3.117 \(\int \cos (x) \sec (3 x) \, dx\)

Optimal. Leaf size=44 \[ \frac {\log \left (\sqrt {3} \sin (x)+\cos (x)\right )}{2 \sqrt {3}}-\frac {\log \left (\cos (x)-\sqrt {3} \sin (x)\right )}{2 \sqrt {3}} \]

[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)

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Rubi [A]  time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {206} \[ \frac {\log \left (\sqrt {3} \sin (x)+\cos (x)\right )}{2 \sqrt {3}}-\frac {\log \left (\cos (x)-\sqrt {3} \sin (x)\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

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

Rubi steps

\begin {align*} \int \cos (x) \sec (3 x) \, dx &=\operatorname {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*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 0.34 \[ \frac {\tanh ^{-1}\left (\sqrt {3} \tan (x)\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.49, size = 53, normalized size = 1.20 \[ \frac {1}{12} \, \sqrt {3} \log \left (-\frac {8 \, \cos \relax (x)^{4} + 4 \, {\left (2 \, \sqrt {3} \cos \relax (x)^{3} - 3 \, \sqrt {3} \cos \relax (x)\right )} \sin \relax (x) - 9}{16 \, \cos \relax (x)^{4} - 24 \, \cos \relax (x)^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.17, size = 31, normalized size = 0.70 \[ -\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 6 \, \tan \relax (x) \right |}}{{\left | 2 \, \sqrt {3} + 6 \, \tan \relax (x) \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [A]  time = 0.18, size = 13, normalized size = 0.30 \[ \frac {\sqrt {3}\, \arctanh \left (\tan \relax (x ) \sqrt {3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.43, size = 76, normalized size = 1.73 \[ \frac {1}{12} \, \sqrt {3} {\left (\log \left (\frac {4}{3} \, \cos \left (2 \, x\right )^{2} + \frac {4}{3} \, \sin \left (2 \, x\right )^{2} + \frac {4}{3} \, \sqrt {3} \sin \left (2 \, x\right ) - \frac {4}{3} \, \cos \left (2 \, x\right ) + \frac {4}{3}\right ) - \log \left (\frac {4}{3} \, \cos \left (2 \, x\right )^{2} + \frac {4}{3} \, \sin \left (2 \, x\right )^{2} - \frac {4}{3} \, \sqrt {3} \sin \left (2 \, x\right ) - \frac {4}{3} \, \cos \left (2 \, x\right ) + \frac {4}{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 2.66, size = 16, normalized size = 0.36 \[ \frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sin \relax (x)}{\cos \relax (x)}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\relax (x )} \sec {\left (3 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)*sec(3*x), x)

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