3.1.20 \(\int \sqrt {c \cos (a+b x)} \, dx\) [20]

Optimal. Leaf size=38 \[ \frac {2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {\cos (a+b x)}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2721, 2719} \begin {gather*} \frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{b \sqrt {\cos (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*Cos[a + b*x]],x]

[Out]

(2*Sqrt[c*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(b*Sqrt[Cos[a + b*x]])

Rule 2719

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

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {\cos (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*Cos[a + b*x]],x]

[Out]

(2*Sqrt[c*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(b*Sqrt[Cos[a + b*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs. \(2(60)=120\).
time = 0.07, size = 142, normalized size = 3.74

method	result	size
default	\(\frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, c \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\)	\(142\)
risch	\(-\frac {i \sqrt {2}\, \sqrt {c \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{-i \left (b x +a \right )}}}{b}-\frac {i \left (-\frac {2 \left (c \,{\mathrm e}^{2 i \left (b x +a \right )}+c \right )}{c \sqrt {{\mathrm e}^{i \left (b x +a \right )} \left (c \,{\mathrm e}^{2 i \left (b x +a \right )}+c \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (b x +a \right )}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {c \,{\mathrm e}^{3 i \left (b x +a \right )}+c \,{\mathrm e}^{i \left (b x +a \right )}}}\right ) \sqrt {2}\, \sqrt {c \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{-i \left (b x +a \right )}}\, \sqrt {c \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{i \left (b x +a \right )}}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}\)	\(299\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(c*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*c*(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))/(-c*(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)/(c*(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*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 = 63, normalized size = 1.66 \begin {gather*} \frac {i \, \sqrt {2} \sqrt {c} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - i \, \sqrt {2} \sqrt {c} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*sqrt(2)*sqrt(c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a))) - I*sqrt(
2)*sqrt(c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(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 \sqrt {c \cos {\left (a + b x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sqrt(c*cos(b*x + a)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {c\,\cos \left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(a + b*x))^(1/2),x)

[Out]

int((c*cos(a + b*x))^(1/2), x)

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