Integrand size = 18, antiderivative size = 85 \[ \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx=-\frac {4 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{15 b^2}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {4 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{15 b^2} \] Output:
-4/15*cos(b*x+a)*csc(b*x+a)^(3/2)/b^2-2/5*x*csc(b*x+a)^(5/2)/b+4/15*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^2
Time = 0.37 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx=-\frac {2 \sqrt {\csc (a+b x)} \left (2 \cot (a+b x)+3 b x \csc ^2(a+b x)+2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}\right )}{15 b^2} \] Input:
Integrate[x*Cos[a + b*x]*Csc[a + b*x]^(7/2),x]
Output:
(-2*Sqrt[Csc[a + b*x]]*(2*Cot[a + b*x] + 3*b*x*Csc[a + b*x]^2 + 2*Elliptic F[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]]))/(15*b^2)
Time = 0.39 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4701, 3042, 4255, 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.
\(\displaystyle \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx\) |
\(\Big \downarrow \) 4701 |
\(\displaystyle \frac {2 \int \csc ^{\frac {5}{2}}(a+b x)dx}{5 b}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \int \csc (a+b x)^{5/2}dx}{5 b}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {2 \left (\frac {1}{3} \int \sqrt {\csc (a+b x)}dx-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b}\right )}{5 b}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {1}{3} \int \sqrt {\csc (a+b x)}dx-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b}\right )}{5 b}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {2 \left (\frac {1}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b}\right )}{5 b}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {1}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b}\right )}{5 b}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 \left (\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b}-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b}\right )}{5 b}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}\) |
Input:
Int[x*Cos[a + b*x]*Csc[a + b*x]^(7/2),x]
Output:
(-2*x*Csc[a + b*x]^(5/2))/(5*b) + (2*((-2*Cos[a + b*x]*Csc[a + b*x]^(3/2)) /(3*b) + (2*Sqrt[Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(3*b)))/(5*b)
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[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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]
\[\int x \cos \left (b x +a \right ) \csc \left (b x +a \right )^{\frac {7}{2}}d x\]
Input:
int(x*cos(b*x+a)*csc(b*x+a)^(7/2),x)
Output:
int(x*cos(b*x+a)*csc(b*x+a)^(7/2),x)
Exception generated. \[ \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*cos(b*x+a)*csc(b*x+a)^(7/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx=\text {Timed out} \] Input:
integrate(x*cos(b*x+a)*csc(b*x+a)**(7/2),x)
Output:
Timed out
\[ \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \csc \left (b x + a\right )^{\frac {7}{2}} \,d x } \] Input:
integrate(x*cos(b*x+a)*csc(b*x+a)^(7/2),x, algorithm="maxima")
Output:
integrate(x*cos(b*x + a)*csc(b*x + a)^(7/2), x)
\[ \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \csc \left (b x + a\right )^{\frac {7}{2}} \,d x } \] Input:
integrate(x*cos(b*x+a)*csc(b*x+a)^(7/2),x, algorithm="giac")
Output:
integrate(x*cos(b*x + a)*csc(b*x + a)^(7/2), x)
Timed out. \[ \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx=\int x\,\cos \left (a+b\,x\right )\,{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{7/2} \,d x \] Input:
int(x*cos(a + b*x)*(1/sin(a + b*x))^(7/2),x)
Output:
int(x*cos(a + b*x)*(1/sin(a + b*x))^(7/2), x)
\[ \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx=\int \sqrt {\csc \left (b x +a \right )}\, \cos \left (b x +a \right ) \csc \left (b x +a \right )^{3} x d x \] Input:
int(x*cos(b*x+a)*csc(b*x+a)^(7/2),x)
Output:
int(sqrt(csc(a + b*x))*cos(a + b*x)*csc(a + b*x)**3*x,x)