Integrand size = 22, antiderivative size = 106 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=-\frac {14 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{15 b}-\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {14 \cos (2 a+2 b x)}{15 b \sqrt {\sin (2 a+2 b x)}} \]
14/15*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi +b*x),2^(1/2))/b-14/45*cos(2*b*x+2*a)/b/sin(2*b*x+2*a)^(5/2)-1/9*csc(b*x+a )^2/b/sin(2*b*x+2*a)^(5/2)-14/15*cos(2*b*x+2*a)/b/sin(2*b*x+2*a)^(1/2)
Time = 1.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.80 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=-\frac {336 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\frac {(-9+98 \cos (2 (a+b x))-28 \cos (4 (a+b x))-42 \cos (6 (a+b x))+21 \cos (8 (a+b x))) \csc ^2(a+b x)}{\sin ^{\frac {5}{2}}(2 (a+b x))}}{360 b} \]
-1/360*(336*EllipticE[a - Pi/4 + b*x, 2] + ((-9 + 98*Cos[2*(a + b*x)] - 28 *Cos[4*(a + b*x)] - 42*Cos[6*(a + b*x)] + 21*Cos[8*(a + b*x)])*Csc[a + b*x ]^2)/Sin[2*(a + b*x)]^(5/2))/b
Time = 0.43 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4788, 3042, 3116, 3042, 3116, 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.
| \(\displaystyle \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (a+b x)^2 \sin (2 a+2 b x)^{7/2}}dx\) |
| \(\Big \downarrow \) 4788 |
| \(\displaystyle \frac {14}{9} \int \frac {1}{\sin ^{\frac {7}{2}}(2 a+2 b x)}dx-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {14}{9} \int \frac {1}{\sin (2 a+2 b x)^{7/2}}dx-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {14}{9} \left (\frac {3}{5} \int \frac {1}{\sin ^{\frac {3}{2}}(2 a+2 b x)}dx-\frac {\cos (2 a+2 b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {14}{9} \left (\frac {3}{5} \int \frac {1}{\sin (2 a+2 b x)^{3/2}}dx-\frac {\cos (2 a+2 b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {14}{9} \left (\frac {3}{5} \left (-\int \sqrt {\sin (2 a+2 b x)}dx-\frac {\cos (2 a+2 b x)}{b \sqrt {\sin (2 a+2 b x)}}\right )-\frac {\cos (2 a+2 b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {14}{9} \left (\frac {3}{5} \left (-\int \sqrt {\sin (2 a+2 b x)}dx-\frac {\cos (2 a+2 b x)}{b \sqrt {\sin (2 a+2 b x)}}\right )-\frac {\cos (2 a+2 b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {14}{9} \left (\frac {3}{5} \left (-\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b}-\frac {\cos (2 a+2 b x)}{b \sqrt {\sin (2 a+2 b x)}}\right )-\frac {\cos (2 a+2 b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\right )-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}\) |
(14*((3*(-(EllipticE[a - Pi/4 + b*x, 2]/b) - Cos[2*a + 2*b*x]/(b*Sqrt[Sin[ 2*a + 2*b*x]])))/5 - Cos[2*a + 2*b*x]/(5*b*Sin[2*a + 2*b*x]^(5/2))))/9 - C sc[a + b*x]^2/(9*b*Sin[2*a + 2*b*x]^(5/2))
3.2.13.3.1 Defintions of rubi rules used
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]
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]
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]
Timed out.
\[\int \frac {\csc \left (x b +a \right )^{2}}{\sin \left (2 x b +2 a \right )^{\frac {7}{2}}}d x\]
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 346, normalized size of antiderivative = 3.26 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=-\frac {168 \, \sqrt {2 i} {\left (i \, \cos \left (b x + a\right )^{7} - 2 i \, \cos \left (b x + a\right )^{5} + i \, \cos \left (b x + a\right )^{3}\right )} E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 168 \, \sqrt {-2 i} {\left (-i \, \cos \left (b x + a\right )^{7} + 2 i \, \cos \left (b x + a\right )^{5} - i \, \cos \left (b x + a\right )^{3}\right )} E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 168 \, \sqrt {2 i} {\left (-i \, \cos \left (b x + a\right )^{7} + 2 i \, \cos \left (b x + a\right )^{5} - i \, \cos \left (b x + a\right )^{3}\right )} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 168 \, \sqrt {-2 i} {\left (i \, \cos \left (b x + a\right )^{7} - 2 i \, \cos \left (b x + a\right )^{5} + i \, \cos \left (b x + a\right )^{3}\right )} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + \sqrt {2} {\left (336 \, \cos \left (b x + a\right )^{8} - 840 \, \cos \left (b x + a\right )^{6} + 644 \, \cos \left (b x + a\right )^{4} - 126 \, \cos \left (b x + a\right )^{2} - 9\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{360 \, {\left (b \cos \left (b x + a\right )^{7} - 2 \, b \cos \left (b x + a\right )^{5} + b \cos \left (b x + a\right )^{3}\right )} \sin \left (b x + a\right )} \]
-1/360*(168*sqrt(2*I)*(I*cos(b*x + a)^7 - 2*I*cos(b*x + a)^5 + I*cos(b*x + a)^3)*elliptic_e(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin(b*x + a) + 168*sqrt(-2*I)*(-I*cos(b*x + a)^7 + 2*I*cos(b*x + a)^5 - I*cos(b*x + a)^ 3)*elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) + 16 8*sqrt(2*I)*(-I*cos(b*x + a)^7 + 2*I*cos(b*x + a)^5 - I*cos(b*x + a)^3)*el liptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin(b*x + a) + 168*sqr t(-2*I)*(I*cos(b*x + a)^7 - 2*I*cos(b*x + a)^5 + I*cos(b*x + a)^3)*ellipti c_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) + sqrt(2)*(336 *cos(b*x + a)^8 - 840*cos(b*x + a)^6 + 644*cos(b*x + a)^4 - 126*cos(b*x + a)^2 - 9)*sqrt(cos(b*x + a)*sin(b*x + a)))/((b*cos(b*x + a)^7 - 2*b*cos(b* x + a)^5 + b*cos(b*x + a)^3)*sin(b*x + a))
Timed out. \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \]
\[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{7/2}} \,d x \]