\(\int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx\) [396]

Optimal result

Integrand size = 21, antiderivative size = 124 \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=-\frac {7 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {7 b \csc (e+f x) \sqrt {b \sec (e+f x)}}{6 f}-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}+\frac {7 b \sqrt {b \sec (e+f x)} \sin (e+f x)}{2 f} \] Output:

-7/2*b^2*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1/2)/(b*sec(f 
*x+e))^(1/2)-7/6*b*csc(f*x+e)*(b*sec(f*x+e))^(1/2)/f-1/3*b*csc(f*x+e)^3*(b 
*sec(f*x+e))^(1/2)/f+7/2*b*(b*sec(f*x+e))^(1/2)*sin(f*x+e)/f
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.62 \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=-\frac {b \left (-21+7 \csc ^2(e+f x)+2 \csc ^4(e+f x)+21 \sqrt {\cos (e+f x)} \csc (e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right ) \sqrt {b \sec (e+f x)} \sin (e+f x)}{6 f} \] Input:

Integrate[Csc[e + f*x]^4*(b*Sec[e + f*x])^(3/2),x]
 

Output:

-1/6*(b*(-21 + 7*Csc[e + f*x]^2 + 2*Csc[e + f*x]^4 + 21*Sqrt[Cos[e + f*x]] 
*Csc[e + f*x]*EllipticE[(e + f*x)/2, 2])*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x] 
)/f
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03, 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, 4255, 3042, 4258, 3042, 3119}

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.

\(\displaystyle \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \csc (e+f x)^4 (b \sec (e+f x))^{3/2}dx\)

\(\Big \downarrow \) 3105

\(\displaystyle \frac {7}{6} \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2}dx-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{6} \int \csc (e+f x)^2 (b \sec (e+f x))^{3/2}dx-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 3105

\(\displaystyle \frac {7}{6} \left (\frac {3}{2} \int (b \sec (e+f x))^{3/2}dx-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\right )-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{6} \left (\frac {3}{2} \int \left (b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\right )-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {7}{6} \left (\frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-b^2 \int \frac {1}{\sqrt {b \sec (e+f x)}}dx\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\right )-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{6} \left (\frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-b^2 \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\right )-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {7}{6} \left (\frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-\frac {b^2 \int \sqrt {\cos (e+f x)}dx}{\sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\right )-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{6} \left (\frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-\frac {b^2 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\right )-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {7}{6} \left (\frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-\frac {2 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\right )-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}\)

Input:

Int[Csc[e + f*x]^4*(b*Sec[e + f*x])^(3/2),x]
 

Output:

-1/3*(b*Csc[e + f*x]^3*Sqrt[b*Sec[e + f*x]])/f + (7*(-((b*Csc[e + f*x]*Sqr 
t[b*Sec[e + f*x]])/f) + (3*((-2*b^2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos 
[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (2*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]) 
/f))/2))/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 3119

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

rule 4255

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

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 = 4.05 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.36

method	result	size
default	\(\frac {\left (-\frac {\csc \left (f x +e \right )^{3}}{3}-\frac {7 \cot \left (f x +e \right )}{2}+\frac {7 \csc \left (f x +e \right )}{3}+\frac {7 i \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )}{2}-\frac {7 i \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )}{2}\right ) b \sqrt {b \sec \left (f x +e \right )}}{f}\)	\(169\)

Input:

int(csc(f*x+e)^4*(b*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/f*(-1/3*csc(f*x+e)^3-7/2*cot(f*x+e)+7/3*csc(f*x+e)+7/2*I*(cos(f*x+e)+1)* 
(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(co 
t(f*x+e)-csc(f*x+e)),I)-7/2*I*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)*(1/(c 
os(f*x+e)+1))^(1/2)*(cos(f*x+e)+1)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2))*b*(b 
*sec(f*x+e))^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.35 \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=-\frac {21 \, \sqrt {2} {\left (i \, b \cos \left (f x + e\right )^{2} - i \, b\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, b \cos \left (f x + e\right )^{2} + i \, b\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (21 \, b \cos \left (f x + e\right )^{4} - 35 \, b \cos \left (f x + e\right )^{2} + 12 \, b\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \] Input:

integrate(csc(f*x+e)^4*(b*sec(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

-1/12*(21*sqrt(2)*(I*b*cos(f*x + e)^2 - I*b)*sqrt(b)*sin(f*x + e)*weierstr 
assZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) 
+ 21*sqrt(2)*(-I*b*cos(f*x + e)^2 + I*b)*sqrt(b)*sin(f*x + e)*weierstrassZ 
eta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + 2* 
(21*b*cos(f*x + e)^4 - 35*b*cos(f*x + e)^2 + 12*b)*sqrt(b/cos(f*x + e)))/( 
(f*cos(f*x + e)^2 - f)*sin(f*x + e))
 

Sympy [F(-1)]

Timed out. \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate(csc(f*x+e)**4*(b*sec(f*x+e))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \csc \left (f x + e\right )^{4} \,d x } \] Input:

integrate(csc(f*x+e)^4*(b*sec(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*sec(f*x + e))^(3/2)*csc(f*x + e)^4, x)
 

Giac [F]

\[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \csc \left (f x + e\right )^{4} \,d x } \] Input:

integrate(csc(f*x+e)^4*(b*sec(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

integrate((b*sec(f*x + e))^(3/2)*csc(f*x + e)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=\int \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^4} \,d x \] Input:

int((b/cos(e + f*x))^(3/2)/sin(e + f*x)^4,x)
 

Output:

int((b/cos(e + f*x))^(3/2)/sin(e + f*x)^4, x)
 

Reduce [F]

\[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=\sqrt {b}\, \left (\int \sqrt {\sec \left (f x +e \right )}\, \csc \left (f x +e \right )^{4} \sec \left (f x +e \right )d x \right ) b \] Input:

int(csc(f*x+e)^4*(b*sec(f*x+e))^(3/2),x)
 

Output:

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