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

Optimal. Leaf size=111 \[ -\frac{23 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{20 \sqrt{7} d}+\frac{9 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{20 d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.149713, antiderivative size = 111, normalized size of antiderivative = 1., 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, 2661, 2653} \[ -\frac{23 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{20 \sqrt{7} d}+\frac{9 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{20 d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d} \]

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

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]

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.170112, size = 81, normalized size = 0.73 \[ \frac{-23 \sqrt{7} F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )+63 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )+7 (\sin (2 (c+d x))-2 \sin (c+d x)) \sqrt{4 \cos (c+d x)+3}}{140 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(63*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] - 23*Sqrt[7]*EllipticF[(c + d*x)/2, 8/7] + 7*Sqrt[3 + 4*Cos[c + d*x]]*
(-2*Sin[c + d*x] + Sin[2*(c + d*x)]))/(140*d)

________________________________________________________________________________________

Maple [A]  time = 2.559, size = 231, normalized size = 2.1 \begin{align*} -{\frac{1}{20\,d}\sqrt{ \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -64\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -23\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2\,\sqrt{2} \right ) -9\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2\,\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+7\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \end{align*}

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]

-1/20*((8*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-64*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+56*
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)*(8*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*Ellip
ticF(cos(1/2*d*x+1/2*c),2*2^(1/2))-9*(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(c
os(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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{3}}{\sqrt{4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}

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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (d x + c\right )^{3}}{\sqrt{4 \, \cos \left (d x + c\right ) + 3}}, x\right ) \end{align*}

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{3}}{\sqrt{4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}

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)