3.1.84 \(\int \frac {1}{\sqrt {\cos (a+b x)}} \, dx\) [84]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2720} \begin {gather*} \frac {2 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Cos[a + b*x]],x]

[Out]

(2*EllipticF[(a + b*x)/2, 2])/b

Rule 2720

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

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.00, size = 18, normalized size = 1.12

method	result	size
default	\(\frac {2 \,\mathrm {am}^{-1}\left (\frac {b x}{2}+\frac {a}{2}| \sqrt {2}\right )}{b}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.09, size = 51, normalized size = 3.19 \begin {gather*} \frac {-i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-I*sqrt(2)*weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a)) + I*sqrt(2)*weierstrassPInverse(-4, 0, c
os(b*x + a) - I*sin(b*x + a)))/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\cos {\left (a + b x \right )}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.20, size = 15, normalized size = 0.94 \begin {gather*} \frac {2\,\mathrm {F}\left (\frac {a}{2}+\frac {b\,x}{2}\middle |2\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(a + b*x)^(1/2),x)

[Out]

(2*ellipticF(a/2 + (b*x)/2, 2))/b

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