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

Optimal. Leaf size=43 \[ \frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x)-\frac{1}{3} \sqrt{5} \sinh ^{-1}\left (\sqrt{5} \cos (3 x)\right ) \]

[Out]

-(Sqrt[5]*ArcSinh[Sqrt[5]*Cos[3*x]])/3 + (Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3

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Rubi [A]  time = 0.0949236, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {277, 215} \[ \frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x)-\frac{1}{3} \sqrt{5} \sinh ^{-1}\left (\sqrt{5} \cos (3 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[5]*ArcSinh[Sqrt[5]*Cos[3*x]])/3 + (Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3

Rule 277

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0573059, size = 43, normalized size = 1. \[ \frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x)-\frac{1}{3} \sqrt{5} \sinh ^{-1}\left (\sqrt{5} \cos (3 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[5]*ArcSinh[Sqrt[5]*Cos[3*x]])/3 + (Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3

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Maple [A]  time = 0.043, size = 66, normalized size = 1.5 \begin{align*} -{\frac{\sec \left ( 3\,x \right ) }{3}\sqrt{{\frac{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}+5}{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}}}} \left ( \sqrt{5}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}+5}}}} \right ) -\sqrt{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}+5} \right ){\frac{1}{\sqrt{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}+5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46382, size = 47, normalized size = 1.09 \begin{align*} -\frac{1}{3} \, \sqrt{5} \operatorname{arsinh}\left (\sqrt{5} \cos \left (3 \, x\right )\right ) + \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)*(1+5*cos(3*x)^2)^(1/2)*tan(3*x),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.76936, size = 354, normalized size = 8.23 \begin{align*} \frac{\sqrt{5} \cos \left (3 \, x\right ) \log \left (80000 \, \cos \left (3 \, x\right )^{8} + 32000 \, \cos \left (3 \, x\right )^{6} + 4000 \, \cos \left (3 \, x\right )^{4} + 160 \, \cos \left (3 \, x\right )^{2} - 8 \,{\left (2000 \, \sqrt{5} \cos \left (3 \, x\right )^{7} + 600 \, \sqrt{5} \cos \left (3 \, x\right )^{5} + 50 \, \sqrt{5} \cos \left (3 \, x\right )^{3} + \sqrt{5} \cos \left (3 \, x\right )\right )} \sqrt{5 \, \cos \left (3 \, x\right )^{2} + 1} + 1\right ) + 8 \, \sqrt{5 \, \cos \left (3 \, x\right )^{2} + 1}}{24 \, \cos \left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(sqrt(5)*cos(3*x)*log(80000*cos(3*x)^8 + 32000*cos(3*x)^6 + 4000*cos(3*x)^4 + 160*cos(3*x)^2 - 8*(2000*sq
rt(5)*cos(3*x)^7 + 600*sqrt(5)*cos(3*x)^5 + 50*sqrt(5)*cos(3*x)^3 + sqrt(5)*cos(3*x))*sqrt(5*cos(3*x)^2 + 1) +
1) + 8*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 \sqrt{5 \cos ^{2}{\left (3 x \right )} + 1} \tan{\left (3 x \right )} \sec{\left (3 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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