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]

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

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Rubi [A]  time = 0.09, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 1.00 \[ \frac {1}{3} \sqrt {5 \cos ^2(3 x)+1} \sec (3 x) \]

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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fricas [A]  time = 0.67, size = 20, normalized size = 0.91 \[ \frac {\sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1}}{3 \, \cos \left (3 \, x\right )} \]

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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giac [A]  time = 0.15, size = 34, normalized size = 1.55 \[ -\frac {2 \, \sqrt {5}}{3 \, {\left ({\left (\sqrt {5} \cos \left (3 \, x\right ) - \sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1}\right )}^{2} - 1\right )}} \]

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(5)/((sqrt(5)*cos(3*x) - sqrt(5*cos(3*x)^2 + 1))^2 - 1)

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maple [A]  time = 0.11, size = 34, normalized size = 1.55 \[ \frac {\sec ^{2}\left (3 x \right )+5}{3 \sqrt {\frac {\sec ^{2}\left (3 x \right )+5}{\sec \left (3 x \right )^{2}}}\, \sec \left (3 x \right )} \]

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 = 0.42, size = 20, normalized size = 0.91 \[ \frac {\sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1}}{3 \, \cos \left (3 \, x\right )} \]

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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mupad [B]  time = 3.03, size = 18, normalized size = 0.82 \[ \frac {\sqrt {\frac {5\,\cos \left (6\,x\right )}{2}+\frac {7}{2}}}{3\,\cos \left (3\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*cos(6*x))/2 + 7/2)^(1/2)/(3*cos(3*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan {\left (3 x \right )} \sec {\left (3 x \right )}}{\sqrt {5 \cos ^{2}{\left (3 x \right )} + 1}}\, dx \]

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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