\(\int \frac {\cot ^4(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [473]

Optimal result

Integrand size = 25, antiderivative size = 344 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(5 a+8 b) \cot (e+f x)}{3 a^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a+2 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 b (a+2 b) \cos (e+f x) \sin (e+f x)}{3 a^4 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a+2 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^4 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(5 a+8 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^3 f \sqrt {a+b \sin ^2(e+f x)}} \] Output:

1/3*(a+b)*cot(f*x+e)*csc(f*x+e)^2/a/b/f/(a+b*sin(f*x+e)^2)^(3/2)+1/3*(5*a+ 
8*b)*cot(f*x+e)/a^3/f/(a+b*sin(f*x+e)^2)^(1/2)-1/3*(a+2*b)*cot(f*x+e)*csc( 
f*x+e)^2/a^2/b/f/(a+b*sin(f*x+e)^2)^(1/2)+8/3*b*(a+2*b)*cos(f*x+e)*sin(f*x 
+e)/a^4/f/(a+b*sin(f*x+e)^2)^(1/2)+8/3*(a+2*b)*(cos(f*x+e)^2)^(1/2)*Ellipt 
icE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^4/f/(1+ 
b*sin(f*x+e)^2/a)^(1/2)-1/3*(5*a+8*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^3/f/(a+b*sin(f 
*x+e)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 3.99 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 a^2 b \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \left (8 (a+2 b) E\left (e+f x\left |-\frac {b}{a}\right .\right )-(5 a+8 b) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )\right )+\sqrt {2} b \left (4 (a+2 b) (2 a+b-b \cos (2 (e+f x)))^2 \cot (e+f x)-a (2 a+b-b \cos (2 (e+f x)))^2 \cot (e+f x) \csc ^2(e+f x)+2 a b (a+b) \sin (2 (e+f x))+4 b (a+2 b) (2 a+b-b \cos (2 (e+f x))) \sin (2 (e+f x))\right )}{6 a^4 b f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \] Input:

Integrate[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]
 

Output:

(2*a^2*b*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*(8*(a + 2*b)*EllipticE[e 
 + f*x, -(b/a)] - (5*a + 8*b)*EllipticF[e + f*x, -(b/a)]) + Sqrt[2]*b*(4*( 
a + 2*b)*(2*a + b - b*Cos[2*(e + f*x)])^2*Cot[e + f*x] - a*(2*a + b - b*Co 
s[2*(e + f*x)])^2*Cot[e + f*x]*Csc[e + f*x]^2 + 2*a*b*(a + b)*Sin[2*(e + f 
*x)] + 4*b*(a + 2*b)*(2*a + b - b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))/(6* 
a^4*b*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3675, 370, 25, 441, 27, 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.

\(\displaystyle \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^4 \left (a+b \sin (e+f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 3675

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\csc ^4(e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2}}{\left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 370

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {\csc ^4(e+f x) \left (3 (a+2 b)-(2 a+5 b) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a b}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int \frac {\csc ^4(e+f x) \left (3 (a+2 b)-(2 a+5 b) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 441

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {3 \left (-\frac {b \left (-\frac {8 (a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-a (5 a+8 b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {8 (a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a+3 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}}{3 a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {3 \left (-\frac {b \left (-\frac {8 (a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {a (5 a+8 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {8 (a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a+3 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}}{3 a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {3 \left (-\frac {b \left (-\frac {8 (a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {a (5 a+8 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {8 (a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a+3 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}}{3 a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {3 \left (-\frac {b \left (-\frac {\frac {8 (a+2 b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (5 a+8 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {8 (a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a+3 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}}{3 a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {3 \left (-\frac {b \left (-\frac {\frac {8 (a+2 b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (5 a+8 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {8 (a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a+3 b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}}{3 a b}+\frac {(a+b) \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x)}{3 a b \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\)

Input:

Int[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),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])/(3*a*b*(a + b*Sin[e + f*x]^2)^(3/2)) + ((2*(a + 3*b)*Csc[e + f 
*x]^3*Sqrt[1 - Sin[e + f*x]^2])/(a*Sqrt[a + b*Sin[e + f*x]^2]) + (3*(-1/3* 
((3*a + 8*b)*Csc[e + f*x]^3*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b*Sin[e + f* 
x]^2])/a - (b*((-8*(a + 2*b)*Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a 
+ b*Sin[e + f*x]^2])/a - ((8*(a + 2*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*(5*a 
+ 8*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)))/a)/(3*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 441

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(632\) vs. \(2(314)=628\).

Time = 5.35 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.84

method	result	size
default	\(-\frac {5 \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} b \sin \left (f x +e \right )^{5}+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 {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \sin \left (f x +e \right )^{5}-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^{2} b \sin \left (f x +e \right )^{5}-16 \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^{2} \sin \left (f x +e \right )^{5}+8 \sin \left (f x +e \right )^{8} a \,b^{2}+16 \sin \left (f x +e \right )^{8} b^{3}+5 \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^{3} \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 {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \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^{3} \sin \left (f x +e \right )^{3}-16 \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} b \sin \left (f x +e \right )^{3}+13 \sin \left (f x +e \right )^{6} a^{2} b +16 \sin \left (f x +e \right )^{6} a \,b^{2}-16 \sin \left (f x +e \right )^{6} b^{3}+4 \sin \left (f x +e \right )^{4} a^{3}-7 a^{2} b \sin \left (f x +e \right )^{4}-24 \sin \left (f x +e \right )^{4} a \,b^{2}-5 \sin \left (f x +e \right )^{2} a^{3}-6 \sin \left (f x +e \right )^{2} a^{2} b +a^{3}}{3 \sin \left (f x +e \right )^{3} a^{4} \left (a +b \sin \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \cos \left (f x +e \right ) f}\)	\(633\)

Input:

int(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(5*(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*b*sin(f*x+e)^5+8*(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*b^2*sin(f*x+e)^5-8*(c 
os(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*b*sin(f*x+e)^5-16*(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^2*sin(f*x+e)^5+8*sin(f*x+e)^ 
8*a*b^2+16*sin(f*x+e)^8*b^3+5*(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^3*sin(f*x+e)^3+8*(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 
*b*sin(f*x+e)^3-8*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*Ellipt 
icE(sin(f*x+e),(-b/a)^(1/2))*a^3*sin(f*x+e)^3-16*(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*b*sin(f*x+ 
e)^3+13*sin(f*x+e)^6*a^2*b+16*sin(f*x+e)^6*a*b^2-16*sin(f*x+e)^6*b^3+4*sin 
(f*x+e)^4*a^3-7*a^2*b*sin(f*x+e)^4-24*sin(f*x+e)^4*a*b^2-5*sin(f*x+e)^2*a^ 
3-6*sin(f*x+e)^2*a^2*b+a^3)/sin(f*x+e)^3/a^4/(a+b*sin(f*x+e)^2)^(3/2)/cos( 
f*x+e)/f
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 1971, normalized size of antiderivative = 5.73 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

Output:

-1/3*(4*(2*((-I*a*b^4 - 2*I*b^5)*cos(f*x + e)^6 + I*a^3*b^2 + 4*I*a^2*b^3 
+ 5*I*a*b^4 + 2*I*b^5 + (2*I*a^2*b^3 + 7*I*a*b^4 + 6*I*b^5)*cos(f*x + e)^4 
 + (-I*a^3*b^2 - 6*I*a^2*b^3 - 11*I*a*b^4 - 6*I*b^5)*cos(f*x + e)^2)*sqrt( 
-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + ((-2*I*a^2*b^3 - 5*I*a*b^4 - 2*I* 
b^5)*cos(f*x + e)^6 + 2*I*a^4*b + 9*I*a^3*b^2 + 14*I*a^2*b^3 + 9*I*a*b^4 + 
 2*I*b^5 + (4*I*a^3*b^2 + 16*I*a^2*b^3 + 19*I*a*b^4 + 6*I*b^5)*cos(f*x + e 
)^4 + (-2*I*a^4*b - 13*I*a^3*b^2 - 28*I*a^2*b^3 - 23*I*a*b^4 - 6*I*b^5)*co 
s(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)*sq 
rt((a^2 + a*b)/b^2))/b^2) + 4*(2*((I*a*b^4 + 2*I*b^5)*cos(f*x + e)^6 - I*a 
^3*b^2 - 4*I*a^2*b^3 - 5*I*a*b^4 - 2*I*b^5 + (-2*I*a^2*b^3 - 7*I*a*b^4 - 6 
*I*b^5)*cos(f*x + e)^4 + (I*a^3*b^2 + 6*I*a^2*b^3 + 11*I*a*b^4 + 6*I*b^5)* 
cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + ((2*I*a^2*b^ 
3 + 5*I*a*b^4 + 2*I*b^5)*cos(f*x + e)^6 - 2*I*a^4*b - 9*I*a^3*b^2 - 14*I*a 
^2*b^3 - 9*I*a*b^4 - 2*I*b^5 + (-4*I*a^3*b^2 - 16*I*a^2*b^3 - 19*I*a*b^4 - 
 6*I*b^5)*cos(f*x + e)^4 + (2*I*a^4*b + 13*I*a^3*b^2 + 28*I*a^2*b^3 + 23*I 
*a*b^4 + 6*I*b^5)*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...
 

Sympy [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(cot(f*x+e)**4/(a+b*sin(f*x+e)**2)**(5/2),x)
 

Output:

Integral(cot(e + f*x)**4/(a + b*sin(e + f*x)**2)**(5/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \] Input:

int(cot(e + f*x)^4/(a + b*sin(e + f*x)^2)^(5/2),x)
 

Output:

int(cot(e + f*x)^4/(a + b*sin(e + f*x)^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right )^{2} b +a}\, \cot \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{6} b^{3}+3 \sin \left (f x +e \right )^{4} a \,b^{2}+3 \sin \left (f x +e \right )^{2} a^{2} b +a^{3}}d x \] Input:

int(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int((sqrt(sin(e + f*x)**2*b + a)*cot(e + f*x)**4)/(sin(e + f*x)**6*b**3 + 
3*sin(e + f*x)**4*a*b**2 + 3*sin(e + f*x)**2*a**2*b + a**3),x)