\(\int \frac {\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx\) [256]

Optimal result

Integrand size = 23, antiderivative size = 318 \[ \int \frac {\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx=\frac {4 a \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) (a+b)^{3/2} f}-\frac {(3 a-b) \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) (a+b)^{3/2} f}-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {4 a b^2 \tan (e+f x)}{\left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}} \] Output:

4*a*cot(f*x+e)*EllipticE((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^ 
(1/2))*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b))^(1/2)/(a-b 
)/(a+b)^(3/2)/f-(3*a-b)*cot(f*x+e)*EllipticF((a+b*sec(f*x+e))^(1/2)/(a+b)^ 
(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e 
))/(a-b))^(1/2)/(a-b)/(a+b)^(3/2)/f-cot(f*x+e)/f/(a+b*sec(f*x+e))^(3/2)+b^ 
2*tan(f*x+e)/(a^2-b^2)/f/(a+b*sec(f*x+e))^(3/2)+4*a*b^2*tan(f*x+e)/(a^2-b^ 
2)^2/f/(a+b*sec(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 5.87 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.81 \[ \int \frac {\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx=\frac {-((a-b) ((3 a-b) b+a (a-3 b) \cos (e+f x)) \csc (e+f x))+8 a b (a+b) \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sec (e+f x) \sqrt {\frac {1}{1+\sec (e+f x)}}-2 b \left (3 a^2+4 a b+b^2\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sec (e+f x) \sqrt {\frac {1}{1+\sec (e+f x)}}}{\left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}} \] Input:

Integrate[Csc[e + f*x]^2/(a + b*Sec[e + f*x])^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 4363, 25, 3042, 4320, 27, 3042, 4491, 27, 3042, 4493, 3042, 4319, 4492}

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.

\(\displaystyle \int \frac {\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cos \left (e+f x-\frac {\pi }{2}\right )^2 \left (a-b \csc \left (e+f x-\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 4363

\(\displaystyle \frac {3}{2} b \int -\frac {\sec (e+f x)}{(a+b \sec (e+f x))^{5/2}}dx-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3}{2} b \int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^{5/2}}dx-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3}{2} b \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4320

\(\displaystyle -\frac {3}{2} b \left (-\frac {2 \int -\frac {\sec (e+f x) (3 a-b \sec (e+f x))}{2 (a+b \sec (e+f x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {2 b \tan (e+f x)}{3 f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}\right )-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{2} b \left (\frac {\int \frac {\sec (e+f x) (3 a-b \sec (e+f x))}{(a+b \sec (e+f x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {2 b \tan (e+f x)}{3 f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}\right )-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3}{2} b \left (\frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (3 a-b \csc \left (e+f x+\frac {\pi }{2}\right )\right )}{\left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {2 b \tan (e+f x)}{3 f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}\right )-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4491

\(\displaystyle -\frac {3}{2} b \left (\frac {-\frac {2 \int -\frac {\sec (e+f x) \left (3 a^2+4 b \sec (e+f x) a+b^2\right )}{2 \sqrt {a+b \sec (e+f x)}}dx}{a^2-b^2}-\frac {8 a b \tan (e+f x)}{f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}}{3 \left (a^2-b^2\right )}-\frac {2 b \tan (e+f x)}{3 f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}\right )-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{2} b \left (\frac {\frac {\int \frac {\sec (e+f x) \left (3 a^2+4 b \sec (e+f x) a+b^2\right )}{\sqrt {a+b \sec (e+f x)}}dx}{a^2-b^2}-\frac {8 a b \tan (e+f x)}{f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}}{3 \left (a^2-b^2\right )}-\frac {2 b \tan (e+f x)}{3 f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}\right )-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3}{2} b \left (\frac {\frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (3 a^2+4 b \csc \left (e+f x+\frac {\pi }{2}\right ) a+b^2\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {8 a b \tan (e+f x)}{f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}}{3 \left (a^2-b^2\right )}-\frac {2 b \tan (e+f x)}{3 f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}\right )-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4493

\(\displaystyle -\frac {3}{2} b \left (\frac {\frac {(a-b) (3 a-b) \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx+4 a b \int \frac {\sec (e+f x) (\sec (e+f x)+1)}{\sqrt {a+b \sec (e+f x)}}dx}{a^2-b^2}-\frac {8 a b \tan (e+f x)}{f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}}{3 \left (a^2-b^2\right )}-\frac {2 b \tan (e+f x)}{3 f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}\right )-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3}{2} b \left (\frac {\frac {(a-b) (3 a-b) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+4 a b \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {8 a b \tan (e+f x)}{f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}}{3 \left (a^2-b^2\right )}-\frac {2 b \tan (e+f x)}{3 f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}\right )-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4319

\(\displaystyle -\frac {3}{2} b \left (\frac {\frac {4 a b \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) (3 a-b) \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b f}}{a^2-b^2}-\frac {8 a b \tan (e+f x)}{f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}}{3 \left (a^2-b^2\right )}-\frac {2 b \tan (e+f x)}{3 f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}\right )-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4492

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

Input:

Int[Csc[e + f*x]^2/(a + b*Sec[e + f*x])^(3/2),x]
 

Output:

-(Cot[e + f*x]/(f*(a + b*Sec[e + f*x])^(3/2))) - (3*b*((-2*b*Tan[e + f*x]) 
/(3*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^(3/2)) + (((-8*a*(a - b)*Sqrt[a + b 
]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a 
+ b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + 
f*x]))/(a - b))])/(b*f) + (2*(a - b)*(3*a - b)*Sqrt[a + b]*Cot[e + f*x]*El 
lipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqr 
t[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]) 
/(b*f))/(a^2 - b^2) - (8*a*b*Tan[e + f*x])/((a^2 - b^2)*f*Sqrt[a + b*Sec[e 
 + f*x]]))/(3*(a^2 - b^2))))/2
 

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 3042

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

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4320

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_ 
Symbol] :> Simp[(-b)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(f*(m + 1)* 
(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2))   Int[Csc[e + f*x]*(a + b* 
Csc[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + 2)*Csc[e + f*x]), x], x] /; FreeQ 
[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
 

rule 4363

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)/cos[(e_.) + (f_.)*(x_)]^2, 
x_Symbol] :> Simp[Tan[e + f*x]*((a + b*Csc[e + f*x])^m/f), x] + Simp[b*m 
Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, 
 m}, x]
 

rule 4491

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs 
c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e 
 + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1 
/((m + 1)*(a^2 - b^2))   Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp 
[(a*A - b*B)*(m + 1) - (A*b - a*B)*(m + 2)*Csc[e + f*x], x], x], x] /; Free 
Q[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m 
, -1]
 

rule 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

rule 4493

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(A - B)   Int[Csc[e 
 + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[B   Int[Csc[e + f*x]*((1 + 
Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B} 
, x] && NeQ[a^2 - b^2, 0] && NeQ[A^2 - B^2, 0]
 

Maple [A] (verified)

Time = 6.56 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.78

method	result	size
default	\(-\frac {\sqrt {a +b \sec \left (f x +e \right )}\, \left (\left (4 \cos \left (f x +e \right )+4\right ) \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, a^{2} b +\left (4 \cos \left (f x +e \right )+4\right ) \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, a \,b^{2}+\left (-3 \cos \left (f x +e \right )-3\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b +\left (-4 \cos \left (f x +e \right )-4\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{2}+\left (-\cos \left (f x +e \right )-1\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b^{3}+a^{3} \cos \left (f x +e \right ) \cot \left (f x +e \right )+\left (-4 \cos \left (f x +e \right )+3\right ) a^{2} b \cot \left (f x +e \right )+\left (3 \cos \left (f x +e \right )-4\right ) a \,b^{2} \cot \left (f x +e \right )+b^{3} \cot \left (f x +e \right )\right )}{f \left (a -b \right )^{2} \left (a +b \right )^{2} \left (a \cos \left (f x +e \right )+b \right )}\)	\(565\)

Input:

int(csc(f*x+e)^2/(a+b*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/f/(a-b)^2/(a+b)^2*(a+b*sec(f*x+e))^(1/2)/(a*cos(f*x+e)+b)*((4*cos(f*x+e 
)+4)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(cos(f*x+e)/(cos 
(f*x+e)+1))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^(1/2)*a^2*b+(4 
*cos(f*x+e)+4)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(cos(f 
*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^(1/2 
)*a*b^2+(-3*cos(f*x+e)-3)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(a+b)*(a*co 
s(f*x+e)+b)/(cos(f*x+e)+1))^(1/2)*EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/( 
a+b))^(1/2))*a^2*b+(-4*cos(f*x+e)-4)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/ 
(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^(1/2)*EllipticF(cot(f*x+e)-csc(f*x+ 
e),((a-b)/(a+b))^(1/2))*a*b^2+(-cos(f*x+e)-1)*(cos(f*x+e)/(cos(f*x+e)+1))^ 
(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^(1/2)*EllipticF(cot(f*x+e) 
-csc(f*x+e),((a-b)/(a+b))^(1/2))*b^3+a^3*cos(f*x+e)*cot(f*x+e)+(-4*cos(f*x 
+e)+3)*a^2*b*cot(f*x+e)+(3*cos(f*x+e)-4)*a*b^2*cot(f*x+e)+b^3*cot(f*x+e))
                                                                                    
                                                                                    
 

Fricas [F]

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

integrate(csc(f*x+e)^2/(a+b*sec(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*sec(f*x + e) + a)*csc(f*x + e)^2/(b^2*sec(f*x + e)^2 + 2*a 
*b*sec(f*x + e) + a^2), x)
 

Sympy [F]

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

integrate(csc(f*x+e)**2/(a+b*sec(f*x+e))**(3/2),x)
 

Output:

Integral(csc(e + f*x)**2/(a + b*sec(e + f*x))**(3/2), x)
 

Maxima [F]

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

integrate(csc(f*x+e)^2/(a+b*sec(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

integrate(csc(f*x + e)^2/(b*sec(f*x + e) + a)^(3/2), x)
 

Giac [F]

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

integrate(csc(f*x+e)^2/(a+b*sec(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

integrate(csc(f*x + e)^2/(b*sec(f*x + e) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^2\,{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \] Input:

int(1/(sin(e + f*x)^2*(a + b/cos(e + f*x))^(3/2)),x)
 

Output:

int(1/(sin(e + f*x)^2*(a + b/cos(e + f*x))^(3/2)), x)
 

Reduce [F]

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

int(csc(f*x+e)^2/(a+b*sec(f*x+e))^(3/2),x)
 

Output:

int((sqrt(sec(e + f*x)*b + a)*csc(e + f*x)**2)/(sec(e + f*x)**2*b**2 + 2*s 
ec(e + f*x)*a*b + a**2),x)