3.734 \(\int \frac{\sec (3 x) \tan (3 x)}{\sqrt{1+5 \cos ^2(3 x)}} \, dx\)

Optimal. Leaf size=22 \[ \frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x) \]

[Out]

(Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3

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Rubi [A]  time = 0.0909895, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {264} \[ \frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x) \]

Antiderivative was successfully verified.

[In]

Int[(Sec[3*x]*Tan[3*x])/Sqrt[1 + 5*Cos[3*x]^2],x]

[Out]

(Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sec (3 x) \tan (3 x)}{\sqrt{1+5 \cos ^2(3 x)}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+5 x^2}} \, dx,x,\cos (3 x)\right )\right )\\ &=\frac{1}{3} \sqrt{1+5 \cos ^2(3 x)} \sec (3 x)\\ \end{align*}

Mathematica [A]  time = 0.0330922, size = 22, normalized size = 1. \[ \frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[3*x]*Tan[3*x])/Sqrt[1 + 5*Cos[3*x]^2],x]

[Out]

(Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3

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Maple [A]  time = 0.032, size = 34, normalized size = 1.6 \begin{align*}{\frac{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}+5}{3\,\sec \left ( 3\,x \right ) }{\frac{1}{\sqrt{{\frac{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}+5}{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(3*x)*tan(3*x)/(1+5*cos(3*x)^2)^(1/2),x)

[Out]

1/3/((sec(3*x)^2+5)/sec(3*x)^2)^(1/2)/sec(3*x)*(sec(3*x)^2+5)

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Maxima [A]  time = 1.43293, size = 27, normalized size = 1.23 \begin{align*} \frac{\sqrt{5 \, \cos \left (3 \, x\right )^{2} + 1}}{3 \, \cos \left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(3*x)*tan(3*x)/(1+5*cos(3*x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(5*cos(3*x)^2 + 1)/cos(3*x)

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Fricas [A]  time = 2.44364, size = 50, normalized size = 2.27 \begin{align*} \frac{\sqrt{5 \, \cos \left (3 \, x\right )^{2} + 1}}{3 \, \cos \left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(3*x)*tan(3*x)/(1+5*cos(3*x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(5*cos(3*x)^2 + 1)/cos(3*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan{\left (3 x \right )} \sec{\left (3 x \right )}}{\sqrt{5 \cos ^{2}{\left (3 x \right )} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(3*x)*tan(3*x)/(1+5*cos(3*x)**2)**(1/2),x)

[Out]

Integral(tan(3*x)*sec(3*x)/sqrt(5*cos(3*x)**2 + 1), x)

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Giac [B]  time = 1.24747, size = 159, normalized size = 7.23 \begin{align*} \frac{2 \, \sqrt{2}{\left (\sqrt{3} \tan \left (\frac{3}{2} \, x\right )^{2} + \sqrt{3} - \sqrt{3 \, \tan \left (\frac{3}{2} \, x\right )^{4} - 4 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 3}\right )}}{3 \,{\left ({\left (\sqrt{3} \tan \left (\frac{3}{2} \, x\right )^{2} - \sqrt{3 \, \tan \left (\frac{3}{2} \, x\right )^{4} - 4 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 3}\right )}^{2} - 2 \, \sqrt{3}{\left (\sqrt{3} \tan \left (\frac{3}{2} \, x\right )^{2} - \sqrt{3 \, \tan \left (\frac{3}{2} \, x\right )^{4} - 4 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 3}\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(3*x)*tan(3*x)/(1+5*cos(3*x)^2)^(1/2),x, algorithm="giac")

[Out]

2/3*sqrt(2)*(sqrt(3)*tan(3/2*x)^2 + sqrt(3) - sqrt(3*tan(3/2*x)^4 - 4*tan(3/2*x)^2 + 3))/((sqrt(3)*tan(3/2*x)^
2 - sqrt(3*tan(3/2*x)^4 - 4*tan(3/2*x)^2 + 3))^2 - 2*sqrt(3)*(sqrt(3)*tan(3/2*x)^2 - sqrt(3*tan(3/2*x)^4 - 4*t
an(3/2*x)^2 + 3)) + 1)