\(\int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx\) [16]

Optimal result

Integrand size = 10, antiderivative size = 90 \[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \cos (a+b x)}{21 b \sqrt {\csc (a+b x)}}+\frac {10 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{21 b} \] Output:

-2/7*cos(b*x+a)/b/csc(b*x+a)^(5/2)-10/21*cos(b*x+a)/b/csc(b*x+a)^(1/2)+10/ 
21*csc(b*x+a)^(1/2)*InverseJacobiAM(1/2*a-1/4*Pi+1/2*b*x,2^(1/2))*sin(b*x+ 
a)^(1/2)/b
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {\sqrt {\csc (a+b x)} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+26 \sin (2 (a+b x))-3 \sin (4 (a+b x))\right )}{84 b} \] Input:

Integrate[Csc[a + b*x]^(-7/2),x]
 

Output:

-1/84*(Sqrt[Csc[a + b*x]]*(40*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin 
[a + b*x]] + 26*Sin[2*(a + b*x)] - 3*Sin[4*(a + b*x)]))/b
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4256, 3042, 4256, 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.

\(\displaystyle \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\csc (a+b x)^{7/2}}dx\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {5}{7} \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)}dx-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5}{7} \int \frac {1}{\csc (a+b x)^{3/2}}dx-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {5}{7} \left (\frac {1}{3} \int \sqrt {\csc (a+b x)}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5}{7} \left (\frac {1}{3} \int \sqrt {\csc (a+b x)}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {5}{7} \left (\frac {1}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5}{7} \left (\frac {1}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {5}{7} \left (\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b}-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\)

Input:

Int[Csc[a + b*x]^(-7/2),x]
 

Output:

(-2*Cos[a + b*x])/(7*b*Csc[a + b*x]^(5/2)) + (5*((-2*Cos[a + b*x])/(3*b*Sq 
rt[Csc[a + b*x]]) + (2*Sqrt[Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2] 
*Sqrt[Sin[a + b*x]])/(3*b)))/7
 

Defintions of rubi rules used

rule 3042

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

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 4256

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

Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16

method	result	size
default	\(\frac {\frac {2 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )}{7}+\frac {5 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{21}-\frac {16 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )}{21}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\)	\(104\)

Input:

int(1/csc(b*x+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

(2/7*cos(b*x+a)^4*sin(b*x+a)+5/21*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^( 
1/2)*(-sin(b*x+a))^(1/2)*EllipticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))-16/21 
*cos(b*x+a)^2*sin(b*x+a))/cos(b*x+a)/sin(b*x+a)^(1/2)/b
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 8 \, \cos \left (b x + a\right )\right )} \sqrt {\sin \left (b x + a\right )} - 5 i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{21 \, b} \] Input:

integrate(1/csc(b*x+a)^(7/2),x, algorithm="fricas")
 

Output:

1/21*(2*(3*cos(b*x + a)^3 - 8*cos(b*x + a))*sqrt(sin(b*x + a)) - 5*I*sqrt( 
2*I)*weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a)) + 5*I*sqrt(- 
2*I)*weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a)))/b
 

Sympy [F]

\[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {1}{\csc ^{\frac {7}{2}}{\left (a + b x \right )}}\, dx \] Input:

integrate(1/csc(b*x+a)**(7/2),x)
 

Output:

Integral(csc(a + b*x)**(-7/2), x)
 

Maxima [F]

\[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/csc(b*x+a)^(7/2),x, algorithm="maxima")
 

Output:

integrate(csc(b*x + a)^(-7/2), x)
 

Giac [F]

\[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/csc(b*x+a)^(7/2),x, algorithm="giac")
 

Output:

integrate(csc(b*x + a)^(-7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{7/2}} \,d x \] Input:

int(1/(1/sin(a + b*x))^(7/2),x)
 

Output:

int(1/(1/sin(a + b*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {\sqrt {\csc \left (b x +a \right )}}{\csc \left (b x +a \right )^{4}}d x \] Input:

int(1/csc(b*x+a)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

int(sqrt(csc(a + b*x))/csc(a + b*x)**4,x)