3.548 \(\int \frac {\cos ^2(c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx\)

Optimal. Leaf size=78 \[ \frac {17 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{12 \sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{4 d}+\frac {\sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{6 d} \]

[Out]

17/84*(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)-1/4
*(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)+1/6*sin(
d*x+c)*(3+4*cos(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.10, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2791, 2752, 2661, 2653} \[ \frac {17 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{12 \sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{4 d}+\frac {\sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2/Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

-(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(4*d) + (17*EllipticF[(c + d*x)/2, 8/7])/(12*Sqrt[7]*d) + (Sqrt[3 + 4*C
os[c + d*x]]*Sin[c + d*x])/(6*d)

Rule 2653

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

Rule 2661

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

Rule 2752

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]

Rule 2791

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

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx &=\frac {\sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{6 d}+\frac {1}{6} \int \frac {2-3 \cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx\\ &=\frac {\sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{6 d}-\frac {1}{8} \int \sqrt {3+4 \cos (c+d x)} \, dx+\frac {17}{24} \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 )}{4 d}+\frac {17 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{12 \sqrt {7} d}+\frac {\sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 70, normalized size = 0.90 \[ \frac {17 \sqrt {7} F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )-21 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )+14 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{84 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2/Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

(-21*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] + 17*Sqrt[7]*EllipticF[(c + d*x)/2, 8/7] + 14*Sqrt[3 + 4*Cos[c + d*x]
]*Sin[c + d*x])/(84*d)

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fricas [F]  time = 1.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (d x + c\right )^{2}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(d*x + c)^2/sqrt(4*cos(d*x + c) + 3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{2}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.59, size = 231, normalized size = 2.96 \[ -\frac {\sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (32 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+17 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )-28 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 \sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*((8*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(32*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+17*(
sin(1/2*d*x+1/2*c)^2)^(1/2)*(8*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2*2^(1/2))+3*(sin(1/
2*d*x+1/2*c)^2)^(1/2)*(8*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2*2^(1/2))-28*sin(1/2*d*x+
1/2*c)^2*cos(1/2*d*x+1/2*c))/(-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 \[ \int \frac {\cos \left (d x + c\right )^{2}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.09, size = 78, normalized size = 1.00 \[ \frac {\sin \left (c+d\,x\right )\,\sqrt {4\,\cos \left (c+d\,x\right )+3}}{6\,d}-\frac {\sqrt {\frac {4\,\cos \left (c+d\,x\right )}{7}+\frac {3}{7}}\,\left (42\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {8}{7}\right )-34\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {8}{7}\right )\right )}{24\,d\,\sqrt {4\,\cos \left (c+d\,x\right )+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sin(c + d*x)*(4*cos(c + d*x) + 3)^(1/2))/(6*d) - (((4*cos(c + d*x))/7 + 3/7)^(1/2)*(42*ellipticE(c/2 + (d*x)/
2, 8/7) - 34*ellipticF(c/2 + (d*x)/2, 8/7)))/(24*d*(4*cos(c + d*x) + 3)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}{\left (c + d x \right )}}{\sqrt {4 \cos {\left (c + d x \right )} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(c + d*x)**2/sqrt(4*cos(c + d*x) + 3), x)

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