\(\int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx\) [515]

Optimal result

Integrand size = 22, antiderivative size = 106 \[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\frac {6 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{7 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b} \] Output:

-6/5*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))/b-2/5*cos(2*b*x+2*a)*sin(2*b*x+2 
*a)^(3/2)/b-2/7*cos(2*b*x+2*a)*sin(2*b*x+2*a)^(7/2)/b+1/7*csc(b*x+a)^2*sin 
(2*b*x+2*a)^(11/2)/b
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\frac {84 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\sqrt {\sin (2 (a+b x))} (15 \sin (2 (a+b x))-14 \sin (4 (a+b x))-5 \sin (6 (a+b x)))}{70 b} \] Input:

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

Output:

(84*EllipticE[a - Pi/4 + b*x, 2] + Sqrt[Sin[2*(a + b*x)]]*(15*Sin[2*(a + b 
*x)] - 14*Sin[4*(a + b*x)] - 5*Sin[6*(a + b*x)]))/(70*b)
 

Rubi [A] (verified)

Time = 0.43 (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, 3115, 3042, 3115, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4788

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

\(\displaystyle \frac {18}{7} \left (\frac {7}{9} \int \sin ^{\frac {5}{2}}(2 a+2 b x)dx-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{9 b}\right )+\frac {\sin ^{\frac {11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {18}{7} \left (\frac {7}{9} \int \sin (2 a+2 b x)^{5/2}dx-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{9 b}\right )+\frac {\sin ^{\frac {11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {18}{7} \left (\frac {7}{9} \left (\frac {3}{5} \int \sqrt {\sin (2 a+2 b x)}dx-\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}\right )-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{9 b}\right )+\frac {\sin ^{\frac {11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {18}{7} \left (\frac {7}{9} \left (\frac {3}{5} \int \sqrt {\sin (2 a+2 b x)}dx-\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}\right )-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{9 b}\right )+\frac {\sin ^{\frac {11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\sin ^{\frac {11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b}+\frac {18}{7} \left (\frac {7}{9} \left (\frac {3 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{5 b}-\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}\right )-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{9 b}\right )\)

Input:

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

Output:

(Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(11/2))/(7*b) + (18*(-1/9*(Cos[2*a + 2*b* 
x]*Sin[2*a + 2*b*x]^(7/2))/b + (7*((3*EllipticE[a - Pi/4 + b*x, 2])/(5*b) 
- (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(3/2))/(5*b)))/9))/7
 

Defintions of rubi rules used

rule 3042

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

rule 3115

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

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(203\) vs. \(2(93)=186\).

Time = 49.23 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.92

method	result	size
default	\(\frac {8 \sqrt {2}\, \left (\frac {\sqrt {2}\, \sin \left (2 b x +2 a \right )^{\frac {7}{2}}}{56}-\frac {\sqrt {2}\, \left (6 \sqrt {\sin \left (2 b x +2 a \right )+1}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (2 b x +2 a \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\sin \left (2 b x +2 a \right )+1}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (2 b x +2 a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (2 b x +2 a \right )^{4}+2 \sin \left (2 b x +2 a \right )^{2}\right )}{80 \cos \left (2 b x +2 a \right ) \sqrt {\sin \left (2 b x +2 a \right )}}\right )}{b}\)	\(204\)

Input:

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

Output:

8*2^(1/2)*(1/56*2^(1/2)*sin(2*b*x+2*a)^(7/2)-1/80*2^(1/2)*(6*(sin(2*b*x+2* 
a)+1)^(1/2)*(-2*sin(2*b*x+2*a)+2)^(1/2)*(-sin(2*b*x+2*a))^(1/2)*EllipticE( 
(sin(2*b*x+2*a)+1)^(1/2),1/2*2^(1/2))-3*(sin(2*b*x+2*a)+1)^(1/2)*(-2*sin(2 
*b*x+2*a)+2)^(1/2)*(-sin(2*b*x+2*a))^(1/2)*EllipticF((sin(2*b*x+2*a)+1)^(1 
/2),1/2*2^(1/2))-2*sin(2*b*x+2*a)^4+2*sin(2*b*x+2*a)^2)/cos(2*b*x+2*a)/sin 
(2*b*x+2*a)^(1/2))/b
 

Fricas [F]

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

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

Output:

integral((cos(2*b*x + 2*a)^4 - 2*cos(2*b*x + 2*a)^2 + 1)*csc(b*x + a)^2*sq 
rt(sin(2*b*x + 2*a)), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

\[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\int { \csc \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}} \,d x } \] 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(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\int \sqrt {\sin \left (2 b x +2 a \right )}\, \csc \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{4}d x \] 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)**4,x)