\(\int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx\) [524]

Optimal result

Integrand size = 22, antiderivative size = 106 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {30 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )}{77 b}-\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {30 \cos (2 a+2 b x)}{77 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \] Output:

30/77*InverseJacobiAM(a-1/4*Pi+b*x,2^(1/2))/b-18/77*cos(2*b*x+2*a)/b/sin(2 
*b*x+2*a)^(7/2)-1/11*csc(b*x+a)^2/b/sin(2*b*x+2*a)^(7/2)-30/77*cos(2*b*x+2 
*a)/b/sin(2*b*x+2*a)^(3/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.81 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {480 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )+\left (-141 \csc ^2(a+b x)-32 \csc ^4(a+b x)-7 \csc ^6(a+b x)+11 \sec ^2(a+b x) \left (9+\sec ^2(a+b x)\right )\right ) \sqrt {\sin (2 (a+b x))}}{1232 b} \] Input:

Integrate[Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(9/2),x]
 

Output:

(480*EllipticF[a - Pi/4 + b*x, 2] + (-141*Csc[a + b*x]^2 - 32*Csc[a + b*x] 
^4 - 7*Csc[a + b*x]^6 + 11*Sec[a + b*x]^2*(9 + Sec[a + b*x]^2))*Sqrt[Sin[2 
*(a + b*x)]])/(1232*b)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4788, 3042, 3116, 3042, 3116, 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.

\(\displaystyle \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (a+b x)^2 \sin (2 a+2 b x)^{9/2}}dx\)

\(\Big \downarrow \) 4788

\(\displaystyle \frac {18}{11} \int \frac {1}{\sin ^{\frac {9}{2}}(2 a+2 b x)}dx-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {18}{11} \int \frac {1}{\sin (2 a+2 b x)^{9/2}}dx-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 3116

\(\displaystyle \frac {18}{11} \left (\frac {5}{7} \int \frac {1}{\sin ^{\frac {5}{2}}(2 a+2 b x)}dx-\frac {\cos (2 a+2 b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {18}{11} \left (\frac {5}{7} \int \frac {1}{\sin (2 a+2 b x)^{5/2}}dx-\frac {\cos (2 a+2 b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 3116

\(\displaystyle \frac {18}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx-\frac {\cos (2 a+2 b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\right )-\frac {\cos (2 a+2 b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {18}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx-\frac {\cos (2 a+2 b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\right )-\frac {\cos (2 a+2 b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {18}{11} \left (\frac {5}{7} \left (\frac {\operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{3 b}-\frac {\cos (2 a+2 b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\right )-\frac {\cos (2 a+2 b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}\)

Input:

Int[Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(9/2),x]
 

Output:

(18*((5*(EllipticF[a - Pi/4 + b*x, 2]/(3*b) - Cos[2*a + 2*b*x]/(3*b*Sin[2* 
a + 2*b*x]^(3/2))))/7 - Cos[2*a + 2*b*x]/(7*b*Sin[2*a + 2*b*x]^(7/2))))/11 
 - Csc[a + b*x]^2/(11*b*Sin[2*a + 2*b*x]^(7/2))
 

Defintions of rubi rules used

rule 3042

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

rule 3116

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

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 4788

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p 
_), x_Symbol] :> Simp[(e*Sin[a + b*x])^m*((g*Sin[c + d*x])^(p + 1)/(2*b*g*( 
m + p + 1))), x] + Simp[(m + 2*p + 2)/(e^2*(m + p + 1))   Int[(e*Sin[a + b* 
x])^(m + 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] & 
& EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ[p] && LtQ[m, -1] && NeQ[m 
+ 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
 

Maple [F(-1)]

Timed out.

\[\int \frac {\csc \left (b x +a \right )^{2}}{\sin \left (2 b x +2 a \right )^{\frac {9}{2}}}d x\]

Input:

int(csc(b*x+a)^2/sin(2*b*x+2*a)^(9/2),x)
 

Output:

int(csc(b*x+a)^2/sin(2*b*x+2*a)^(9/2),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.22 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=-\frac {240 \, \sqrt {2 i} {\left (\cos \left (b x + a\right )^{10} - 3 \, \cos \left (b x + a\right )^{8} + 3 \, \cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + 240 \, \sqrt {-2 i} {\left (\cos \left (b x + a\right )^{10} - 3 \, \cos \left (b x + a\right )^{8} + 3 \, \cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - \sqrt {2} {\left (240 \, \cos \left (b x + a\right )^{8} - 600 \, \cos \left (b x + a\right )^{6} + 444 \, \cos \left (b x + a\right )^{4} - 66 \, \cos \left (b x + a\right )^{2} - 11\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{1232 \, {\left (b \cos \left (b x + a\right )^{10} - 3 \, b \cos \left (b x + a\right )^{8} + 3 \, b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )}} \] Input:

integrate(csc(b*x+a)^2/sin(2*b*x+2*a)^(9/2),x, algorithm="fricas")
 

Output:

-1/1232*(240*sqrt(2*I)*(cos(b*x + a)^10 - 3*cos(b*x + a)^8 + 3*cos(b*x + a 
)^6 - cos(b*x + a)^4)*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1 
) + 240*sqrt(-2*I)*(cos(b*x + a)^10 - 3*cos(b*x + a)^8 + 3*cos(b*x + a)^6 
- cos(b*x + a)^4)*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) - 
sqrt(2)*(240*cos(b*x + a)^8 - 600*cos(b*x + a)^6 + 444*cos(b*x + a)^4 - 66 
*cos(b*x + a)^2 - 11)*sqrt(cos(b*x + a)*sin(b*x + a)))/(b*cos(b*x + a)^10 
- 3*b*cos(b*x + a)^8 + 3*b*cos(b*x + a)^6 - b*cos(b*x + a)^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \] Input:

integrate(csc(b*x+a)**2/sin(2*b*x+2*a)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate(csc(b*x+a)^2/sin(2*b*x+2*a)^(9/2),x, algorithm="maxima")
 

Output:

integrate(csc(b*x + a)^2/sin(2*b*x + 2*a)^(9/2), x)
 

Giac [F]

\[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate(csc(b*x+a)^2/sin(2*b*x+2*a)^(9/2),x, algorithm="giac")
 

Output:

integrate(csc(b*x + a)^2/sin(2*b*x + 2*a)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{9/2}} \,d x \] Input:

int(1/(sin(a + b*x)^2*sin(2*a + 2*b*x)^(9/2)),x)
 

Output:

int(1/(sin(a + b*x)^2*sin(2*a + 2*b*x)^(9/2)), x)
 

Reduce [F]

\[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int \frac {\sqrt {\sin \left (2 b x +2 a \right )}\, \csc \left (b x +a \right )^{2}}{\sin \left (2 b x +2 a \right )^{5}}d x \] Input:

int(csc(b*x+a)^2/sin(2*b*x+2*a)^(9/2),x)
 

Output:

int((sqrt(sin(2*a + 2*b*x))*csc(a + b*x)**2)/sin(2*a + 2*b*x)**5,x)