 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (8 a^4+3 a^2 b^2-32 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{28 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 a \left (4 a^2-5 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.04 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.07 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\frac {2 i \left (-8 a^2+247 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}}}{b^2 \sqrt {-\frac {1}{a+b}}}+\frac {8 b \left (125 a^2-16 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (160 a^2+37 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+4 \sqrt {a+b \sin (c+d x)} \left (\left (-24 a^2+22 b^2\right ) \cos (c+d x)+2 b^2 \cos (3 (c+d x))-7 a \cot (c+d x) (9 b+2 a \csc (c+d x))-12 a b \sin (2 (c+d x))\right )}{112 d} \]

[In]

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

[Out]

(((2*I)*(-8*a^2 + 247*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)])/(b^2*Sqrt[-(a + b)^(-1)])
+ (8*b*(125*a^2 - 16*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*a*(160*a^2 + 37*b^2)*EllipticPi[2, (-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]] + 4*Sqrt[a + b*Sin[c + d*x]]*((-24*a^2 + 22*b^2)*Cos[c +
d*x] + 2*b^2*Cos[3*(c + d*x)] - 7*a*Cot[c + d*x]*(9*b + 2*a*Csc[c + d*x]) - 12*a*b*Sin[2*(c + d*x)]))/(112*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1519\) vs. \(2(491)=982\).

Time = 17.62 (sec) , antiderivative size = 1520, normalized size of antiderivative = 3.53

method	result	size
default	\(\text {Expression too large to display}\)	\(1520\)

[In]

int(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/28*(8*b^5*sin(d*x+c)^7+8*((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)^2-258*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)^2+3*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)^2+279*((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^4*sin(d*x+c)^2-32*b^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)^2-8*((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)^2+25
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)*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)^2-247*((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*b^4*sin(d*x+c)^2+84*((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)*b^2*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^3*sin(d*x+c)^2-
84*((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)*b^3*Elliptic
Pi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^2*sin(d*x+c)^2-105*((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^4*sin(d*x+c)^2+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
^5*sin(d*x+c)^2+32*a*b^4*sin(d*x+c)^6+48*a^2*b^3*sin(d*x+c)^5-32*b^5*sin(d*x+c)^5+24*a^3*b^2*sin(d*x+c)^4+7*a*
b^4*sin(d*x+c)^4+29*a^2*b^3*sin(d*x+c)^3+24*b^5*sin(d*x+c)^3-10*a^3*b^2*sin(d*x+c)^2-39*a*b^4*sin(d*x+c)^2-77*
a^2*b^3*sin(d*x+c)-14*a^3*b^2)/b^2/sin(d*x+c)^2/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

Fricas [F(-1)]

Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \cot \left (d x + c\right )^{3} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int \cos \left (c+d\,x\right )\,{\mathrm {cot}\left (c+d\,x\right )}^3\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]

[In]

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

[Out]

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