\(\int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx\) [447]

Optimal result

Integrand size = 33, antiderivative size = 98 \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx=-\frac {2 \sqrt {5} \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {-3+2 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \] Output:

-2/3*5^(1/2)*(-cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticE((-3 
+2*cos(d*x+c))^(1/2)/(-cos(d*x+c))^(1/2),1/5*I*5^(1/2))*(1-sec(d*x+c))^(1/ 
2)*(1+sec(d*x+c))^(1/2)/d
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx=\int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx \] Input:

Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]] 
),x]
 

Output:

Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]] 
), x]
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 3474, 3042, 3473}

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.

\(\displaystyle \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {2 \cos (c+d x)-3}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {2 \sin \left (c+d x+\frac {\pi }{2}\right )-3}}dx\)

\(\Big \downarrow \) 3474

\(\displaystyle -\frac {\sqrt {-\cos (c+d x)} \int \frac {\cos (c+d x)+1}{(-\cos (c+d x))^{3/2} \sqrt {2 \cos (c+d x)-3}}dx}{\sqrt {\cos (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\sqrt {-\cos (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \sqrt {2 \sin \left (c+d x+\frac {\pi }{2}\right )-3}}dx}{\sqrt {\cos (c+d x)}}\)

\(\Big \downarrow \) 3473

\(\displaystyle -\frac {2 \sqrt {5} \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} E\left (\arcsin \left (\frac {\sqrt {2 \cos (c+d x)-3}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{3 d}\)

Input:

Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]]),x]
 

Output:

(-2*Sqrt[5]*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ 
ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -1/5]*Sqrt[1 - Sec[ 
c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3473

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) 
^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* 
(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] 
)/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + 
d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], 
x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && 
PosQ[(c + d)/b]
 

rule 3474

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) 
^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-Sqrt 
[(-b)*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]   Int[(A + B*Sin[e + f*x])/(((-b)* 
Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{b, c, d, e, 
 f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && NegQ[(c + d)/b]
 

Maple [A] (verified)

Time = 14.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.57

method	result	size
default	\(\frac {\left (6-4 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+i \left (-\cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )-1\right ) \sqrt {5}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}, \frac {i \sqrt {5}}{5}\right )}{3 d \sqrt {-3+2 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, \left (\cos \left (d x +c \right )+1\right )}\)	\(154\)
parts	\(-\frac {\left (\left (20 \cos \left (d x +c \right )-30\right ) \sin \left (d x +c \right )+i \left (3 \cos \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )+3\right ) \sqrt {5}\, \sqrt {2}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {5}, \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )+1}}+i \left (-5 \cos \left (d x +c \right )^{2}-10 \cos \left (d x +c \right )-5\right ) \sqrt {5}\, \sqrt {2}\, \operatorname {EllipticE}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {5}, \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-3+2 \cos \left (d x +c \right )}}{15 d \sqrt {\cos \left (d x +c \right )}\, \left (2 \cos \left (d x +c \right )^{2}-\cos \left (d x +c \right )-3\right )}+\frac {i \left (\cos \left (d x +c \right )+1\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {5}, \frac {i \sqrt {5}}{5}\right ) \sqrt {5}}{5 d \sqrt {\cos \left (d x +c \right )}\, \sqrt {-3+2 \cos \left (d x +c \right )}}\)	\(378\)

Input:

int((cos(d*x+c)+1)/cos(d*x+c)^(3/2)/(-3+2*cos(d*x+c))^(1/2),x,method=_RETU 
RNVERBOSE)
 

Output:

1/3/d*((6-4*cos(d*x+c))*sin(d*x+c)+I*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*5^(1/2 
)*2^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(-2*(-3+2*cos(d*x+c))/(cos(d*x 
+c)+1))^(1/2)*EllipticE(I*(csc(d*x+c)-cot(d*x+c))*5^(1/2),1/5*I*5^(1/2)))/ 
(-3+2*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2)/(cos(d*x+c)+1)
 

Fricas [F]

\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {2 \, \cos \left (d x + c\right ) - 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-3+2*cos(d*x+c))^(1/2),x, algor 
ithm="fricas")
 

Output:

integral(sqrt(2*cos(d*x + c) - 3)*(cos(d*x + c) + 1)*sqrt(cos(d*x + c))/(2 
*cos(d*x + c)^3 - 3*cos(d*x + c)^2), x)
 

Sympy [F]

\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx=\int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {2 \cos {\left (c + d x \right )} - 3} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:

integrate((1+cos(d*x+c))/cos(d*x+c)**(3/2)/(-3+2*cos(d*x+c))**(1/2),x)
 

Output:

Integral((cos(c + d*x) + 1)/(sqrt(2*cos(c + d*x) - 3)*cos(c + d*x)**(3/2)) 
, x)
 

Maxima [F]

\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {2 \, \cos \left (d x + c\right ) - 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-3+2*cos(d*x+c))^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate((cos(d*x + c) + 1)/(sqrt(2*cos(d*x + c) - 3)*cos(d*x + c)^(3/2)) 
, x)
 

Giac [F]

\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {2 \, \cos \left (d x + c\right ) - 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-3+2*cos(d*x+c))^(1/2),x, algor 
ithm="giac")
 

Output:

integrate((cos(d*x + c) + 1)/(sqrt(2*cos(d*x + c) - 3)*cos(d*x + c)^(3/2)) 
, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx=\int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {2\,\cos \left (c+d\,x\right )-3}} \,d x \] Input:

int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(2*cos(c + d*x) - 3)^(1/2)),x)
 

Output:

int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(2*cos(c + d*x) - 3)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-3+2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {2 \cos \left (d x +c \right )-3}\, \sqrt {\cos \left (d x +c \right )}}{2 \cos \left (d x +c \right )^{3}-3 \cos \left (d x +c \right )^{2}}d x +\int \frac {\sqrt {2 \cos \left (d x +c \right )-3}\, \sqrt {\cos \left (d x +c \right )}}{2 \cos \left (d x +c \right )^{2}-3 \cos \left (d x +c \right )}d x \] Input:

int((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-3+2*cos(d*x+c))^(1/2),x)
 

Output:

int((sqrt(2*cos(c + d*x) - 3)*sqrt(cos(c + d*x)))/(2*cos(c + d*x)**3 - 3*c 
os(c + d*x)**2),x) + int((sqrt(2*cos(c + d*x) - 3)*sqrt(cos(c + d*x)))/(2* 
cos(c + d*x)**2 - 3*cos(c + d*x)),x)