\(\int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx\) [252]

Optimal result

Integrand size = 23, antiderivative size = 228 \[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=-\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}+\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f} \] Output:

-3*(a-b)*(a+b)^(1/2)*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+3*(a-b)*(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))^(3/2)/f
 

Mathematica [A] (warning: unable to verify)

Time = 10.71 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.21 \[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=\frac {\cos (e+f x) (a+b \sec (e+f x))^{3/2} ((-b-a \cos (e+f x)) \csc (e+f x)+3 b \sin (e+f x))}{f (b+a \cos (e+f x))}+\frac {3 b (a+b \sec (e+f x))^{3/2} \left (-\frac {(a+b) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left (E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )-\operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}-(b+a \cos (e+f x)) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f (b+a \cos (e+f x))^2 \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sec ^{\frac {3}{2}}(e+f x) \sqrt {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)}} \] Input:

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

Output:

(Cos[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*((-b - a*Cos[e + f*x])*Csc[e + f* 
x] + 3*b*Sin[e + f*x]))/(f*(b + a*Cos[e + f*x])) + (3*b*(a + b*Sec[e + f*x 
])^(3/2)*(-(((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[ArcSi 
n[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]))/(f*(b + a*Cos[e + f*x])^2*S 
qrt[Sec[(e + f*x)/2]^2]*Sec[e + f*x]^(3/2)*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + 
 f*x]])
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4363, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4363

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4316

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4319

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

\(\Big \downarrow \) 4492

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

Input:

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

Output:

-((Cot[e + f*x]*(a + b*Sec[e + f*x])^(3/2))/f) + (3*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 
[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
 

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 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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(444\) vs. \(2(208)=416\).

Time = 6.96 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.95

method	result	size
default	\(\frac {\sqrt {a +b \sec \left (f x +e \right )}\, \left (\left (3 \cos \left (f x +e \right )+3\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, a b +\left (3 \cos \left (f x +e \right )+3\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, b^{2}+\left (-3 \cos \left (f x +e \right )-3\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a b +\left (-3 \cos \left (f x +e \right )-3\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b^{2}-a^{2} \cos \left (f x +e \right ) \cot \left (f x +e \right )+\left (-3 \cos \left (f x +e \right )+1\right ) a b \cot \left (f x +e \right )+b^{2} \left (-3 \cot \left (f x +e \right )+2 \csc \left (f x +e \right )\right )\right )}{f \left (a \cos \left (f x +e \right )+b \right )}\)	\(445\)

Input:

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

Output:

1/f*(a+b*sec(f*x+e))^(1/2)/(a*cos(f*x+e)+b)*((3*cos(f*x+e)+3)*(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)) 
*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^(1/2)*a*b+(3*cos(f*x+e)+3)*(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))*(1/(a+b)*(a*cos(f*x+e)+b)/(cos(f*x+e)+1))^(1/2)*b^2+(-3*cos(f*x+e 
)-3)*(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+(-3* 
cos(f*x+e)-3)*(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^2-a^2*cos(f*x+e)*cot(f*x+e)+(-3*cos(f*x+e)+1)*a*b*cot(f*x+e)+b^2*(-3*co 
t(f*x+e)+2*csc(f*x+e)))
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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