3.3.54 \(\int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx\) [254]

3.3.54.1 Optimal result
3.3.54.2 Mathematica [A] (warning: unable to verify)
3.3.54.3 Rubi [A] (verified)
3.3.54.4 Maple [B] (verified)
3.3.54.5 Fricas [F]
3.3.54.6 Sympy [F]
3.3.54.7 Maxima [F]
3.3.54.8 Giac [F]
3.3.54.9 Mupad [F(-1)]

3.3.54.1 Optimal result

Integrand size = 23, antiderivative size = 255 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {\cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{\sqrt {a+b} f}-\frac {\cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{\sqrt {a+b} f}-\frac {\cot (e+f x)}{f \sqrt {a+b \sec (e+f x)}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f \sqrt {a+b \sec (e+f x)}} \]

output
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)/f/(a+b)^ 
(1/2)-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)/f/ 
(a+b)^(1/2)-cot(f*x+e)/f/(a+b*sec(f*x+e))^(1/2)+b^2*tan(f*x+e)/(a^2-b^2)/f 
/(a+b*sec(f*x+e))^(1/2)
 
3.3.54.2 Mathematica [A] (warning: unable to verify)

Time = 7.39 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {\sqrt {\sec (e+f x)} \left (\frac {(b+a \cos (e+f x)) (-a+b \cos (e+f x)) \csc (e+f x)}{\left (a^2-b^2\right ) \sqrt {\sec (e+f x)}}+\frac {b \left (-\frac {(a+b) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left (E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )-\operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}-(b+a \cos (e+f x)) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (-a^2+b^2\right ) \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)}}\right )}{f \sqrt {a+b \sec (e+f x)}} \]

input
Integrate[Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]],x]
 
output
(Sqrt[Sec[e + f*x]]*(((b + a*Cos[e + f*x])*(-a + b*Cos[e + f*x])*Csc[e + f 
*x])/((a^2 - b^2)*Sqrt[Sec[e + f*x]]) + (b*(-(((a + b)*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)] - EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]))/ 
Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) - (b + a*Cos[e + f*x])*Tan[(e + f*x 
)/2]))/((-a^2 + b^2)*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[(e + f*x)/2]^2*Sec[ 
e + f*x]])))/(f*Sqrt[a + b*Sec[e + f*x]])
 
3.3.54.3 Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 4363, 25, 3042, 4320, 27, 3042, 4316, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4363

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4320

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4316

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4319

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

\(\Big \downarrow \) 4492

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

input
Int[Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]],x]
 
output
-(Cot[e + f*x]/(f*Sqrt[a + b*Sec[e + f*x]])) - (b*(((-2*(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)*Sqrt[a + b]*Cot[e + f*x]*EllipticF[Ar 
cSin[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))/(a 
^2 - b^2) - (2*b*Tan[e + f*x])/((a^2 - b^2)*f*Sqrt[a + b*Sec[e + f*x]])))/ 
2
 

3.3.54.3.1 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 4316
Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> 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}, x] && NeQ[a^2 - b^2, 0]
 

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 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]
 
3.3.54.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(758\) vs. \(2(233)=466\).

Time = 5.73 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.98

method result size
default \(-\frac {\sqrt {a +b \sec \left (f x +e \right )}\, \left (-\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a b \cos \left (f x +e \right )-\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, b^{2} \cos \left (f x +e \right )+\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a b \cos \left (f x +e \right )+\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b^{2} \cos \left (f x +e \right )-\sqrt {\frac {b +a \cos \left (f x +e \right )}{\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 ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a b -\sqrt {\frac {b +a \cos \left (f x +e \right )}{\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 ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, b^{2}+\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\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}}\, a b +\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\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}}\, b^{2}+a^{2} \cos \left (f x +e \right ) \cot \left (f x +e \right )-a b \cos \left (f x +e \right ) \cot \left (f x +e \right )+a b \cot \left (f x +e \right )-b^{2} \cot \left (f x +e \right )\right )}{f \left (a -b \right ) \left (a +b \right ) \left (b +a \cos \left (f x +e \right )\right )}\) \(759\)

input
int(csc(f*x+e)^2/(a+b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 
output
-1/f/(a-b)/(a+b)*(a+b*sec(f*x+e))^(1/2)/(b+a*cos(f*x+e))*(-EllipticF(cot(f 
*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e 
)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*a*b*cos(f*x+e)-EllipticF(cot 
(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x 
+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*b^2*cos(f*x+e)+(1/(a+b)*(b 
+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ell 
ipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*a*b*cos(f*x+e)+(1/(a+b)* 
(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*E 
llipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*b^2*cos(f*x+e)-(1/(a+b 
)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*EllipticF(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)*a*b-(1/(a+b)*(b+a*co 
s(f*x+e))/(cos(f*x+e)+1))^(1/2)*EllipticF(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)*b^2+(1/(a+b)*(b+a*cos(f*x+e)) 
/(cos(f*x+e)+1))^(1/2)*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)*a*b+(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x 
+e)+1))^(1/2)*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)*b^2+a^2*cos(f*x+e)*cot(f*x+e)-a*b*cos(f*x+e)*co 
t(f*x+e)+a*b*cot(f*x+e)-b^2*cot(f*x+e))
 
3.3.54.5 Fricas [F]

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

input
integrate(csc(f*x+e)^2/(a+b*sec(f*x+e))^(1/2),x, algorithm="fricas")
 
output
integral(csc(f*x + e)^2/sqrt(b*sec(f*x + e) + a), x)
 
3.3.54.6 Sympy [F]

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

input
integrate(csc(f*x+e)**2/(a+b*sec(f*x+e))**(1/2),x)
 
output
Integral(csc(e + f*x)**2/sqrt(a + b*sec(e + f*x)), x)
 
3.3.54.7 Maxima [F]

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

input
integrate(csc(f*x+e)^2/(a+b*sec(f*x+e))^(1/2),x, algorithm="maxima")
 
output
integrate(csc(f*x + e)^2/sqrt(b*sec(f*x + e) + a), x)
 
3.3.54.8 Giac [F]

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

input
integrate(csc(f*x+e)^2/(a+b*sec(f*x+e))^(1/2),x, algorithm="giac")
 
output
integrate(csc(f*x + e)^2/sqrt(b*sec(f*x + e) + a), x)
 
3.3.54.9 Mupad [F(-1)]

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

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