\(\int \cos (x) \sec (3 x) \, dx\) [117]

Optimal result

Integrand size = 7, antiderivative size = 44 \[ \int \cos (x) \sec (3 x) \, dx=-\frac {\log \left (\cos (x)-\sqrt {3} \sin (x)\right )}{2 \sqrt {3}}+\frac {\log \left (\cos (x)+\sqrt {3} \sin (x)\right )}{2 \sqrt {3}} \]

[Out]

-1/6*ln(cos(x)-sin(x)*3^(1/2))*3^(1/2)+1/6*ln(cos(x)+sin(x)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {212} \[ \int \cos (x) \sec (3 x) \, dx=\frac {\log \left (\sqrt {3} \sin (x)+\cos (x)\right )}{2 \sqrt {3}}-\frac {\log \left (\cos (x)-\sqrt {3} \sin (x)\right )}{2 \sqrt {3}} \]

[In]

Int[Cos[x]*Sec[3*x],x]

[Out]

-1/2*Log[Cos[x] - Sqrt[3]*Sin[x]]/Sqrt[3] + Log[Cos[x] + Sqrt[3]*Sin[x]]/(2*Sqrt[3])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\tan (x)\right ) \\ & = -\frac {\log \left (\cos (x)-\sqrt {3} \sin (x)\right )}{2 \sqrt {3}}+\frac {\log \left (\cos (x)+\sqrt {3} \sin (x)\right )}{2 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.34 \[ \int \cos (x) \sec (3 x) \, dx=\frac {\text {arctanh}\left (\sqrt {3} \tan (x)\right )}{\sqrt {3}} \]

[In]

Integrate[Cos[x]*Sec[3*x],x]

[Out]

ArcTanh[Sqrt[3]*Tan[x]]/Sqrt[3]

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.30

method	result	size
default	\(\frac {\sqrt {3}\, \operatorname {arctanh}\left (\tan \left (x \right ) \sqrt {3}\right )}{3}\)	\(13\)
risch	\(\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{6}-\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{6}\)	\(40\)

[In]

int(cos(x)*sec(3*x),x,method=_RETURNVERBOSE)

[Out]

1/3*3^(1/2)*arctanh(tan(x)*3^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.20 \[ \int \cos (x) \sec (3 x) \, dx=\frac {1}{12} \, \sqrt {3} \log \left (-\frac {8 \, \cos \left (x\right )^{4} + 4 \, {\left (2 \, \sqrt {3} \cos \left (x\right )^{3} - 3 \, \sqrt {3} \cos \left (x\right )\right )} \sin \left (x\right ) - 9}{16 \, \cos \left (x\right )^{4} - 24 \, \cos \left (x\right )^{2} + 9}\right ) \]

[In]

integrate(cos(x)*sec(3*x),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*log(-(8*cos(x)^4 + 4*(2*sqrt(3)*cos(x)^3 - 3*sqrt(3)*cos(x))*sin(x) - 9)/(16*cos(x)^4 - 24*cos(x)
^2 + 9))

Sympy [F]

\[ \int \cos (x) \sec (3 x) \, dx=\int \cos {\left (x \right )} \sec {\left (3 x \right )}\, dx \]

[In]

integrate(cos(x)*sec(3*x),x)

[Out]

Integral(cos(x)*sec(3*x), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (32) = 64\).

Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.73 \[ \int \cos (x) \sec (3 x) \, dx=\frac {1}{12} \, \sqrt {3} {\left (\log \left (\frac {4}{3} \, \cos \left (2 \, x\right )^{2} + \frac {4}{3} \, \sin \left (2 \, x\right )^{2} + \frac {4}{3} \, \sqrt {3} \sin \left (2 \, x\right ) - \frac {4}{3} \, \cos \left (2 \, x\right ) + \frac {4}{3}\right ) - \log \left (\frac {4}{3} \, \cos \left (2 \, x\right )^{2} + \frac {4}{3} \, \sin \left (2 \, x\right )^{2} - \frac {4}{3} \, \sqrt {3} \sin \left (2 \, x\right ) - \frac {4}{3} \, \cos \left (2 \, x\right ) + \frac {4}{3}\right )\right )} \]

[In]

integrate(cos(x)*sec(3*x),x, algorithm="maxima")

[Out]

1/12*sqrt(3)*(log(4/3*cos(2*x)^2 + 4/3*sin(2*x)^2 + 4/3*sqrt(3)*sin(2*x) - 4/3*cos(2*x) + 4/3) - log(4/3*cos(2
*x)^2 + 4/3*sin(2*x)^2 - 4/3*sqrt(3)*sin(2*x) - 4/3*cos(2*x) + 4/3))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.70 \[ \int \cos (x) \sec (3 x) \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 6 \, \tan \left (x\right ) \right |}}{{\left | 2 \, \sqrt {3} + 6 \, \tan \left (x\right ) \right |}}\right ) \]

[In]

integrate(cos(x)*sec(3*x),x, algorithm="giac")

[Out]

-1/6*sqrt(3)*log(abs(-2*sqrt(3) + 6*tan(x))/abs(2*sqrt(3) + 6*tan(x)))

Mupad [B] (verification not implemented)

Time = 26.79 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.36 \[ \int \cos (x) \sec (3 x) \, dx=\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sin \left (x\right )}{\cos \left (x\right )}\right )}{3} \]

[In]

int(cos(x)/cos(3*x),x)

[Out]

(3^(1/2)*atanh((3^(1/2)*sin(x))/cos(x)))/3