Integrand size = 19, antiderivative size = 67 \[ \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{5 b} \] Output:
2/5*cos(b*x+a)/b/csc(b*x+a)^(3/2)-4/5*csc(b*x+a)^(1/2)*EllipticE(cos(1/2*a +1/4*Pi+1/2*b*x),2^(1/2))*sin(b*x+a)^(1/2)/b
Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=-\frac {2 \sqrt {\csc (a+b x)} \left (2 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}-\cos (a+b x) \sin ^2(a+b x)\right )}{5 b} \] Input:
Integrate[Cos[a + b*x]^2/Sqrt[Csc[a + b*x]],x]
Output:
(-2*Sqrt[Csc[a + b*x]]*(2*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] - Cos[a + b*x]*Sin[a + b*x]^2))/(5*b)
Time = 0.35 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 3108, 3042, 4258, 3042, 3119}
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.
\(\displaystyle \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {\csc (a+b x)} \sec (a+b x)^2}dx\) |
\(\Big \downarrow \) 3108 |
\(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {\csc (a+b x)}}dx+\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {\csc (a+b x)}}dx+\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {2}{5} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \sqrt {\sin (a+b x)}dx+\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \sqrt {\sin (a+b x)}dx+\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b}\) |
Input:
Int[Cos[a + b*x]^2/Sqrt[Csc[a + b*x]],x]
Output:
(2*Cos[a + b*x])/(5*b*Csc[a + b*x]^(3/2)) + (4*Sqrt[Csc[a + b*x]]*Elliptic E[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(5*b)
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*((b_.)*sec[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[(-a)*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n + 1)/(b*f*(m + n))), x] + Simp[(n + 1)/(b^2*(m + n)) Int[(a*Csc[e + f*x])^ m*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, - 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs. \(2(57)=114\).
Time = 0.76 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.12
method | result | size |
default | \(\frac {-\frac {2 \sin \left (b x +a \right )^{4}}{5}+\frac {2 \sin \left (b x +a \right )^{2}}{5}-\frac {4 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{5}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\) | \(142\) |
Input:
int(cos(b*x+a)^2/csc(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-2/5*sin(b*x+a)^4+2/5*sin(b*x+a)^2-4/5*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a )+2)^(1/2)*(-sin(b*x+a))^(1/2)*EllipticE((sin(b*x+a)+1)^(1/2),1/2*2^(1/2)) +2/5*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*Elli pticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2)))/cos(b*x+a)/sin(b*x+a)^(1/2)/b
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.24 \[ \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \, {\left (\sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - \frac {\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )}{\sqrt {\sin \left (b x + a\right )}}\right )}}{5 \, b} \] Input:
integrate(cos(b*x+a)^2/csc(b*x+a)^(1/2),x, algorithm="fricas")
Output:
2/5*(sqrt(2*I)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a ) + I*sin(b*x + a))) + sqrt(-2*I)*weierstrassZeta(4, 0, weierstrassPInvers e(4, 0, cos(b*x + a) - I*sin(b*x + a))) - (cos(b*x + a)^3 - cos(b*x + a))/ sqrt(sin(b*x + a)))/b
\[ \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {\cos ^{2}{\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \] Input:
integrate(cos(b*x+a)**2/csc(b*x+a)**(1/2),x)
Output:
Integral(cos(a + b*x)**2/sqrt(csc(a + b*x)), x)
\[ \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {\cos \left (b x + a\right )^{2}}{\sqrt {\csc \left (b x + a\right )}} \,d x } \] Input:
integrate(cos(b*x+a)^2/csc(b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(cos(b*x + a)^2/sqrt(csc(b*x + a)), x)
\[ \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {\cos \left (b x + a\right )^{2}}{\sqrt {\csc \left (b x + a\right )}} \,d x } \] Input:
integrate(cos(b*x+a)^2/csc(b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(cos(b*x + a)^2/sqrt(csc(b*x + a)), x)
Timed out. \[ \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \] Input:
int(cos(a + b*x)^2/(1/sin(a + b*x))^(1/2),x)
Output:
int(cos(a + b*x)^2/(1/sin(a + b*x))^(1/2), x)
\[ \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {\sqrt {\csc \left (b x +a \right )}\, \cos \left (b x +a \right )^{2}}{\csc \left (b x +a \right )}d x \] Input:
int(cos(b*x+a)^2/csc(b*x+a)^(1/2),x)
Output:
int((sqrt(csc(a + b*x))*cos(a + b*x)**2)/csc(a + b*x),x)