\(\int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx\) [384]

Optimal result

Integrand size = 21, antiderivative size = 123 \[ \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {3 b \csc (e+f x)}{4 f \sqrt {b \sec (e+f x)}}-\frac {3 b \csc ^3(e+f x)}{10 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}+\frac {3 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{4 f} \] Output:

-3/4*b*csc(f*x+e)/f/(b*sec(f*x+e))^(1/2)-3/10*b*csc(f*x+e)^3/f/(b*sec(f*x+ 
e))^(1/2)-1/5*b*csc(f*x+e)^5/f/(b*sec(f*x+e))^(1/2)+3/4*cos(f*x+e)^(1/2)*I 
nverseJacobiAM(1/2*f*x+1/2*e,2^(1/2))*(b*sec(f*x+e))^(1/2)/f
 

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59 \[ \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\left (-\cot (e+f x) \left (15+6 \csc ^2(e+f x)+4 \csc ^4(e+f x)\right )+15 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )\right ) \sqrt {b \sec (e+f x)}}{20 f} \] Input:

Integrate[Csc[e + f*x]^6*Sqrt[b*Sec[e + f*x]],x]
 

Output:

((-(Cot[e + f*x]*(15 + 6*Csc[e + f*x]^2 + 4*Csc[e + f*x]^4)) + 15*Sqrt[Cos 
[e + f*x]]*EllipticF[(e + f*x)/2, 2])*Sqrt[b*Sec[e + f*x]])/(20*f)
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3105, 3042, 3105, 3042, 3105, 3042, 4258, 3042, 3120}

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.

\(\displaystyle \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \csc (e+f x)^6 \sqrt {b \sec (e+f x)}dx\)

\(\Big \downarrow \) 3105

\(\displaystyle \frac {9}{10} \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)}dx-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9}{10} \int \csc (e+f x)^4 \sqrt {b \sec (e+f x)}dx-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3105

\(\displaystyle \frac {9}{10} \left (\frac {5}{6} \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)}dx-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9}{10} \left (\frac {5}{6} \int \csc (e+f x)^2 \sqrt {b \sec (e+f x)}dx-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3105

\(\displaystyle \frac {9}{10} \left (\frac {5}{6} \left (\frac {1}{2} \int \sqrt {b \sec (e+f x)}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9}{10} \left (\frac {5}{6} \left (\frac {1}{2} \int \sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {9}{10} \left (\frac {5}{6} \left (\frac {1}{2} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9}{10} \left (\frac {5}{6} \left (\frac {1}{2} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {9}{10} \left (\frac {5}{6} \left (\frac {\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f}-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}\)

Input:

Int[Csc[e + f*x]^6*Sqrt[b*Sec[e + f*x]],x]
 

Output:

-1/5*(b*Csc[e + f*x]^5)/(f*Sqrt[b*Sec[e + f*x]]) + (9*(-1/3*(b*Csc[e + f*x 
]^3)/(f*Sqrt[b*Sec[e + f*x]]) + (5*(-((b*Csc[e + f*x])/(f*Sqrt[b*Sec[e + f 
*x]])) + (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x] 
])/f))/6))/10
 

Defintions of rubi rules used

rule 3042

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

rule 3105

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n 
_.), x_Symbol] :> Simp[(-a)*b*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n 
 - 1)/(f*(m - 1))), x] + Simp[a^2*((m + n - 2)/(m - 1))   Int[(a*Csc[e + f* 
x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[ 
m, 1] && IntegersQ[2*m, 2*n] &&  !GtQ[n, m]
 

rule 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 
)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^n*Sin[c + d*x]^n   Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && 
 EqQ[n^2, 1/4]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.84 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {\left (\frac {\left (-15 \cos \left (f x +e \right )^{4}+36 \cos \left (f x +e \right )^{2}-25\right ) \cot \left (f x +e \right ) \csc \left (f x +e \right )^{4}}{20}+\frac {3 i \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )}{4}\right ) \sqrt {b \sec \left (f x +e \right )}}{f}\)	\(117\)

Input:

int(csc(f*x+e)^6*(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/f*(1/20*(-15*cos(f*x+e)^4+36*cos(f*x+e)^2-25)*cot(f*x+e)*csc(f*x+e)^4+3/ 
4*I*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f 
*x+e)+1)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2))*(b*sec(f*x+e))^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.52 \[ \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {15 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{4} - 2 i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 15 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{4} + 2 i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (15 \, \cos \left (f x + e\right )^{5} - 36 \, \cos \left (f x + e\right )^{3} + 25 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{40 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )} \sin \left (f x + e\right )} \] Input:

integrate(csc(f*x+e)^6*(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

-1/40*(15*sqrt(2)*(I*cos(f*x + e)^4 - 2*I*cos(f*x + e)^2 + I)*sqrt(b)*sin( 
f*x + e)*weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e)) + 15*sq 
rt(2)*(-I*cos(f*x + e)^4 + 2*I*cos(f*x + e)^2 - I)*sqrt(b)*sin(f*x + e)*we 
ierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e)) + 2*(15*cos(f*x + 
e)^5 - 36*cos(f*x + e)^3 + 25*cos(f*x + e))*sqrt(b/cos(f*x + e)))/((f*cos( 
f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)*sin(f*x + e))
 

Sympy [F(-1)]

Timed out. \[ \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx=\text {Timed out} \] Input:

integrate(csc(f*x+e)**6*(b*sec(f*x+e))**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{6} \,d x } \] Input:

integrate(csc(f*x+e)^6*(b*sec(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*sec(f*x + e))*csc(f*x + e)^6, x)
 

Giac [F]

\[ \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{6} \,d x } \] Input:

integrate(csc(f*x+e)^6*(b*sec(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*sec(f*x + e))*csc(f*x + e)^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^6} \,d x \] Input:

int((b/cos(e + f*x))^(1/2)/sin(e + f*x)^6,x)
 

Output:

int((b/cos(e + f*x))^(1/2)/sin(e + f*x)^6, x)
 

Reduce [F]

\[ \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx=\sqrt {b}\, \left (\int \sqrt {\sec \left (f x +e \right )}\, \csc \left (f x +e \right )^{6}d x \right ) \] Input:

int(csc(f*x+e)^6*(b*sec(f*x+e))^(1/2),x)
                                                                                    
                                                                                    
 

Output:

sqrt(b)*int(sqrt(sec(e + f*x))*csc(e + f*x)**6,x)