Integrand size = 27, antiderivative size = 54 \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {-\cos (c+d x)}} \] Output:
2/5*cos(d*x+c)^(1/2)*EllipticF(sin(d*x+c)/(1+cos(d*x+c)),1/5*5^(1/2))*5^(1 /2)/d/(-cos(d*x+c))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(150\) vs. \(2(54)=108\).
Time = 9.75 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.78 \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=-\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \] Input:
Integrate[1/(Sqrt[-Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]),x]
Output:
(-4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt [(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sq rt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4]*Sin[(c + d*x)/2]^4)/(d *Sqrt[-Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]])
Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, 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.
\(\displaystyle \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {3 \cos (c+d x)+2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {-\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {3 \sin \left (c+d x+\frac {\pi }{2}\right )+2}}dx\) |
\(\Big \downarrow \) 3293 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3 \cos (c+d x)+2}}dx}{\sqrt {-\cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {3 \sin \left (c+d x+\frac {\pi }{2}\right )+2}}dx}{\sqrt {-\cos (c+d x)}}\) |
\(\Big \downarrow \) 3292 |
\(\displaystyle \frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {-\cos (c+d x)}}\) |
Input:
Int[1/(Sqrt[-Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]),x]
Output:
(2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 1 /5])/(Sqrt[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. \(110\) vs. \(2(49)=98\).
Time = 9.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.06
method | result | size |
default | \(\frac {\left (\cos \left (d x +c \right )+1\right ) \operatorname {EllipticF}\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {5}}{5 d \sqrt {2+3 \cos \left (d x +c \right )}\, \sqrt {-\cos \left (d x +c \right )}}\) | \(111\) |
Input:
int(1/(-cos(d*x+c))^(1/2)/(2+3*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/5/d*(cos(d*x+c)+1)*EllipticF(1/5*(csc(d*x+c)-cot(d*x+c))*5^(1/2),5^(1/2) )*2^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(co s(d*x+c)+1))^(1/2)/(2+3*cos(d*x+c))^(1/2)/(-cos(d*x+c))^(1/2)*5^(1/2)
\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {3 \, \cos \left (d x + c\right ) + 2}} \,d x } \] Input:
integrate(1/(-cos(d*x+c))^(1/2)/(2+3*cos(d*x+c))^(1/2),x, algorithm="frica s")
Output:
integral(-sqrt(-cos(d*x + c))*sqrt(3*cos(d*x + c) + 2)/(3*cos(d*x + c)^2 + 2*cos(d*x + c)), x)
\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {- \cos {\left (c + d x \right )}} \sqrt {3 \cos {\left (c + d x \right )} + 2}}\, dx \] Input:
integrate(1/(-cos(d*x+c))**(1/2)/(2+3*cos(d*x+c))**(1/2),x)
Output:
Integral(1/(sqrt(-cos(c + d*x))*sqrt(3*cos(c + d*x) + 2)), x)
\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {3 \, \cos \left (d x + c\right ) + 2}} \,d x } \] Input:
integrate(1/(-cos(d*x+c))^(1/2)/(2+3*cos(d*x+c))^(1/2),x, algorithm="maxim a")
Output:
integrate(1/(sqrt(-cos(d*x + c))*sqrt(3*cos(d*x + c) + 2)), x)
\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {3 \, \cos \left (d x + c\right ) + 2}} \,d x } \] Input:
integrate(1/(-cos(d*x+c))^(1/2)/(2+3*cos(d*x+c))^(1/2),x, algorithm="giac" )
Output:
integrate(1/(sqrt(-cos(d*x + c))*sqrt(3*cos(d*x + c) + 2)), x)
Timed out. \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {3\,\cos \left (c+d\,x\right )+2}} \,d x \] 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)
\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=-\left (\int \frac {\sqrt {3 \cos \left (d x +c \right )+2}\, \sqrt {\cos \left (d x +c \right )}}{3 \cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )}d x \right ) i \] Input:
int(1/(-cos(d*x+c))^(1/2)/(2+3*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)*i