\(\int \frac {\cos (c+d x) \cot ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\) [1173]

Optimal result

Integrand size = 29, antiderivative size = 307 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {3 b \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}+\frac {\left (8 a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{4 a^2 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (8 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a b d \sqrt {a+b \sin (c+d x)}}-\frac {3 \left (4 a^2-b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^2 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2972, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=-\frac {\left (8 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{4 a b d \sqrt {a+b \sin (c+d x)}}+\frac {\left (8 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{4 a^2 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {3 \left (4 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{4 a^2 d \sqrt {a+b \sin (c+d x)}}+\frac {3 b \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d} \]

[In]

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

[Out]

(3*b*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(
2*a*d) + ((8*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(4*a^2*b*d*Sq
rt[(a + b*Sin[c + d*x])/(a + b)]) - ((8*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Si
n[c + d*x])/(a + b)])/(4*a*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*(4*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2,
(2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a^2*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 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) \sqrt {a+b \sin (c+d x)}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}-\frac {\int \frac {\csc (c+d x) \left (\frac {3}{4} \left (4 a^2-b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {1}{4} \left (8 a^2+3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{2 a^2} \\ & = \frac {3 b \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}+\frac {\int \frac {\csc (c+d x) \left (-\frac {3}{4} b \left (4 a^2-b^2\right )-\frac {1}{4} a \left (8 a^2+b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{2 a^2 b}-\frac {\left (-8 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{8 a^2 b} \\ & = \frac {3 b \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}-\frac {1}{8} \left (\frac {8 a}{b}+\frac {b}{a}\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {1}{8} \left (3 \left (4-\frac {b^2}{a^2}\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {\left (\left (-8 a^2-3 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^2 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \\ & = \frac {3 b \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}+\frac {\left (8 a^2+3 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^2 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (\left (\frac {8 a}{b}+\frac {b}{a}\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 \sqrt {a+b \sin (c+d x)}}-\frac {\left (3 \left (4-\frac {b^2}{a^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 {3 b \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}+\frac {\left (8 a^2+3 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^2 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (\frac {8 a}{b}+\frac {b}{a}\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 d \sqrt {a+b \sin (c+d x)}}-\frac {3 \left (4-\frac {b^2}{a^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 = 2.35 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.44 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {\frac {2 i \left (8 a^2+3 b^2\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a^3 b^2 \sqrt {-\frac {1}{a+b}} \left (-2+\csc ^2(c+d x)\right )}-\frac {4 \cot (c+d x) (-3 b+2 a \csc (c+d x)) \sqrt {a+b \sin (c+d x)}}{a^2}-\frac {8 b \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a \sqrt {a+b \sin (c+d x)}}+\frac {2 \left (16 a^2-9 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a^2 \sqrt {a+b \sin (c+d x)}}}{16 d} \]

[In]

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

[Out]

(((2*I)*(8*a^2 + 3*b^2)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*S
qrt[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^3*b^2*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (4*Cot[c + d*x]*(-3*b + 2*a*Csc[c + d*x])*Sqrt[a + b*
Sin[c + d*x]])/a^2 - (8*b*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/
(a*Sqrt[a + b*Sin[c + d*x]]) + (2*(16*a^2 - 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a
+ b*Sin[c + d*x])/(a + b)])/(a^2*Sqrt[a + b*Sin[c + d*x]]))/(16*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(912\) vs. \(2(380)=760\).

Time = 1.50 (sec) , antiderivative size = 913, normalized size of antiderivative = 2.97

method	result	size
default	\(\frac {\sqrt {-\left (-b \sin \left (d x +c \right )-a \right ) \left (\cos ^{2}\left (d x +c \right )\right )}\, \left (\frac {2 \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {\frac {\left (1-\sin \left (d x +c \right )\right ) b}{a +b}}\, \sqrt {\frac {\left (-1-\sin \left (d x +c \right )\right ) b}{a -b}}\, \left (\left (-\frac {a}{b}-1\right ) E\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )+F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )\right )}{\sqrt {-\left (-b \sin \left (d x +c \right )-a \right ) \left (\cos ^{2}\left (d x +c \right )\right )}}-\frac {\sqrt {-\left (-b \sin \left (d x +c \right )-a \right ) \left (\cos ^{2}\left (d x +c \right )\right )}}{2 a \sin \left (d x +c \right )^{2}}+\frac {3 b \sqrt {-\left (-b \sin \left (d x +c \right )-a \right ) \left (\cos ^{2}\left (d x +c \right )\right )}}{4 a^{2} \sin \left (d x +c \right )}+\frac {b \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {\frac {\left (1-\sin \left (d x +c \right )\right ) b}{a +b}}\, \sqrt {\frac {\left (-1-\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )}{2 a \sqrt {-\left (-b \sin \left (d x +c \right )-a \right ) \left (\cos ^{2}\left (d x +c \right )\right )}}+\frac {3 b^{2} \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {\frac {\left (1-\sin \left (d x +c \right )\right ) b}{a +b}}\, \sqrt {\frac {\left (-1-\sin \left (d x +c \right )\right ) b}{a -b}}\, \left (\left (-\frac {a}{b}-1\right ) E\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )+F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )\right )}{4 a^{2} \sqrt {-\left (-b \sin \left (d x +c \right )-a \right ) \left (\cos ^{2}\left (d x +c \right )\right )}}-\frac {\left (4 a^{2}+3 b^{2}\right ) \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {\frac {\left (1-\sin \left (d x +c \right )\right ) b}{a +b}}\, \sqrt {\frac {\left (-1-\sin \left (d x +c \right )\right ) b}{a -b}}\, b \Pi \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, -\frac {\left (-\frac {a}{b}+1\right ) b}{a}, \sqrt {\frac {a -b}{a +b}}\right )}{4 a^{3} \sqrt {-\left (-b \sin \left (d x +c \right )-a \right ) \left (\cos ^{2}\left (d x +c \right )\right )}}+\frac {4 \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {\frac {\left (1-\sin \left (d x +c \right )\right ) b}{a +b}}\, \sqrt {\frac {\left (-1-\sin \left (d x +c \right )\right ) b}{a -b}}\, b \Pi \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, -\frac {\left (-\frac {a}{b}+1\right ) b}{a}, \sqrt {\frac {a -b}{a +b}}\right )}{\sqrt {-\left (-b \sin \left (d x +c \right )-a \right ) \left (\cos ^{2}\left (d x +c \right )\right )}\, a}\right )}{\cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\)	\(913\)

[In]

int(cos(d*x+c)*cot(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*(2*(1/b*a-1)*((a+b*sin(d*x+c))/(a-b))^(1/2)*((1-sin(d*x+c))*b/(a+b))^(
1/2)*((-1-sin(d*x+c))*b/(a-b))^(1/2)/(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*((-1/b*a-1)*EllipticE(((a+b*sin(d
*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))+EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2)))-1/2/a*
(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)/sin(d*x+c)^2+3/4*b/a^2*(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)/sin(d*x
+c)+1/2/a*b*(1/b*a-1)*((a+b*sin(d*x+c))/(a-b))^(1/2)*((1-sin(d*x+c))*b/(a+b))^(1/2)*((-1-sin(d*x+c))*b/(a-b))^
(1/2)/(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))+3/
4*b^2/a^2*(1/b*a-1)*((a+b*sin(d*x+c))/(a-b))^(1/2)*((1-sin(d*x+c))*b/(a+b))^(1/2)*((-1-sin(d*x+c))*b/(a-b))^(1
/2)/(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*((-1/b*a-1)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))
^(1/2))+EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2)))-1/4*(4*a^2+3*b^2)/a^3*(1/b*a-1)*((a+b*s
in(d*x+c))/(a-b))^(1/2)*((1-sin(d*x+c))*b/(a+b))^(1/2)*((-1-sin(d*x+c))*b/(a-b))^(1/2)/(-(-b*sin(d*x+c)-a)*cos
(d*x+c)^2)^(1/2)*b*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),-(-1/b*a+1)*b/a,((a-b)/(a+b))^(1/2))+4*(1/b*a-1)*
((a+b*sin(d*x+c))/(a-b))^(1/2)*((1-sin(d*x+c))*b/(a+b))^(1/2)*((-1-sin(d*x+c))*b/(a-b))^(1/2)/(-(-b*sin(d*x+c)
-a)*cos(d*x+c)^2)^(1/2)*b/a*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),-(-1/b*a+1)*b/a,((a-b)/(a+b))^(1/2)))/co
s(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

Fricas [F(-1)]

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

[In]

integrate(cos(d*x+c)*cot(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

integrate(cos(d*x+c)*cot(d*x+c)**3/(a+b*sin(d*x+c))**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate(cos(d*x+c)*cot(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(cos(d*x+c)*cot(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((cos(c + d*x)*cot(c + d*x)^3)/(a + b*sin(c + d*x))^(1/2),x)

[Out]

int((cos(c + d*x)*cot(c + d*x)^3)/(a + b*sin(c + d*x))^(1/2), x)