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

Optimal. Leaf size=113 \[ \frac {23 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{20 \sqrt {7} d}-\frac {9 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{20 d}-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{10 d} \]

[Out]

-23/140*(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)+9
/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)-1/10*
sin(d*x+c)*(3-4*cos(d*x+c))^(1/2)/d-1/10*cos(d*x+c)*sin(d*x+c)*(3-4*cos(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.15, antiderivative size = 113, 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 = {2793, 3023, 2752, 2662, 2654} \[ \frac {23 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{20 \sqrt {7} d}-\frac {9 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{20 d}-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{10 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(20*d) + (23*EllipticF[(c + Pi + d*x)/2, 8/7])/(20*Sqrt[7]*d) -
(Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(10*d) - (Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]*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 2793

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d
*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a*d
*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n -
2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] ||
(EqQ[a, 0] && NeQ[c, 0])))

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*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[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\cos ^3(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx &=-\frac {\sqrt {3-4 \cos (c+d x)} \cos (c+d x) \sin (c+d x)}{10 d}-\frac {1}{10} \int \frac {3-6 \cos (c+d x)-6 \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)}{10 d}-\frac {\sqrt {3-4 \cos (c+d x)} \cos (c+d x) \sin (c+d x)}{10 d}+\frac {1}{60} \int \frac {-6+54 \cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=-\frac {\sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{10 d}-\frac {\sqrt {3-4 \cos (c+d x)} \cos (c+d x) \sin (c+d x)}{10 d}-\frac {9}{40} \int \sqrt {3-4 \cos (c+d x)} \, dx+\frac {23}{40} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=-\frac {9 \sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 d}+\frac {23 F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}-\frac {\sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{10 d}-\frac {\sqrt {3-4 \cos (c+d x)} \cos (c+d x) \sin (c+d x)}{10 d}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 102, normalized size = 0.90 \[ \frac {-4 \sin (c+d x)+\sin (2 (c+d x))+2 \sin (3 (c+d x))+23 \sqrt {4 \cos (c+d x)-3} F\left (\left .\frac {1}{2} (c+d x)\right |8\right )+9 \sqrt {4 \cos (c+d x)-3} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )}{20 d \sqrt {3-4 \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] + 23*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8
] - 4*Sin[c + d*x] + Sin[2*(c + d*x)] + 2*Sin[3*(c + d*x)])/(20*d*Sqrt[3 - 4*Cos[c + d*x]])

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fricas [F]  time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{3}}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.63, size = 254, normalized size = 2.25 \[ -\frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-448 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+23 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-63 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/140*(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-448*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+
504*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+23*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*
EllipticF(cos(1/2*d*x+1/2*c),2/7*14^(1/2))-63*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*E
llipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))-56*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 \[ \int \frac {\cos \left (d x + c\right )^{3}}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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