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

Optimal. Leaf size=105 \[ -\frac{\sqrt{7} F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{20 d}+\frac{21 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{20 d}+\frac{\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{5 d} \]

[Out]

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

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Rubi [A]  time = 0.137643, antiderivative size = 105, 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 = {2791, 2753, 2752, 2661, 2653} \[ -\frac{\sqrt{7} F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{20 d}+\frac{21 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{20 d}+\frac{\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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 \cos ^2(c+d x) \sqrt{3+4 \cos (c+d x)} \, dx &=\frac{(3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d}+\frac{1}{10} \int (6-3 \cos (c+d x)) \sqrt{3+4 \cos (c+d x)} \, dx\\ &=-\frac{\sqrt{3+4 \cos (c+d x)} \sin (c+d x)}{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+d x)|\frac{8}{7}\right )}{20 d}-\frac{\sqrt{7} F\left (\frac{1}{2} (c+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*}

Mathematica [A]  time = 0.139423, size = 81, normalized size = 0.77 \[ \frac{-\sqrt{7} F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )+21 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )+2 (\sin (c+d x)+2 \sin (2 (c+d x))) \sqrt{4 \cos (c+d x)+3}}{20 d} \]

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

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Maple [A]  time = 2.188, size = 253, normalized size = 2.4 \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 ( -256\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +384\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -7\,\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 ) -21\,\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 ) -140\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/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)^2*(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)*(-256*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+38
4*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-7*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(8*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*Elli
pticF(cos(1/2*d*x+1/2*c),2*2^(1/2))-21*(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))-140*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., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2}, x\right ) \end{align*}

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

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)