\(\int \frac {\cot ^4(e+f x)}{(a+b \sin ^2(e+f x))^{3/2}} \, dx\) [462]

Optimal result

Integrand size = 25, antiderivative size = 297 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {(7 a+8 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {4 (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \] Output:

(a+b)*cot(f*x+e)*csc(f*x+e)^2/a/b/f/(a+b*sin(f*x+e)^2)^(1/2)+1/3*(7*a+8*b) 
*cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^3/f-1/3*(3*a+4*b)*cot(f*x+e)*csc(f* 
x+e)^2*(a+b*sin(f*x+e)^2)^(1/2)/a^2/b/f+1/3*(7*a+8*b)*(cos(f*x+e)^2)^(1/2) 
*EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^ 
3/f/(1+b*sin(f*x+e)^2/a)^(1/2)-4/3*(a+b)*(cos(f*x+e)^2)^(1/2)*EllipticF(si 
n(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(1+b*sin(f*x+e)^2/a)^(1/2)/a^2/f/(a+b*si 
n(f*x+e)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 4.62 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.67 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\frac {\left (8 a^2+37 a b+24 b^2-4 \left (4 a^2+11 a b+8 b^2\right ) \cos (2 (e+f x))+b (7 a+8 b) \cos (4 (e+f x))\right ) \cot (e+f x) \csc ^2(e+f x)}{2 \sqrt {2}}+2 a (7 a+8 b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-8 a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{6 a^3 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \] Input:

Integrate[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]
 

Output:

(((8*a^2 + 37*a*b + 24*b^2 - 4*(4*a^2 + 11*a*b + 8*b^2)*Cos[2*(e + f*x)] + 
 b*(7*a + 8*b)*Cos[4*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^2)/(2*Sqrt[2]) 
+ 2*a*(7*a + 8*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x 
, -(b/a)] - 8*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e 
 + f*x, -(b/a)])/(6*a^3*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3675, 370, 25, 445, 27, 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.

\(\displaystyle \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^4 \left (a+b \sin (e+f x)^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 3675

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

\(\Big \downarrow \) 370

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 445

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {-\frac {\int \frac {b \csc ^2(e+f x) \left (-\left ((3 a+4 b) \sin ^2(e+f x)\right )+7 a+8 b\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{3 a}-\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}}{a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a b \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {-\frac {b \int \frac {\csc ^2(e+f x) \left (-\left ((3 a+4 b) \sin ^2(e+f x)\right )+7 a+8 b\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{3 a}-\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}}{a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a b \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {-\frac {b \left (-\frac {\int \frac {b (7 a+8 b) \sin ^2(e+f x)+a (3 a+4 b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}}{a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a b \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {-\frac {b \left (-\frac {(7 a+8 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-4 a (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}}{a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a b \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {-\frac {b \left (-\frac {(7 a+8 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}}{a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a b \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {-\frac {b \left (-\frac {(7 a+8 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}}{a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a b \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {-\frac {b \left (-\frac {\frac {(7 a+8 b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}}{a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a b \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {-\frac {b \left (-\frac {\frac {(7 a+8 b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}}{a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a b \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\)

Input:

Int[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]
 

Output:

(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*(((a + b)*Csc[e + f*x]^3*Sqrt[1 - Sin[e 
 + f*x]^2])/(a*b*Sqrt[a + b*Sin[e + f*x]^2]) + (-1/3*((3*a + 4*b)*Csc[e + 
f*x]^3*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b*Sin[e + f*x]^2])/a - (b*(-(((7* 
a + 8*b)*Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b*Sin[e + f*x]^2]) 
/a) - (((7*a + 8*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] - (4*a*(a + b)*EllipticF[ArcS 
in[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))/(a*b)))/f
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 370

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + 
 d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(e*x) 
^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a 
*d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], 
 x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] 
&& GtQ[q, 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 = 3.93 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.19

method	result	size
default	\(-\frac {4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \sin \left (f x +e \right )^{3}+4 b \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \sin \left (f x +e \right )^{3}-7 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \sin \left (f x +e \right )^{3}-8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (f x +e \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \sin \left (f x +e \right )^{3}+7 a b \sin \left (f x +e \right )^{6}+8 b^{2} \sin \left (f x +e \right )^{6}+4 \sin \left (f x +e \right )^{4} a^{2}-3 a b \sin \left (f x +e \right )^{4}-8 b^{2} \sin \left (f x +e \right )^{4}-5 a^{2} \sin \left (f x +e \right )^{2}-4 a b \sin \left (f x +e \right )^{2}+a^{2}}{3 a^{3} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )^{2}}\, f}\)	\(353\)

Input:

int(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(4*(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+4*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)^3-7*(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)^3-8*(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*b*sin(f*x+e)^3+7*a*b*sin(f*x+e)^6+8*b 
^2*sin(f*x+e)^6+4*sin(f*x+e)^4*a^2-3*a*b*sin(f*x+e)^4-8*b^2*sin(f*x+e)^4-5 
*a^2*sin(f*x+e)^2-4*a*b*sin(f*x+e)^2+a^2)/a^3/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 = 0.16 (sec) , antiderivative size = 1465, normalized size of antiderivative = 4.93 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

1/6*((2*((7*I*a*b^3 + 8*I*b^4)*cos(f*x + e)^4 + 7*I*a^2*b^2 + 15*I*a*b^3 + 
 8*I*b^4 + (-7*I*a^2*b^2 - 22*I*a*b^3 - 16*I*b^4)*cos(f*x + e)^2)*sqrt(-b) 
*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - ((-14*I*a^2*b^2 - 23*I*a*b^3 - 8*I*b 
^4)*cos(f*x + e)^4 - 14*I*a^3*b - 37*I*a^2*b^2 - 31*I*a*b^3 - 8*I*b^4 + (1 
4*I*a^3*b + 51*I*a^2*b^2 + 54*I*a*b^3 + 16*I*b^4)*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(ar 
csin(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*((-7*I*a*b^3 - 8*I*b^4)*cos(f*x + e)^4 - 7*I*a^2*b^2 - 15*I*a*b^3 
 - 8*I*b^4 + (7*I*a^2*b^2 + 22*I*a*b^3 + 16*I*b^4)*cos(f*x + e)^2)*sqrt(-b 
)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - ((14*I*a^2*b^2 + 23*I*a*b^3 + 8*I*b 
^4)*cos(f*x + e)^4 + 14*I*a^3*b + 37*I*a^2*b^2 + 31*I*a*b^3 + 8*I*b^4 + (- 
14*I*a^3*b - 51*I*a^2*b^2 - 54*I*a*b^3 - 16*I*b^4)*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(a 
rcsin(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*((3*I*a^2*b^2 + 11*I*a*b^3 + 8*I*b^4)*cos(f*x + e)^4 + 3*I*a^3 
*b + 14*I*a^2*b^2 + 19*I*a*b^3 + 8*I*b^4 + (-3*I*a^3*b - 17*I*a^2*b^2 - 30 
*I*a*b^3 - 16*I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f* 
x + e) + ((-6*I*a^3*b - 11*I*a^2*b^2 - 4*I*a*b^3)*cos(f*x + e)^4 - 6*I*...
 

Sympy [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(cot(f*x+e)**4/(a+b*sin(f*x+e)**2)**(3/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(cot(e + f*x)**4/(a + b*sin(e + f*x)**2)**(3/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\cot \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="giac")
 

Output:

integrate(cot(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \] Input:

int(cot(e + f*x)^4/(a + b*sin(e + f*x)^2)^(3/2),x)
 

Output:

int(cot(e + f*x)^4/(a + b*sin(e + f*x)^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right )^{2} b +a}\, \cot \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{4} b^{2}+2 \sin \left (f x +e \right )^{2} a b +a^{2}}d x \] Input:

int(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x)
 

Output:

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