\(\int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx\) [427]

Optimal result

Integrand size = 39, antiderivative size = 112 \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\frac {5 \arcsin \left (2 \sqrt {\frac {2}{7}} \sin (x)\right )}{4 \sqrt {2}}+\frac {3}{4} \arcsin \left (\frac {2 \sin (x)}{\sqrt {3}}\right )-\frac {3}{4} \arctan \left (\frac {\sin (x)}{\sqrt {-1+4 \cos ^2(x)}}\right )-\frac {3}{4} \arctan \left (\frac {\sin (x)}{\sqrt {-1+8 \cos ^2(x)}}\right )-\frac {1}{2} \sqrt {-1+4 \cos ^2(x)} \sin (x)-\frac {1}{2} \sqrt {-1+8 \cos ^2(x)} \sin (x) \] Output:

3/4*arcsin(2/3*sin(x)*3^(1/2))-3/4*arctan(sin(x)/(-1+4*cos(x)^2)^(1/2))-3/ 
4*arctan(sin(x)/(-1+8*cos(x)^2)^(1/2))+5/8*arcsin(2/7*sin(x)*14^(1/2))*2^( 
1/2)-1/2*sin(x)*(-1+4*cos(x)^2)^(1/2)-1/2*sin(x)*(-1+8*cos(x)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.39 \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\frac {1}{8} \left (5 \sqrt {2} \arcsin \left (2 \sqrt {\frac {2}{7}} \sin (x)\right )+6 \arcsin \left (\frac {2 \sin (x)}{\sqrt {3}}\right )+3 \arctan \left (\frac {7-8 \sin (x)}{\sqrt {3+4 \cos (2 x)}}\right )+3 \arctan \left (\frac {3-4 \sin (x)}{\sqrt {1+2 \cos (2 x)}}\right )-3 \arctan \left (\frac {3+4 \sin (x)}{\sqrt {1+2 \cos (2 x)}}\right )-3 \arctan \left (\frac {7+8 \sin (x)}{\sqrt {3+4 \cos (2 x)}}\right )-4 \sqrt {1+2 \cos (2 x)} \sin (x)-4 \sqrt {3+4 \cos (2 x)} \sin (x)\right ) \] Input:

Integrate[Cos[3*x]/(-Sqrt[-1 + 8*Cos[x]^2] + Sqrt[3*Cos[x]^2 - Sin[x]^2]), 
x]
 

Output:

(5*Sqrt[2]*ArcSin[2*Sqrt[2/7]*Sin[x]] + 6*ArcSin[(2*Sin[x])/Sqrt[3]] + 3*A 
rcTan[(7 - 8*Sin[x])/Sqrt[3 + 4*Cos[2*x]]] + 3*ArcTan[(3 - 4*Sin[x])/Sqrt[ 
1 + 2*Cos[2*x]]] - 3*ArcTan[(3 + 4*Sin[x])/Sqrt[1 + 2*Cos[2*x]]] - 3*ArcTa 
n[(7 + 8*Sin[x])/Sqrt[3 + 4*Cos[2*x]]] - 4*Sqrt[1 + 2*Cos[2*x]]*Sin[x] - 4 
*Sqrt[3 + 4*Cos[2*x]]*Sin[x])/8
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3042, 4878, 25, 7293, 2009}

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.

\(\displaystyle \int \frac {\cos (3 x)}{\sqrt {3 \cos ^2(x)-\sin ^2(x)}-\sqrt {8 \cos ^2(x)-1}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (3 x)}{\sqrt {3 \cos (x)^2-\sin (x)^2}-\sqrt {8 \cos (x)^2-1}}dx\)

\(\Big \downarrow \) 4878

\(\displaystyle \int -\frac {1-4 \sin ^2(x)}{\sqrt {7-8 \sin ^2(x)}-\sqrt {3-4 \sin ^2(x)}}d\sin (x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {1-4 \sin ^2(x)}{\sqrt {7-8 \sin ^2(x)}-\sqrt {3-4 \sin ^2(x)}}d\sin (x)\)

\(\Big \downarrow \) 7293

\(\displaystyle -\int \left (\frac {1}{\sqrt {7-8 \sin ^2(x)}-\sqrt {3-4 \sin ^2(x)}}-\frac {4 \sin ^2(x)}{\sqrt {7-8 \sin ^2(x)}-\sqrt {3-4 \sin ^2(x)}}\right )d\sin (x)\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 \arcsin \left (2 \sqrt {\frac {2}{7}} \sin (x)\right )}{4 \sqrt {2}}+\frac {3}{4} \arcsin \left (\frac {2 \sin (x)}{\sqrt {3}}\right )-\frac {3}{4} \arctan \left (\frac {\sin (x)}{\sqrt {7-8 \sin ^2(x)}}\right )-\frac {3}{4} \arctan \left (\frac {\sin (x)}{\sqrt {3-4 \sin ^2(x)}}\right )-\frac {1}{2} \sin (x) \sqrt {7-8 \sin ^2(x)}-\frac {1}{2} \sin (x) \sqrt {3-4 \sin ^2(x)}\)

Input:

Int[Cos[3*x]/(-Sqrt[-1 + 8*Cos[x]^2] + Sqrt[3*Cos[x]^2 - Sin[x]^2]),x]
 

Output:

(5*ArcSin[2*Sqrt[2/7]*Sin[x]])/(4*Sqrt[2]) + (3*ArcSin[(2*Sin[x])/Sqrt[3]] 
)/4 - (3*ArcTan[Sin[x]/Sqrt[7 - 8*Sin[x]^2]])/4 - (3*ArcTan[Sin[x]/Sqrt[3 
- 4*Sin[x]^2]])/4 - (Sin[x]*Sqrt[7 - 8*Sin[x]^2])/2 - (Sin[x]*Sqrt[3 - 4*S 
in[x]^2])/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4878

Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa 
ctors[Sin[v], x]}, d/Coefficient[v, x, 1]   Subst[Int[SubstFor[1, Sin[v]/d, 
 u/Cos[v], x], x], x, Sin[v]/d]], x] /;  !FalseQ[v] && FunctionOfQ[NonfreeF 
actors[Sin[v], x], u/Cos[v], x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int \frac {\cos \left (3 x \right )}{-\sqrt {-1+8 \cos \left (x \right )^{2}}+\sqrt {3 \cos \left (x \right )^{2}-\sin \left (x \right )^{2}}}d x\]

Input:

int(cos(3*x)/(-(-1+8*cos(x)^2)^(1/2)+(3*cos(x)^2-sin(x)^2)^(1/2)),x)
 

Output:

int(cos(3*x)/(-(-1+8*cos(x)^2)^(1/2)+(3*cos(x)^2-sin(x)^2)^(1/2)),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (84) = 168\).

Time = 0.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.72 \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=-\frac {5}{32} \, \sqrt {2} \arctan \left (\frac {{\left (512 \, \sqrt {2} \cos \left (x\right )^{4} - 576 \, \sqrt {2} \cos \left (x\right )^{2} + 113 \, \sqrt {2}\right )} \sqrt {8 \, \cos \left (x\right )^{2} - 1}}{16 \, {\left (128 \, \cos \left (x\right )^{4} - 88 \, \cos \left (x\right )^{2} + 9\right )} \sin \left (x\right )}\right ) - \frac {1}{2} \, \sqrt {8 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right ) - \frac {1}{2} \, \sqrt {4 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right ) - \frac {3}{4} \, \arctan \left (\frac {\sin \left (x\right )}{\sqrt {4 \, \cos \left (x\right )^{2} - 1}}\right ) + \frac {3}{8} \, \arctan \left (\frac {4 \, {\left (8 \, \cos \left (x\right )^{2} - 5\right )} \sqrt {4 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right ) - 9 \, \cos \left (x\right ) \sin \left (x\right )}{64 \, \cos \left (x\right )^{4} - 71 \, \cos \left (x\right )^{2} + 16}\right ) + \frac {3}{8} \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) + \frac {3}{8} \, \arctan \left (\frac {9 \, \cos \left (x\right )^{2} - 2}{2 \, \sqrt {8 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right )}\right ) \] Input:

integrate(cos(3*x)/(-(-1+8*cos(x)^2)^(1/2)+(3*cos(x)^2-sin(x)^2)^(1/2)),x, 
 algorithm="fricas")
 

Output:

-5/32*sqrt(2)*arctan(1/16*(512*sqrt(2)*cos(x)^4 - 576*sqrt(2)*cos(x)^2 + 1 
13*sqrt(2))*sqrt(8*cos(x)^2 - 1)/((128*cos(x)^4 - 88*cos(x)^2 + 9)*sin(x)) 
) - 1/2*sqrt(8*cos(x)^2 - 1)*sin(x) - 1/2*sqrt(4*cos(x)^2 - 1)*sin(x) - 3/ 
4*arctan(sin(x)/sqrt(4*cos(x)^2 - 1)) + 3/8*arctan((4*(8*cos(x)^2 - 5)*sqr 
t(4*cos(x)^2 - 1)*sin(x) - 9*cos(x)*sin(x))/(64*cos(x)^4 - 71*cos(x)^2 + 1 
6)) + 3/8*arctan(sin(x)/cos(x)) + 3/8*arctan(1/2*(9*cos(x)^2 - 2)/(sqrt(8* 
cos(x)^2 - 1)*sin(x)))
 

Sympy [F]

\[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\int \frac {\cos {\left (3 x \right )}}{\sqrt {- \sin ^{2}{\left (x \right )} + 3 \cos ^{2}{\left (x \right )}} - \sqrt {8 \cos ^{2}{\left (x \right )} - 1}}\, dx \] Input:

integrate(cos(3*x)/(-(-1+8*cos(x)**2)**(1/2)+(3*cos(x)**2-sin(x)**2)**(1/2 
)),x)
 

Output:

Integral(cos(3*x)/(sqrt(-sin(x)**2 + 3*cos(x)**2) - sqrt(8*cos(x)**2 - 1)) 
, x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\text {Timed out} \] Input:

integrate(cos(3*x)/(-(-1+8*cos(x)^2)^(1/2)+(3*cos(x)^2-sin(x)^2)^(1/2)),x, 
 algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\int { -\frac {\cos \left (3 \, x\right )}{\sqrt {8 \, \cos \left (x\right )^{2} - 1} - \sqrt {3 \, \cos \left (x\right )^{2} - \sin \left (x\right )^{2}}} \,d x } \] Input:

integrate(cos(3*x)/(-(-1+8*cos(x)^2)^(1/2)+(3*cos(x)^2-sin(x)^2)^(1/2)),x, 
 algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate(-cos(3*x)/(sqrt(8*cos(x)^2 - 1) - sqrt(3*cos(x)^2 - sin(x)^2)), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=-\int -\frac {\cos \left (3\,x\right )}{\sqrt {3\,{\cos \left (x\right )}^2-{\sin \left (x\right )}^2}-\sqrt {8\,{\cos \left (x\right )}^2-1}} \,d x \] Input:

int(cos(3*x)/((3*cos(x)^2 - sin(x)^2)^(1/2) - (8*cos(x)^2 - 1)^(1/2)),x)
 

Output:

-int(-cos(3*x)/((3*cos(x)^2 - sin(x)^2)^(1/2) - (8*cos(x)^2 - 1)^(1/2)), x 
)
 

Reduce [F]

\[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=-\left (\int \frac {\sqrt {8 \cos \left (x \right )^{2}-1}\, \cos \left (3 x \right )}{5 \cos \left (x \right )^{2}+\sin \left (x \right )^{2}-1}d x \right )-\left (\int \frac {\sqrt {3 \cos \left (x \right )^{2}-\sin \left (x \right )^{2}}\, \cos \left (3 x \right )}{5 \cos \left (x \right )^{2}+\sin \left (x \right )^{2}-1}d x \right ) \] Input:

int(cos(3*x)/(-(-1+8*cos(x)^2)^(1/2)+(3*cos(x)^2-sin(x)^2)^(1/2)),x)
 

Output:

 - (int((sqrt(8*cos(x)**2 - 1)*cos(3*x))/(5*cos(x)**2 + sin(x)**2 - 1),x) 
+ int((sqrt(3*cos(x)**2 - sin(x)**2)*cos(3*x))/(5*cos(x)**2 + sin(x)**2 - 
1),x))