\(\int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx\) [383]

Optimal result

Integrand size = 21, antiderivative size = 95 \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{6 f} \] Output:

-5/6*b*csc(f*x+e)/f/(b*sec(f*x+e))^(1/2)-1/3*b*csc(f*x+e)^3/f/(b*sec(f*x+e 
))^(1/2)+5/6*cos(f*x+e)^(1/2)*InverseJacobiAM(1/2*f*x+1/2*e,2^(1/2))*(b*se 
c(f*x+e))^(1/2)/f
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\left (-\cot (e+f x) \left (5+2 \csc ^2(e+f x)\right )+5 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )\right ) \sqrt {b \sec (e+f x)}}{6 f} \] Input:

Integrate[Csc[e + f*x]^4*Sqrt[b*Sec[e + f*x]],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {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.

\(\displaystyle \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \csc (e+f x)^4 \sqrt {b \sec (e+f x)}dx\)

\(\Big \downarrow \) 3105

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3105

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

Input:

Int[Csc[e + f*x]^4*Sqrt[b*Sec[e + f*x]],x]
 

Output:

-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
 

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 = 3.58 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.11

method	result	size
default	\(\frac {\left (\frac {5 \cot \left (f x +e \right )^{3}}{6}-\frac {7 \cot \left (f x +e \right ) \csc \left (f x +e \right )^{2}}{6}+\frac {5 i \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )}{6}\right ) \sqrt {b \sec \left (f x +e \right )}}{f}\)	\(105\)

Input:

int(csc(f*x+e)^4*(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/f*(5/6*cot(f*x+e)^3-7/6*cot(f*x+e)*csc(f*x+e)^2+5/6*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 = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.56 \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {5 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (5 \, \cos \left (f x + e\right )^{3} - 7 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \] Input:

integrate(csc(f*x+e)^4*(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

-1/12*(5*sqrt(2)*(I*cos(f*x + e)^2 - I)*sqrt(b)*sin(f*x + e)*weierstrassPI 
nverse(-4, 0, cos(f*x + e) + I*sin(f*x + e)) + 5*sqrt(2)*(-I*cos(f*x + e)^ 
2 + I)*sqrt(b)*sin(f*x + e)*weierstrassPInverse(-4, 0, cos(f*x + e) - I*si 
n(f*x + e)) + 2*(5*cos(f*x + e)^3 - 7*cos(f*x + e))*sqrt(b/cos(f*x + e)))/ 
((f*cos(f*x + e)^2 - f)*sin(f*x + e))
 

Sympy [F]

\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \sqrt {b \sec {\left (e + f x \right )}} \csc ^{4}{\left (e + f x \right )}\, dx \] Input:

integrate(csc(f*x+e)**4*(b*sec(f*x+e))**(1/2),x)
 

Output:

Integral(sqrt(b*sec(e + f*x))*csc(e + f*x)**4, x)
 

Maxima [F]

\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4} \,d x } \] Input:

integrate(csc(f*x+e)^4*(b*sec(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*sec(f*x + e))*csc(f*x + e)^4, x)
 

Giac [F]

\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4} \,d x } \] Input:

integrate(csc(f*x+e)^4*(b*sec(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*sec(f*x + e))*csc(f*x + e)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^4} \,d x \] Input:

int((b/cos(e + f*x))^(1/2)/sin(e + f*x)^4,x)
 

Output:

int((b/cos(e + f*x))^(1/2)/sin(e + f*x)^4, x)
 

Reduce [F]

\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\sqrt {b}\, \left (\int \sqrt {\sec \left (f x +e \right )}\, \csc \left (f x +e \right )^{4}d x \right ) \] Input:

int(csc(f*x+e)^4*(b*sec(f*x+e))^(1/2),x)
                                                                                    
                                                                                    
 

Output:

sqrt(b)*int(sqrt(sec(e + f*x))*csc(e + f*x)**4,x)