3.6.49 \(\int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx\) [549]

Optimal. Leaf size=51 \[ \frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {3 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 \sqrt {7} d} \]

[Out]

-3/14*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)+1/2
*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2831, 2740, 2732} \begin {gather*} \frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {3 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 \sqrt {7} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]/Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(2*d) - (3*EllipticF[(c + d*x)/2, 8/7])/(2*Sqrt[7]*d)

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx &=\frac {1}{4} \int \sqrt {3+4 \cos (c+d x)} \, dx-\frac {3}{4} \int \frac {1}{\sqrt {3+4 \cos (c+d x)}} \, dx\\ &=\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {3 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 \sqrt {7} d}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 43, normalized size = 0.84 \begin {gather*} \frac {7 E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )-3 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 \sqrt {7} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]/Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

(7*EllipticE[(c + d*x)/2, 8/7] - 3*EllipticF[(c + d*x)/2, 8/7])/(2*Sqrt[7]*d)

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Maple [A]
time = 0.18, size = 155, normalized size = 3.04

method	result	size
default	\(\frac {\sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (3 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )+\EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )\right )}{2 \sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(155\)
risch	\(-\frac {i \left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 d \sqrt {\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{-i \left (d x +c \right )}}}-\frac {i \left (-\frac {2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2}{\sqrt {\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{i \left (d x +c \right )}}}+\frac {2 \left (\frac {3}{4}+\frac {i \sqrt {7}}{4}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}+\frac {i \sqrt {7}}{4}}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}}\, \sqrt {14}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{-\frac {3}{4}-\frac {i \sqrt {7}}{4}}}\, \left (-\frac {i \sqrt {7}\, \EllipticE \left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}+\frac {i \sqrt {7}}{4}}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}}, \frac {\sqrt {14}\, \sqrt {i \left (-\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}}{7}\right )}{2}+\left (-\frac {3}{4}+\frac {i \sqrt {7}}{4}\right ) \EllipticF \left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}+\frac {i \sqrt {7}}{4}}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}}, \frac {\sqrt {14}\, \sqrt {i \left (-\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}}{7}\right )\right )}{7 \sqrt {2 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{i \left (d x +c \right )}}\, {\mathrm e}^{-i \left (d x +c \right )}}{d \sqrt {\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{-i \left (d x +c \right )}}}\)	\(497\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)/(3+4*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*((8*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-8*cos(1/2*d*x+1/2*c
)^2+1)^(1/2)*(3*EllipticF(cos(1/2*d*x+1/2*c),2*2^(1/2))+EllipticE(cos(1/2*d*x+1/2*c),2*2^(1/2)))/(-8*sin(1/2*d
*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(8*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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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(cos(d*x+c)/(3+4*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)/sqrt(4*cos(d*x + c) + 3), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 108, normalized size = 2.12 \begin {gather*} \frac {i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) - i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 2 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 2 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(3+4*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/4*(I*sqrt(2)*weierstrassPInverse(-1, 1, cos(d*x + c) + I*sin(d*x + c) + 1/2) - I*sqrt(2)*weierstrassPInverse
(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2) + 2*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, c
os(d*x + c) + I*sin(d*x + c) + 1/2)) - 2*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, cos(d*x +
c) - I*sin(d*x + c) + 1/2)))/d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (c + d x \right )}}{\sqrt {4 \cos {\left (c + d x \right )} + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(3+4*cos(d*x+c))**(1/2),x)

[Out]

Integral(cos(c + d*x)/sqrt(4*cos(c + d*x) + 3), 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(cos(d*x+c)/(3+4*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)/sqrt(4*cos(d*x + c) + 3), x)

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Mupad [B]
time = 0.63, size = 54, normalized size = 1.06 \begin {gather*} \frac {\sqrt {\frac {4\,\cos \left (c+d\,x\right )}{7}+\frac {3}{7}}\,\left (7\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {8}{7}\right )-3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {8}{7}\right )\right )}{2\,d\,\sqrt {4\,\cos \left (c+d\,x\right )+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)/(4*cos(c + d*x) + 3)^(1/2),x)

[Out]

(((4*cos(c + d*x))/7 + 3/7)^(1/2)*(7*ellipticE(c/2 + (d*x)/2, 8/7) - 3*ellipticF(c/2 + (d*x)/2, 8/7)))/(2*d*(4
*cos(c + d*x) + 3)^(1/2))

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