\(\int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx\) [442]

Optimal result

Integrand size = 33, antiderivative size = 70 \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=-\frac {\sqrt {5} \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {-2+3 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 70, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3473

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

Input:

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

Output:

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

Leaf count of result is larger than twice the leaf count of optimal. \(215\) vs. \(2(61)=122\).

Time = 11.94 (sec) , antiderivative size = 216, normalized size of antiderivative = 3.09

method	result	size
default	\(-\frac {\left (-2+3 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (4 \cos \left (d x +c \right )^{2}+8 \cos \left (d x +c \right )+4\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )}{d \sqrt {-2+3 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, \left (\cos \left (d x +c \right )+1\right )}\)	\(216\)
parts	\(-\frac {\left (\left (-2+3 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (2+2 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )\right ) \sqrt {-2+3 \cos \left (d x +c \right )}}{d \sqrt {\cos \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )-2\right )}-\frac {2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )}{d \sqrt {\cos \left (d x +c \right )}\, \sqrt {-2+3 \cos \left (d x +c \right )}}\)	\(321\)

Input:

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

Output:

-1/d*((-2+3*cos(d*x+c))*sin(d*x+c)+(4*cos(d*x+c)^2+8*cos(d*x+c)+4)*(cos(d* 
x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),5^(1/2))*((-2+3 
*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)+(cos(d*x+c)^2+2*cos(d*x+c)+1)*(cos(d*x+ 
c)/(cos(d*x+c)+1))^(1/2)*((-2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Elliptic 
E(cot(d*x+c)-csc(d*x+c),5^(1/2)))/(-2+3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2) 
/(cos(d*x+c)+1)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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