3.861 \(\int \sec (x) \sqrt {\sec (x)+\tan (x)} \, dx\)

Optimal. Leaf size=13 \[ 2 \sqrt {(\sin (x)+1) \sec (x)} \]

[Out]

2*(sec(x)*(1+sin(x)))^(1/2)

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Rubi [A]  time = 0.15, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4397, 4400, 2705, 2671} \[ 2 \sqrt {(\sin (x)+1) \sec (x)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]*Sqrt[Sec[x] + Tan[x]],x]

[Out]

2*Sqrt[Sec[x]*(1 + Sin[x])]

Rule 2671

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rule 2705

Int[((g_.)*sec[(e_.) + (f_.)*(x_)])^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[g^(2*
IntPart[p])*(g*Cos[e + f*x])^FracPart[p]*(g*Sec[e + f*x])^FracPart[p], Int[(a + b*Sin[e + f*x])^m/(g*Cos[e + f
*x])^p, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] &&  !IntegerQ[p]

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 4400

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p)
, x], x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  !InertTrigFreeQ[w])

Rubi steps

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

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Mathematica [B]  time = 0.04, size = 37, normalized size = 2.85 \[ 2 \sqrt {\frac {\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]*Sqrt[Sec[x] + Tan[x]],x]

[Out]

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

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fricas [A]  time = 0.89, size = 21, normalized size = 1.62 \[ 2 \, \sqrt {\frac {\cos \relax (x) + \sin \relax (x) + 1}{\cos \relax (x) - \sin \relax (x) + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt((cos(x) + sin(x) + 1)/(cos(x) - sin(x) + 1))

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giac [B]  time = 0.32, size = 55, normalized size = 4.23 \[ -\frac {4 \, \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{3} - \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) - 1\right ) \mathrm {sgn}\left (\cos \relax (x)\right )}{\frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{2} + 1} - 1}{\tan \left (\frac {1}{2} \, x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sgn(-tan(1/2*x)^3 - tan(1/2*x)^2 - tan(1/2*x) - 1)*sgn(cos(x))/((sqrt(-tan(1/2*x)^2 + 1) - 1)/tan(1/2*x) +
1)

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maple [A]  time = 0.14, size = 10, normalized size = 0.77 \[ 2 \sqrt {\sec \relax (x )+\tan \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sec(x)+tan(x))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sec \relax (x) + \tan \relax (x)} \sec \relax (x)\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sec(x) + tan(x))*sec(x), x)

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mupad [B]  time = 0.29, size = 14, normalized size = 1.08 \[ 2\,\sqrt {\frac {1}{\cos \relax (x)}}\,\sqrt {\sin \relax (x)+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(1/cos(x))^(1/2)*(sin(x) + 1)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\tan {\relax (x )} + \sec {\relax (x )}} \sec {\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(tan(x) + sec(x))*sec(x), x)

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