3.25 \(\int \frac{1}{1-3 \cos ^2(2+3 x)} \, dx\)

Optimal. Leaf size=61 \[ \frac{\log \left (\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right )}{6 \sqrt{2}}-\frac{\log \left (\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right )}{6 \sqrt{2}} \]

[Out]

Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])

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Rubi [A]  time = 0.0191013, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3181, 206} \[ \frac{\log \left (\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right )}{6 \sqrt{2}}-\frac{\log \left (\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right )}{6 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 3*Cos[2 + 3*x]^2)^(-1),x]

[Out]

Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

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 \frac{1}{1-3 \cos ^2(2+3 x)} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\cot (2+3 x)\right )\right )\\ &=\frac{\log \left (\sqrt{2} \cos (2+3 x)-\sin (2+3 x)\right )}{6 \sqrt{2}}-\frac{\log \left (\sqrt{2} \cos (2+3 x)+\sin (2+3 x)\right )}{6 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0578985, size = 22, normalized size = 0.36 \[ -\frac{\tanh ^{-1}\left (\frac{\tan (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 3*Cos[2 + 3*x]^2)^(-1),x]

[Out]

-ArcTanh[Tan[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])

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Maple [A]  time = 0.018, size = 18, normalized size = 0.3 \begin{align*} -{\frac{\sqrt{2}}{6}{\it Artanh} \left ({\frac{\tan \left ( 2+3\,x \right ) \sqrt{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*2^(1/2)*arctanh(1/2*tan(2+3*x)*2^(1/2))

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Maxima [A]  time = 1.66264, size = 43, normalized size = 0.7 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \tan \left (3 \, x + 2\right )}{\sqrt{2} + \tan \left (3 \, x + 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(2)*log(-(sqrt(2) - tan(3*x + 2))/(sqrt(2) + tan(3*x + 2)))

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Fricas [A]  time = 1.63888, size = 230, normalized size = 3.77 \begin{align*} \frac{1}{24} \, \sqrt{2} \log \left (-\frac{7 \, \cos \left (3 \, x + 2\right )^{4} - 10 \, \cos \left (3 \, x + 2\right )^{2} + 4 \,{\left (\sqrt{2} \cos \left (3 \, x + 2\right )^{3} + \sqrt{2} \cos \left (3 \, x + 2\right )\right )} \sin \left (3 \, x + 2\right ) - 1}{9 \, \cos \left (3 \, x + 2\right )^{4} - 6 \, \cos \left (3 \, x + 2\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(2)*log(-(7*cos(3*x + 2)^4 - 10*cos(3*x + 2)^2 + 4*(sqrt(2)*cos(3*x + 2)^3 + sqrt(2)*cos(3*x + 2))*si
n(3*x + 2) - 1)/(9*cos(3*x + 2)^4 - 6*cos(3*x + 2)^2 + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.20706, size = 53, normalized size = 0.87 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \tan \left (3 \, x + 2\right ) \right |}}{{\left | 2 \, \sqrt{2} + 2 \, \tan \left (3 \, x + 2\right ) \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(2)*log(abs(-2*sqrt(2) + 2*tan(3*x + 2))/abs(2*sqrt(2) + 2*tan(3*x + 2)))