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

Optimal. Leaf size=107 \[ -\frac {\sqrt {7} F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{20 d}+\frac {21 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{20 d}-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}+\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{5 d} \]

[Out]

-1/10*(3-4*cos(d*x+c))^(3/2)*sin(d*x+c)/d-21/20*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)*EllipticE(cos(
1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)+1/20*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)*EllipticF(cos(1/2*
d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)+1/5*sin(d*x+c)*(3-4*cos(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.14, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2791, 2753, 2752, 2662, 2654} \[ -\frac {\sqrt {7} F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{20 d}+\frac {21 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{20 d}-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}+\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(20*d) - (Sqrt[7]*EllipticF[(c + Pi + d*x)/2, 8/7])/(20*d) + (Sq
rt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(5*d) - ((3 - 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(10*d)

Rule 2654

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 2662

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 2753

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

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 \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx &=-\frac {(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d}-\frac {1}{10} \int \sqrt {3-4 \cos (c+d x)} (-6-3 \cos (c+d x)) \, dx\\ &=\frac {\sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{5 d}-\frac {(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d}-\frac {1}{15} \int \frac {-21+\frac {63}{2} \cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=\frac {\sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{5 d}-\frac {(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d}-\frac {7}{40} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx+\frac {21}{40} \int \sqrt {3-4 \cos (c+d x)} \, dx\\ &=\frac {21 \sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 d}-\frac {\sqrt {7} F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{5 d}-\frac {(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 104, normalized size = 0.97 \[ -\frac {14 \sin (c+d x)-16 \sin (2 (c+d x))+8 \sin (3 (c+d x))+7 \sqrt {4 \cos (c+d x)-3} F\left (\left .\frac {1}{2} (c+d x)\right |8\right )+21 \sqrt {4 \cos (c+d x)-3} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )}{20 d \sqrt {3-4 \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/20*(21*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] + 7*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x
)/2, 8] + 14*Sin[c + d*x] - 16*Sin[2*(c + d*x)] + 8*Sin[3*(c + d*x)])/(d*Sqrt[3 - 4*Cos[c + d*x]])

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fricas [F]  time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2}, x\right ) \]

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

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

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

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maple [A]  time = 0.72, size = 253, normalized size = 2.36 \[ \frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-256 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-12 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]

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]

1/20*(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-256*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+12
8*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*Ellip
ticF(cos(1/2*d*x+1/2*c),2/7*14^(1/2))-21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*Ellipt
icE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))-12*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-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+7)^(1/2)/d

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

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^2\,\sqrt {3-4\,\cos \left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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