\(\int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [1181]

Optimal result

Integrand size = 29, antiderivative size = 366 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (8 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 \left (4 a^2-5 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^3 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

1/2*(4*a^2-5*b^2)*cot(d*x+c)/a^2/b/d/(a+b*sin(d*x+c))^(1/2)-1/2*cot(d*x+c)*csc(d*x+c)/a/d/(a+b*sin(d*x+c))^(1/
2)-1/4*(8*a^2-15*b^2)*cot(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^3/b/d+1/4*(8*a^2-15*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)/a^3/b/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/4*(8*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)*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/b/d/(a+b*sin(d*x+c))^(1/2)+3/4*(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)/a^3/d/(a+b*sin
(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2969, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}+\frac {\left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{4 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {3 \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{4 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}} \]

[In]

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

[Out]

((4*a^2 - 5*b^2)*Cot[c + d*x])/(2*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*Sqrt[
a + b*Sin[c + d*x]]) - ((8*a^2 - 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(4*a^3*b*d) - ((8*a^2 - 15*b^2
)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(4*a^3*b*d*Sqrt[(a + b*Sin[c + d*x])/
(a + b)]) + ((8*a^2 - 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/
(4*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*(4*a^2 - 5*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqr
t[(a + b*Sin[c + d*x])/(a + b)])/(4*a^3*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 2969

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]*(d*Sin[e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] +
(Dist[1/(a^2*b*d*(n + 1)*(m + 1)), Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)*Simp[a^2*(n + 1)
*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m
+ n + 4))*Sin[e + f*x]^2, x], x], x] - Simp[(a^2*(n + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(d*Sin[e + f*x])^(n +
2)*((a + b*Sin[e + f*x])^(m + 1)/(a^2*b*d^2*f*(n + 1)*(m + 1))), x]) /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2
- b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]

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 3134

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] + D
ist[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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 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 {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}+\frac {\int \frac {\csc ^2(c+d x) \left (\frac {1}{4} \left (8 a^2-15 b^2\right )-\frac {1}{2} a b \sin (c+d x)+\frac {5}{4} b^2 \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{a^2 b} \\ & = \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}+\frac {\int \frac {\csc (c+d x) \left (-\frac {3}{8} b \left (4 a^2-5 b^2\right )+\frac {5}{4} a b^2 \sin (c+d x)-\frac {1}{8} b \left (8 a^2-15 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{a^3 b} \\ & = \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\int \frac {\csc (c+d x) \left (\frac {3}{8} b^2 \left (4 a^2-5 b^2\right )-\frac {1}{8} a b \left (8 a^2-5 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{a^3 b^2}+\frac {\left (-8 a^2+15 b^2\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{8 a^3 b} \\ & = \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (3 \left (4 a^2-5 b^2\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{8 a^3}+\frac {\left (8 a^2-5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{8 a^2 b}+\frac {\left (\left (-8 a^2+15 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}{8 a^3 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \\ & = \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 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)}}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (3 \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 a^3 \sqrt {a+b \sin (c+d x)}}+\frac {\left (\left (8 a^2-5 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}{8 a^2 b \sqrt {a+b \sin (c+d x)}} \\ & = \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 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)}}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (8 a^2-5 b^2\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}}}{4 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 \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 a^3 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.51 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.19 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\frac {\left (-32 a^2+60 b^2\right ) \cos (c+d x)+4 a \cot (c+d x) (5 b-2 a \csc (c+d x))}{a^3 \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {2 i \left (-8 a^2+15 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}}}-\frac {40 a b \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 \left (32 a^2-45 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}}{a^3}}{16 d} \]

[In]

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

[Out]

(((-32*a^2 + 60*b^2)*Cos[c + d*x] + 4*a*Cot[c + d*x]*(5*b - 2*a*Csc[c + d*x]))/(a^3*Sqrt[a + b*Sin[c + d*x]])
+ (((-2*I)*(-8*a^2 + 15*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^2*Sqrt[-(a + b)^(
-1)]) - (40*a*b*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*(32*a^2 - 45*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]])/a^3)/(16*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1348\) vs. \(2(435)=870\).

Time = 1.43 (sec) , antiderivative size = 1349, normalized size of antiderivative = 3.69

method	result	size
default	\(\text {Expression too large to display}\)	\(1349\)

[In]

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

[Out]

-1/4*(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)*Ellipti
cF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b*sin(d*x+c)^2-18*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)^2-5*b^3*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+s
in(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+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*s
in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4*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+23*((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)^2-15*((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^4*sin(d*x+c)^2-12*((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+12*((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*EllipticPi(((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+15*((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)*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)^2-15*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d
*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-
b)/(a+b))^(1/2))*b^5*sin(d*x+c)^2-8*a^3*b^2*sin(d*x+c)^4+15*a*b^4*sin(d*x+c)^4+5*a^2*b^3*sin(d*x+c)^3+6*a^3*b^
2*sin(d*x+c)^2-15*a*b^4*sin(d*x+c)^2-5*a^2*b^3*sin(d*x+c)+2*a^3*b^2)/b^2/sin(d*x+c)^2/a^4/cos(d*x+c)/(a+b*sin(
d*x+c))^(1/2)/d

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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