\(\int x \cos (a+b x) \csc ^{\frac {9}{2}}(a+b x) \, dx\) [352]

Optimal result

Integrand size = 18, antiderivative size = 108 \[ \int x \cos (a+b x) \csc ^{\frac {9}{2}}(a+b x) \, dx=-\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {12 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{35 b^2} \]

[Out]

-4/35*cos(b*x+a)*csc(b*x+a)^(5/2)/b^2-2/7*x*csc(b*x+a)^(7/2)/b-12/35*cos(b*x+a)*csc(b*x+a)^(1/2)/b^2+12/35*(si
n(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*x)*EllipticE(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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4298, 3853, 3856, 2719} \[ \int x \cos (a+b x) \csc ^{\frac {9}{2}}(a+b x) \, dx=-\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {12 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b} \]

[In]

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

[Out]

(-12*Cos[a + b*x]*Sqrt[Csc[a + b*x]])/(35*b^2) - (4*Cos[a + b*x]*Csc[a + b*x]^(5/2))/(35*b^2) - (2*x*Csc[a + b
*x]^(7/2))/(7*b) - (12*Sqrt[Csc[a + b*x]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(35*b^2)

Rule 2719

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 3853

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] + Dist[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]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

Rule 4298

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] + Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \int \csc ^{\frac {7}{2}}(a+b x) \, dx}{7 b} \\ & = -\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {6 \int \csc ^{\frac {3}{2}}(a+b x) \, dx}{35 b} \\ & = -\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {6 \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx}{35 b} \\ & = -\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {\left (6 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx}{35 b} \\ & = -\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {12 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{35 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.68 \[ \int x \cos (a+b x) \csc ^{\frac {9}{2}}(a+b x) \, dx=-\frac {2 \csc ^{\frac {7}{2}}(a+b x) \left (5 b x+6 \cos (a+b x) \sin ^3(a+b x)-6 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {7}{2}}(a+b x)+\sin (2 (a+b x))\right )}{35 b^2} \]

[In]

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

[Out]

(-2*Csc[a + b*x]^(7/2)*(5*b*x + 6*Cos[a + b*x]*Sin[a + b*x]^3 - 6*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a +
b*x]^(7/2) + Sin[2*(a + b*x)]))/(35*b^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int x \cos (a+b x) \csc ^{\frac {9}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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