Integrand size = 39, antiderivative size = 112 \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\frac {5 \arcsin \left (2 \sqrt {\frac {2}{7}} \sin (x)\right )}{4 \sqrt {2}}+\frac {3}{4} \arcsin \left (\frac {2 \sin (x)}{\sqrt {3}}\right )-\frac {3}{4} \arctan \left (\frac {\sin (x)}{\sqrt {-1+4 \cos ^2(x)}}\right )-\frac {3}{4} \arctan \left (\frac {\sin (x)}{\sqrt {-1+8 \cos ^2(x)}}\right )-\frac {1}{2} \sqrt {-1+4 \cos ^2(x)} \sin (x)-\frac {1}{2} \sqrt {-1+8 \cos ^2(x)} \sin (x) \]
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)
Time = 0.72 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.39 \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\frac {1}{8} \left (5 \sqrt {2} \arcsin \left (2 \sqrt {\frac {2}{7}} \sin (x)\right )+6 \arcsin \left (\frac {2 \sin (x)}{\sqrt {3}}\right )+3 \arctan \left (\frac {7-8 \sin (x)}{\sqrt {3+4 \cos (2 x)}}\right )+3 \arctan \left (\frac {3-4 \sin (x)}{\sqrt {1+2 \cos (2 x)}}\right )-3 \arctan \left (\frac {3+4 \sin (x)}{\sqrt {1+2 \cos (2 x)}}\right )-3 \arctan \left (\frac {7+8 \sin (x)}{\sqrt {3+4 \cos (2 x)}}\right )-4 \sqrt {1+2 \cos (2 x)} \sin (x)-4 \sqrt {3+4 \cos (2 x)} \sin (x)\right ) \]
(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
Time = 0.59 (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 |
| \(\displaystyle \frac {5 \arcsin \left (2 \sqrt {\frac {2}{7}} \sin (x)\right )}{4 \sqrt {2}}+\frac {3}{4} \arcsin \left (\frac {2 \sin (x)}{\sqrt {3}}\right )-\frac {3}{4} \arctan \left (\frac {\sin (x)}{\sqrt {7-8 \sin ^2(x)}}\right )-\frac {3}{4} \arctan \left (\frac {\sin (x)}{\sqrt {3-4 \sin ^2(x)}}\right )-\frac {1}{2} \sin (x) \sqrt {7-8 \sin ^2(x)}-\frac {1}{2} \sin (x) \sqrt {3-4 \sin ^2(x)}\) |
(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
3.5.27.3.1 Defintions of rubi rules used
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]]
Leaf count of result is larger than twice the leaf count of optimal. \(232\) vs. \(2(84)=168\).
Time = 1.15 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.08
| method | result | size |
| default | \(\frac {3 \sqrt {-8 \left (\sin \left (x \right )+1\right )^{2}+16 \sin \left (x \right )+15}}{8}+\frac {5 \arcsin \left (\frac {2 \sin \left (x \right ) \sqrt {14}}{7}\right ) \sqrt {2}}{8}-\frac {3 \arctan \left (\frac {14+16 \sin \left (x \right )}{2 \sqrt {-8 \left (\sin \left (x \right )+1\right )^{2}+16 \sin \left (x \right )+15}}\right )}{8}-\frac {3 \sqrt {-8 \left (\sin \left (x \right )-1\right )^{2}-16 \sin \left (x \right )+15}}{8}+\frac {3 \arctan \left (\frac {14-16 \sin \left (x \right )}{2 \sqrt {-8 \left (\sin \left (x \right )-1\right )^{2}-16 \sin \left (x \right )+15}}\right )}{8}+\frac {3 \sqrt {-4 \left (\sin \left (x \right )+1\right )^{2}+8 \sin \left (x \right )+7}}{8}+\frac {3 \arcsin \left (\frac {2 \sin \left (x \right ) \sqrt {3}}{3}\right )}{4}-\frac {3 \arctan \left (\frac {6+8 \sin \left (x \right )}{2 \sqrt {-4 \left (\sin \left (x \right )+1\right )^{2}+8 \sin \left (x \right )+7}}\right )}{8}-\frac {3 \sqrt {-4 \left (\sin \left (x \right )-1\right )^{2}-8 \sin \left (x \right )+7}}{8}+\frac {3 \arctan \left (\frac {6-8 \sin \left (x \right )}{2 \sqrt {-4 \left (\sin \left (x \right )-1\right )^{2}-8 \sin \left (x \right )+7}}\right )}{8}-\frac {\sin \left (x \right ) \sqrt {-8 \left (\sin ^{2}\left (x \right )\right )+7}}{2}-\frac {\sin \left (x \right ) \sqrt {3-4 \left (\sin ^{2}\left (x \right )\right )}}{2}\) | \(233\) |
3/8*(-8*(sin(x)+1)^2+16*sin(x)+15)^(1/2)+5/8*arcsin(2/7*sin(x)*14^(1/2))*2 ^(1/2)-3/8*arctan(1/2*(14+16*sin(x))/(-8*(sin(x)+1)^2+16*sin(x)+15)^(1/2)) -3/8*(-8*(sin(x)-1)^2-16*sin(x)+15)^(1/2)+3/8*arctan(1/2*(14-16*sin(x))/(- 8*(sin(x)-1)^2-16*sin(x)+15)^(1/2))+3/8*(-4*(sin(x)+1)^2+8*sin(x)+7)^(1/2) +3/4*arcsin(2/3*sin(x)*3^(1/2))-3/8*arctan(1/2*(6+8*sin(x))/(-4*(sin(x)+1) ^2+8*sin(x)+7)^(1/2))-3/8*(-4*(sin(x)-1)^2-8*sin(x)+7)^(1/2)+3/8*arctan(1/ 2*(6-8*sin(x))/(-4*(sin(x)-1)^2-8*sin(x)+7)^(1/2))-1/2*sin(x)*(-8*sin(x)^2 +7)^(1/2)-1/2*sin(x)*(3-4*sin(x)^2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (84) = 168\).
Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.74 \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=-\frac {5}{32} \, \sqrt {2} \arctan \left (\frac {{\left (512 \, \sqrt {2} \cos \left (x\right )^{4} - 576 \, \sqrt {2} \cos \left (x\right )^{2} + 113 \, \sqrt {2}\right )} \sqrt {8 \, \cos \left (x\right )^{2} - 1}}{16 \, {\left (128 \, \cos \left (x\right )^{4} - 88 \, \cos \left (x\right )^{2} + 9\right )} \sin \left (x\right )}\right ) - \frac {1}{2} \, \sqrt {8 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right ) - \frac {1}{2} \, \sqrt {4 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right ) + \frac {3}{8} \, \arctan \left (\frac {4 \, {\left (8 \, \cos \left (x\right )^{2} - 5\right )} \sqrt {4 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right ) - 9 \, \cos \left (x\right ) \sin \left (x\right )}{64 \, \cos \left (x\right )^{4} - 71 \, \cos \left (x\right )^{2} + 16}\right ) + \frac {3}{8} \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) + \frac {3}{8} \, \arctan \left (\frac {9 \, \cos \left (x\right )^{2} - 2}{2 \, \sqrt {8 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right )}\right ) + \frac {3}{4} \, \arctan \left (\frac {\sqrt {4 \, \cos \left (x\right )^{2} - 1}}{\sin \left (x\right )}\right ) \]
integrate(cos(3*x)/(-(-1+8*cos(x)^2)^(1/2)+(3*cos(x)^2-sin(x)^2)^(1/2)),x, algorithm="fricas")
-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/ 8*arctan((4*(8*cos(x)^2 - 5)*sqrt(4*cos(x)^2 - 1)*sin(x) - 9*cos(x)*sin(x) )/(64*cos(x)^4 - 71*cos(x)^2 + 16)) + 3/8*arctan(sin(x)/cos(x)) + 3/8*arct an(1/2*(9*cos(x)^2 - 2)/(sqrt(8*cos(x)^2 - 1)*sin(x))) + 3/4*arctan(sqrt(4 *cos(x)^2 - 1)/sin(x))
\[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\int \frac {\cos {\left (3 x \right )}}{\sqrt {- \sin ^{2}{\left (x \right )} + 3 \cos ^{2}{\left (x \right )}} - \sqrt {8 \cos ^{2}{\left (x \right )} - 1}}\, dx \]
Timed out. \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\text {Timed out} \]
integrate(cos(3*x)/(-(-1+8*cos(x)^2)^(1/2)+(3*cos(x)^2-sin(x)^2)^(1/2)),x, algorithm="maxima")
\[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=\int { -\frac {\cos \left (3 \, x\right )}{\sqrt {8 \, \cos \left (x\right )^{2} - 1} - \sqrt {3 \, \cos \left (x\right )^{2} - \sin \left (x\right )^{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=-\int -\frac {\cos \left (3\,x\right )}{\sqrt {3\,{\cos \left (x\right )}^2-{\sin \left (x\right )}^2}-\sqrt {8\,{\cos \left (x\right )}^2-1}} \,d x \]
\[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx=-\left (\int \frac {\sqrt {8 \cos \left (x \right )^{2}-1}\, \cos \left (3 x \right )}{5 \cos \left (x \right )^{2}+\sin \left (x \right )^{2}-1}d x \right )-\left (\int \frac {\sqrt {3 \cos \left (x \right )^{2}-\sin \left (x \right )^{2}}\, \cos \left (3 x \right )}{5 \cos \left (x \right )^{2}+\sin \left (x \right )^{2}-1}d x \right ) \]