\(\int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx\) [250]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [F]
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 23, antiderivative size = 121 \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\frac {\sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {\cot (e+f x) \sqrt {a+b \sec (e+f x)}}{f} \] Output:

(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-cot(f*x+e)*(a+b*sec(f*x+e))^(1/2)/f
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\frac {-\left ((b+a \cos (e+f x)) \csc (e+f x) \sqrt {\frac {1}{1+\sec (e+f x)}}\right )+b \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}}{f \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}} \] Input:

Integrate[Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]],x]
 

Output:

(-((b + a*Cos[e + f*x])*Csc[e + f*x]*Sqrt[(1 + Sec[e + f*x])^(-1)]) + b*El 
lipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x 
])/((a + b)*(1 + Sec[e + f*x]))])/(f*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[a 
+ b*Sec[e + f*x]])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4363, 25, 3042, 4319}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4363

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4319

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

Input:

Int[Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]],x]
 

Output:

(Sqrt[a + b]*Cot[e + f*x]*EllipticF[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))])/f - (Cot[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/f
 

Defintions of rubi rules used

rule 25
Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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]
 
Maple [A] (verified)

Time = 3.75 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.15

method result size
default \(-\frac {\sqrt {a +b \sec \left (f x +e \right )}\, \left (\left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b +a \cos \left (f x +e \right ) \cot \left (f x +e \right )+b \cot \left (f x +e \right )\right )}{f \left (a \cos \left (f x +e \right )+b \right )}\) \(139\)

Input:

int(csc(f*x+e)^2*(a+b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/f*(a+b*sec(f*x+e))^(1/2)/(a*cos(f*x+e)+b)*((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)*Ellip 
ticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*b+a*cos(f*x+e)*cot(f*x+e)+ 
b*cot(f*x+e))
 

Fricas [F]

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

integrate(csc(f*x+e)^2*(a+b*sec(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(csc(f*x+e)**2*(a+b*sec(f*x+e))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(csc(f*x+e)^2*(a+b*sec(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(csc(f*x+e)^2*(a+b*sec(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(csc(f*x+e)^2*(a+b*sec(f*x+e))^(1/2),x)
 

Output:

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