3.13 \(\int \frac{1}{\sqrt{\cos (a+b x)}} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b} \]

[Out]

(2*EllipticF[(a + b*x)/2, 2])/b

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Rubi [A]  time = 0.0091156, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2641} \[ \frac{2 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Cos[a + b*x]],x]

[Out]

(2*EllipticF[(a + b*x)/2, 2])/b

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx &=\frac{2 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0307141, size = 16, normalized size = 1. \[ \frac{2 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Cos[a + b*x]],x]

[Out]

(2*EllipticF[(a + b*x)/2, 2])/b

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Maple [C]  time = 0.037, size = 18, normalized size = 1.1 \begin{align*} 2\,{\frac{{\it InverseJacobiAM} \left ( 1/2\,bx+a/2,\sqrt{2} \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(b*x+a)^(1/2),x)

[Out]

2/b*InverseJacobiAM(1/2*b*x+1/2*a,2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cos \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(cos(b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{\cos \left (b x + a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(cos(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cos{\left (a + b x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(b*x+a)**(1/2),x)

[Out]

Integral(1/sqrt(cos(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cos \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(cos(b*x + a)), x)