\(\int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx\) [357]

Optimal result

Integrand size = 18, antiderivative size = 85 \[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\frac {4 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{9 b^2} \]

[Out]

2/3*x/b/csc(b*x+a)^(3/2)+4/9*cos(b*x+a)/b^2/csc(b*x+a)^(1/2)+4/9*(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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4298, 3854, 3856, 2720} \[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{9 b^2}+\frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)} \]

[In]

Int[(x*Cos[a + b*x])/Sqrt[Csc[a + b*x]],x]

[Out]

(2*x)/(3*b*Csc[a + b*x]^(3/2)) + (4*Cos[a + b*x])/(9*b^2*Sqrt[Csc[a + b*x]]) - (4*Sqrt[Csc[a + b*x]]*EllipticF
[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(9*b^2)

Rule 2720

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]

Rule 3854

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
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx}{3 b} \\ & = \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\frac {2 \int \sqrt {\csc (a+b x)} \, dx}{9 b} \\ & = \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\frac {\left (2 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{9 b} \\ & = \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\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)}}{9 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \sqrt {\csc (a+b x)} \left (2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+3 b x \sin ^2(a+b x)+\sin (2 (a+b x))\right )}{9 b^2} \]

[In]

Integrate[(x*Cos[a + b*x])/Sqrt[Csc[a + b*x]],x]

[Out]

(2*Sqrt[Csc[a + b*x]]*(2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 3*b*x*Sin[a + b*x]^2 + Sin[2
*(a + b*x)]))/(9*b^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*cos(b*x+a)/csc(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

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

[In]

integrate(x*cos(b*x+a)/csc(b*x+a)**(1/2),x)

[Out]

Integral(x*cos(a + b*x)/sqrt(csc(a + b*x)), x)

Maxima [F]

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

[In]

integrate(x*cos(b*x+a)/csc(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(x*cos(b*x+a)/csc(b*x+a)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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