Integrand size = 18, antiderivative size = 85 \[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\frac {4 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{9 b^2} \] Output:
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*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 = 0.36 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \sqrt {\csc (a+b x)} \left (2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+3 b x \sin ^2(a+b x)+\sin (2 (a+b x))\right )}{9 b^2} \] Input:
Integrate[(x*Cos[a + b*x])/Sqrt[Csc[a + b*x]],x]
Output:
(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)
Time = 0.38 (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, 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.
\(\displaystyle \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx\) |
\(\Big \downarrow \) 4701 |
\(\displaystyle \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)}dx}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc (a+b x)^{3/2}}dx}{3 b}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2 \left (\frac {1}{3} \int \sqrt {\csc (a+b x)}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2 \left (\frac {1}{3} \int \sqrt {\csc (a+b x)}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )}{3 b}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2 \left (\frac {1}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2 \left (\frac {1}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )}{3 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2 \left (\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b}-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )}{3 b}\) |
Input:
Int[(x*Cos[a + b*x])/Sqrt[Csc[a + b*x]],x]
Output:
(2*x)/(3*b*Csc[a + b*x]^(3/2)) - (2*((-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)))/(3*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[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]
\[\int \frac {x \cos \left (b x +a \right )}{\sqrt {\csc \left (b x +a \right )}}d x\]
Input:
int(x*cos(b*x+a)/csc(b*x+a)^(1/2),x)
Output:
int(x*cos(b*x+a)/csc(b*x+a)^(1/2),x)
Exception generated. \[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*cos(b*x+a)/csc(b*x+a)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {x \cos {\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \] Input:
integrate(x*cos(b*x+a)/csc(b*x+a)**(1/2),x)
Output:
Integral(x*cos(a + b*x)/sqrt(csc(a + b*x)), x)
\[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sqrt {\csc \left (b x + a\right )}} \,d x } \] Input:
integrate(x*cos(b*x+a)/csc(b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(x*cos(b*x + a)/sqrt(csc(b*x + a)), x)
\[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sqrt {\csc \left (b x + a\right )}} \,d x } \] Input:
integrate(x*cos(b*x+a)/csc(b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(x*cos(b*x + a)/sqrt(csc(b*x + a)), x)
Timed out. \[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {x\,\cos \left (a+b\,x\right )}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \] Input:
int((x*cos(a + b*x))/(1/sin(a + b*x))^(1/2),x)
Output:
int((x*cos(a + b*x))/(1/sin(a + b*x))^(1/2), x)
\[ \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {\sqrt {\csc \left (b x +a \right )}\, \cos \left (b x +a \right ) x}{\csc \left (b x +a \right )}d x \] Input:
int(x*cos(b*x+a)/csc(b*x+a)^(1/2),x)
Output:
int((sqrt(csc(a + b*x))*cos(a + b*x)*x)/csc(a + b*x),x)