Integrand size = 18, antiderivative size = 108 \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{147 b^2} \]
2/7*x/b/csc(b*x+a)^(7/2)+4/49*cos(b*x+a)/b^2/csc(b*x+a)^(5/2)+20/147*cos(b *x+a)/b^2/csc(b*x+a)^(1/2)+20/147*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/sin( 1/2*a+1/4*Pi+1/2*b*x)*EllipticF(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))*csc(b*x +a)^(1/2)*sin(b*x+a)^(1/2)/b^2
Time = 0.51 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.86 \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\frac {\sqrt {\csc (a+b x)} \left (63 b x-84 b x \cos (2 (a+b x))+21 b x \cos (4 (a+b x))+80 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+52 \sin (2 (a+b x))-6 \sin (4 (a+b x))\right )}{588 b^2} \]
(Sqrt[Csc[a + b*x]]*(63*b*x - 84*b*x*Cos[2*(a + b*x)] + 21*b*x*Cos[4*(a + b*x)] + 80*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 52*Sin [2*(a + b*x)] - 6*Sin[4*(a + b*x)]))/(588*b^2)
Time = 0.48 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4701, 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.
\(\displaystyle \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx\) |
\(\Big \downarrow \) 4701 |
\(\displaystyle \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)}dx}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc (a+b x)^{7/2}}dx}{7 b}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)}dx-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \int \frac {1}{\csc (a+b x)^{3/2}}dx-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\csc (a+b x)}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\csc (a+b x)}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \left (\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b}-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
(2*x)/(7*b*Csc[a + b*x]^(7/2)) - (2*((-2*Cos[a + b*x])/(7*b*Csc[a + b*x]^( 5/2)) + (5*((-2*Cos[a + b*x])/(3*b*Sqrt[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))/(7*b )
3.4.59.3.1 Defintions of rubi rules used
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]
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]
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]
Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*Csc[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*(x_)^( m_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Csc[a + b*x^n]^(p - 1)/(b*n*(p - 1))), x] + Simp[(m - n + 1)/(b*n*(p - 1)) Int[x^(m - n)*Csc[a + b*x^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ [p, 1]
\[\int \frac {x \cos \left (x b +a \right )}{\csc \left (x b +a \right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x \cos {\left (a + b x \right )}}{\csc ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x\,\cos \left (a+b\,x\right )}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]