3.76 \(\int \sqrt {\cos (a+b x)} \, dx\)

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

[Out]

2*(cos(1/2*b*x+1/2*a)^2)^(1/2)/cos(1/2*b*x+1/2*a)*EllipticE(sin(1/2*b*x+1/2*a),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 = {2639} \[ \frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cos[a + b*x]],x]

[Out]

(2*EllipticE[(a + b*x)/2, 2])/b

Rule 2639

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

Rubi steps

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

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cos[a + b*x]],x]

[Out]

(2*EllipticE[(a + b*x)/2, 2])/b

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\cos \left (b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(cos(b*x + a)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos \left (b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(cos(b*x + a)), x)

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maple [B]  time = 0.00, size = 133, normalized size = 8.31 \[ \frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cos(1/2*b*x+1/2*a)^
2+1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))/(-2*sin(1/2*b*x+1/2*a)^4+sin(1/2*b*x+1/2*a)^2)^(1/2)/sin(1/2*
b*x+1/2*a)/(2*cos(1/2*b*x+1/2*a)^2-1)^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos \left (b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(cos(b*x + a)), x)

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mupad [B]  time = 0.18, size = 15, normalized size = 0.94 \[ \frac {2\,\mathrm {E}\left (\frac {a}{2}+\frac {b\,x}{2}\middle |2\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^(1/2),x)

[Out]

(2*ellipticE(a/2 + (b*x)/2, 2))/b

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos {\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(cos(a + b*x)), x)

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