Integrand size = 25, antiderivative size = 49 \[ \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),5\right )}{d \sqrt {\cos (c+d x)}} \] Output:
-2*(-cos(d*x+c))^(1/2)*EllipticF(sin(d*x+c)/(1-cos(d*x+c)),5^(1/2))/d/cos( d*x+c)^(1/2)
Time = 1.70 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {-2-3 \cos (c+d x)} \sqrt {\frac {\cos (c+d x)}{2+3 \cos (c+d x)}}} \] Input:
Integrate[1/(Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]
Output:
(2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Tan[(c + d*x)/2]], 1/5])/(Sqrt[5]*d *Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]/(2 + 3*Cos[c + d*x])])
Time = 0.34 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3293, 3042, 3292}
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.
\(\displaystyle \int \frac {1}{\sqrt {-3 \cos (c+d x)-2} \sqrt {\cos (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {-3 \sin \left (c+d x+\frac {\pi }{2}\right )-2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3293 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {-3 \cos (c+d x)-2} \sqrt {-\cos (c+d x)}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {-3 \sin \left (c+d x+\frac {\pi }{2}\right )-2} \sqrt {-\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3292 |
\(\displaystyle -\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),5\right )}{d \sqrt {\cos (c+d x)}}\) |
Input:
Int[1/(Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]
Output:
(-2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 5])/(d*Sqrt[Cos[c + d*x]])
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(d/(f*Sqrt[a + b*d]))*EllipticF[ArcSin[Co s[e + f*x]/(1 + d*Sin[e + f*x])], -(a - b*d)/(a + b*d)], x] /; FreeQ[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && EqQ[d^2, 1] && GtQ[b*d, 0]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[Sqrt[Sign[b]*Sin[e + f*x]]/Sqrt[d*Sin[e + f* x]] Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[Sign[b]*Sin[e + f*x]]), x], x] / ; FreeQ[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && GtQ[b^2, 0] && !(EqQ[d ^2, 1] && GtQ[b*d, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(46)=92\).
Time = 9.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.22
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+1\right ) \operatorname {EllipticF}\left (\frac {\left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {5}}{5 d \sqrt {-2-3 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\) | \(109\) |
Input:
int(1/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/5/d/(-2-3*cos(d*x+c))^(1/2)*2^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(cos(d*x +c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)/cos(d*x+c)^(1/2)*(cos(d*x+ c)+1)*EllipticF(1/5*(cot(d*x+c)-csc(d*x+c))*5^(1/2),5^(1/2))*5^(1/2)
\[ \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:
integrate(1/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),x, algorithm="fricas" )
Output:
integral(-sqrt(-3*cos(d*x + c) - 2)*sqrt(cos(d*x + c))/(3*cos(d*x + c)^2 + 2*cos(d*x + c)), x)
\[ \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {- 3 \cos {\left (c + d x \right )} - 2} \sqrt {\cos {\left (c + d x \right )}}}\, dx \] Input:
integrate(1/(-2-3*cos(d*x+c))**(1/2)/cos(d*x+c)**(1/2),x)
Output:
Integral(1/(sqrt(-3*cos(c + d*x) - 2)*sqrt(cos(c + d*x))), x)
\[ \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:
integrate(1/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),x, algorithm="maxima" )
Output:
integrate(1/(sqrt(-3*cos(d*x + c) - 2)*sqrt(cos(d*x + c))), x)
\[ \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:
integrate(1/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-3*cos(d*x + c) - 2)*sqrt(cos(d*x + c))), x)
Timed out. \[ \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {-3\,\cos \left (c+d\,x\right )-2}} \,d x \] Input:
int(1/(cos(c + d*x)^(1/2)*(- 3*cos(c + d*x) - 2)^(1/2)),x)
Output:
int(1/(cos(c + d*x)^(1/2)*(- 3*cos(c + d*x) - 2)^(1/2)), x)
\[ \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\left (\int \frac {\sqrt {-3 \cos \left (d x +c \right )-2}\, \sqrt {\cos \left (d x +c \right )}}{3 \cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )}d x \right ) \] Input:
int(1/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),x)
Output:
- int((sqrt( - 3*cos(c + d*x) - 2)*sqrt(cos(c + d*x)))/(3*cos(c + d*x)**2 + 2*cos(c + d*x)),x)