3.383 \(\int \cos ^2(x) \sec (3 x) \, dx\)

Optimal. Leaf size=9 \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x)) \]

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Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {206} \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^2*Sec[3*x],x]

[Out]

ArcTanh[2*Sin[x]]/2

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 ^2(x) \sec (3 x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \tanh ^{-1}(2 \sin (x))\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 1.00 \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^2*Sec[3*x],x]

[Out]

ArcTanh[2*Sin[x]]/2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^2(x) \sec (3 x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[x]^2*Sec[3*x],x]

[Out]

Could not integrate

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fricas [B]  time = 1.03, size = 19, normalized size = 2.11 \[ \frac {1}{4} \, \log \left (2 \, \sin \relax (x) + 1\right ) - \frac {1}{4} \, \log \left (-2 \, \sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2/cos(3*x),x, algorithm="fricas")

[Out]

1/4*log(2*sin(x) + 1) - 1/4*log(-2*sin(x) + 1)

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giac [B]  time = 0.65, size = 21, normalized size = 2.33 \[ \frac {1}{4} \, \log \left ({\left | 2 \, \sin \relax (x) + 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | 2 \, \sin \relax (x) - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2/cos(3*x),x, algorithm="giac")

[Out]

1/4*log(abs(2*sin(x) + 1)) - 1/4*log(abs(2*sin(x) - 1))

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maple [B]  time = 0.15, size = 20, normalized size = 2.22

method	result	size
default	\(-\frac {\ln \left (2 \sin \relax (x )-1\right )}{4}+\frac {\ln \left (1+2 \sin \relax (x )\right )}{4}\)	\(20\)
risch	\(\frac {\ln \left (i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{4}-\frac {\ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{4}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2/cos(3*x),x,method=_RETURNVERBOSE)

[Out]

-1/4*ln(2*sin(x)-1)+1/4*ln(1+2*sin(x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \relax (x)^{2}}{\cos \left (3 \, x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2/cos(3*x),x, algorithm="maxima")

[Out]

integrate(cos(x)^2/cos(3*x), x)

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mupad [B]  time = 0.36, size = 7, normalized size = 0.78 \[ \frac {\mathrm {atanh}\left (2\,\sin \relax (x)\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2/cos(3*x),x)

[Out]

atanh(2*sin(x))/2

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sympy [B]  time = 5.46, size = 76, normalized size = 8.44 \[ - \frac {\log {\left (\sin {\left (3 x \right )} - 1 \right )}}{12} + \frac {\log {\left (\sin {\left (3 x \right )} + 1 \right )}}{12} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{6} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{6} - \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} - 4 \tan {\left (\frac {x}{2} \right )} + 1 \right )}}{12} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )} + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**2/cos(3*x),x)

[Out]

-log(sin(3*x) - 1)/12 + log(sin(3*x) + 1)/12 - log(tan(x/2) - 1)/6 + log(tan(x/2) + 1)/6 - log(tan(x/2)**2 - 4
*tan(x/2) + 1)/12 + log(tan(x/2)**2 + 4*tan(x/2) + 1)/12

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