\(\int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx\) [451]

Optimal result

Integrand size = 25, antiderivative size = 240 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}+\frac {2 (2 a+b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a f \sqrt {a+b \sin ^2(e+f x)}} \] Output:

2/3*(2*a+b)*cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^2/f-1/3*cot(f*x+e)*csc(f 
*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2)/a/f+2/3*(2*a+b)*(cos(f*x+e)^2)^(1/2)*Elli 
pticE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^2/f/( 
1+b*sin(f*x+e)^2/a)^(1/2)-1/3*(a+b)*(cos(f*x+e)^2)^(1/2)*EllipticF(sin(f*x 
+e),(-b/a)^(1/2))*sec(f*x+e)*(1+b*sin(f*x+e)^2/a)^(1/2)/a/f/(a+b*sin(f*x+e 
)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 4.75 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.78 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {\frac {\left (-2 \left (4 a^2+5 a b+2 b^2\right ) \cos (2 (e+f x))+(2 a+b) (2 a+3 b+b \cos (4 (e+f x)))\right ) \cot (e+f x) \csc ^2(e+f x)}{\sqrt {2}}+4 a (2 a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-2 a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{6 a^2 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \] Input:

Integrate[Cot[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]
 

Output:

(((-2*(4*a^2 + 5*a*b + 2*b^2)*Cos[2*(e + f*x)] + (2*a + b)*(2*a + 3*b + b* 
Cos[4*(e + f*x)]))*Cot[e + f*x]*Csc[e + f*x]^2)/Sqrt[2] + 4*a*(2*a + b)*Sq 
rt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 2*a*(a + 
 b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6* 
a^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3675, 376, 25, 445, 399, 323, 321, 330, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^4 \sqrt {a+b \sin (e+f x)^2}}dx\)

\(\Big \downarrow \) 3675

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\csc ^4(e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{\sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 376

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int -\frac {\csc ^2(e+f x) \left (2 (2 a+b)-(3 a+b) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{3 a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {\int \frac {\csc ^2(e+f x) \left (2 (2 a+b)-(3 a+b) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{3 a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{f}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {-\frac {\int \frac {2 b (2 a+b) \sin ^2(e+f x)+a (3 a+b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {2 (2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{f}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {-\frac {2 (2 a+b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-a (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {2 (2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{f}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {-\frac {2 (2 a+b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {2 (2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{f}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {-\frac {2 (2 a+b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {2 (2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{f}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {-\frac {\frac {2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {2 (2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{f}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {-\frac {\frac {2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {2 (2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{f}\)

Input:

Int[Cot[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]
 

Output:

(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*(-1/3*(Csc[e + f*x]^3*Sqrt[1 - Sin[e + 
f*x]^2]*Sqrt[a + b*Sin[e + f*x]^2])/a - ((-2*(2*a + b)*Csc[e + f*x]*Sqrt[1 
 - Sin[e + f*x]^2]*Sqrt[a + b*Sin[e + f*x]^2])/a - ((2*(2*a + b)*EllipticE 
[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/Sqrt[1 + (b*Sin 
[e + f*x]^2)/a] - (a*(a + b)*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[ 
1 + (b*Sin[e + f*x]^2)/a])/Sqrt[a + b*Sin[e + f*x]^2])/a)/(3*a)))/f
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

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

rule 327

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

rule 330

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

rule 376

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1 
)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^ 
2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b*c - a*d)*(m + 1) + 2*c*(b*c*(p + 1) + a* 
d*(q - 1)) + d*((b*c - a*d)*(m + 1) + 2*b*c*(p + q))*x^2, x], x], x] /; Fre 
eQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && LtQ[m, -1] & 
& IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3675

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ 
(m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 
)*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x]))   Subst[Int[x^m*((a + b*ff^2*x^2) 
^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b 
, e, f, p}, x] && IntegerQ[m/2] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.46

method	result	size
default	\(-\frac {\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \sin \left (f x +e \right )^{3}+b \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \sin \left (f x +e \right )^{3}-4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \sin \left (f x +e \right )^{3}-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \sin \left (f x +e \right )^{3}+4 a b \sin \left (f x +e \right )^{6}+2 b^{2} \sin \left (f x +e \right )^{6}+4 \sin \left (f x +e \right )^{4} a^{2}-3 a b \sin \left (f x +e \right )^{4}-2 b^{2} \sin \left (f x +e \right )^{4}-5 a^{2} \sin \left (f x +e \right )^{2}-a b \sin \left (f x +e \right )^{2}+a^{2}}{3 a^{2} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )^{2}}\, f}\)	\(351\)

Input:

int(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*((cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticF(sin(f*x+ 
e),(-b/a)^(1/2))*a^2*sin(f*x+e)^3+b*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^ 
2)/a)^(1/2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a*sin(f*x+e)^3-4*(cos(f*x+e 
)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticE(sin(f*x+e),(-b/a)^(1/2)) 
*a^2*sin(f*x+e)^3-2*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*Elli 
pticE(sin(f*x+e),(-b/a)^(1/2))*a*b*sin(f*x+e)^3+4*a*b*sin(f*x+e)^6+2*b^2*s 
in(f*x+e)^6+4*sin(f*x+e)^4*a^2-3*a*b*sin(f*x+e)^4-2*b^2*sin(f*x+e)^4-5*a^2 
*sin(f*x+e)^2-a*b*sin(f*x+e)^2+a^2)/a^2/sin(f*x+e)^3/cos(f*x+e)/(a+b*sin(f 
*x+e)^2)^(1/2)/f
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 1045, normalized size of antiderivative = 4.35 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

1/3*((2*(-2*I*a*b^2 - I*b^3 + (2*I*a*b^2 + I*b^3)*cos(f*x + e)^2)*sqrt(-b) 
*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - (4*I*a^2*b + 4*I*a*b^2 + I*b^3 + (-4 
*I*a^2*b - 4*I*a*b^2 - I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt( 
(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt( 
(a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 
8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*(2*I*a*b^2 
+ I*b^3 + (-2*I*a*b^2 - I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b 
^2)*sin(f*x + e) - (-4*I*a^2*b - 4*I*a*b^2 - I*b^3 + (4*I*a^2*b + 4*I*a*b^ 
2 + I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a* 
b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 
2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2* 
a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*(3*I*a^2*b + 5*I*a*b^2 + 2*I*b 
^3 + (-3*I*a^2*b - 5*I*a*b^2 - 2*I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 
 + a*b)/b^2)*sin(f*x + e) - (6*I*a^3 + 5*I*a^2*b + I*a*b^2 + (-6*I*a^3 - 5 
*I*a^2*b - I*a*b^2)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt( 
(a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a*b 
)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^ 
2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*(-3*I*a^2*b - 5*I*a*b 
^2 - 2*I*b^3 + (3*I*a^2*b + 5*I*a*b^2 + 2*I*b^3)*cos(f*x + e)^2)*sqrt(-b)* 
sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - (-6*I*a^3 - 5*I*a^2*b - I*a*b^2 + ...
 

Sympy [F]

\[ \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \] Input:

integrate(cot(f*x+e)**4/(a+b*sin(f*x+e)**2)**(1/2),x)
 

Output:

Integral(cot(e + f*x)**4/sqrt(a + b*sin(e + f*x)**2), x)
 

Maxima [F]

\[ \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{4}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \] Input:

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")
 

Output:

integrate(cot(f*x + e)^4/sqrt(b*sin(f*x + e)^2 + a), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \] Input:

int(cot(e + f*x)^4/(a + b*sin(e + f*x)^2)^(1/2),x)
 

Output:

int(cot(e + f*x)^4/(a + b*sin(e + f*x)^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right )^{2} b +a}\, \cot \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{2} b +a}d x \] Input:

int(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x)
 

Output:

int((sqrt(sin(e + f*x)**2*b + a)*cot(e + f*x)**4)/(sin(e + f*x)**2*b + a), 
x)