\(\int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx\) [446]

Optimal result

Integrand size = 33, antiderivative size = 74 \[ \int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \sqrt {5} \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \] Output:

2/3*5^(1/2)*cot(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]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3473

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

Input:

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

Output:

(2*Sqrt[5]*Cot[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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (64 ) = 128\).

Time = 12.61 (sec) , antiderivative size = 242, normalized size of antiderivative = 3.27

method	result	size
default	\(-\frac {\left (\left (6-4 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {2}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (-6 \cos \left (d x +c \right )^{2}-12 \cos \left (d x +c \right )-6\right ) \sqrt {2}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {3-2 \cos \left (d x +c \right )}}{3 d \sqrt {\cos \left (d x +c \right )}\, \left (2 \cos \left (d x +c \right )^{2}-\cos \left (d x +c \right )-3\right )}\)	\(242\)
parts	\(\frac {\left (\left (-6+4 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (3 \cos \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )+3\right ) \sqrt {2}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )+1}}+\left (-\cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )-1\right ) \sqrt {2}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {3-2 \cos \left (d x +c \right )}}{3 d \sqrt {\cos \left (d x +c \right )}\, \left (2 \cos \left (d x +c \right )^{2}-\cos \left (d x +c \right )-3\right )}-\frac {\sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+1\right ) \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right )}{d \sqrt {3-2 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\)	\(346\)

Input:

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

Output:

-1/3/d*((6-4*cos(d*x+c))*sin(d*x+c)+(cos(d*x+c)^2+2*cos(d*x+c)+1)*2^(1/2)* 
(-2*(-3+2*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c 
),I*5^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+(-6*cos(d*x+c)^2-12*cos(d*x 
+c)-6)*2^(1/2)*(-2*(-3+2*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d 
*x+c)-csc(d*x+c),I*5^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(3-2*cos(d* 
x+c))^(1/2)/cos(d*x+c)^(1/2)/(2*cos(d*x+c)^2-cos(d*x+c)-3)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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