3.6.10 \(\int \cos ^2(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx\) [510]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 105, 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 = {2870, 2832, 2831, 2740, 2732} \begin {gather*} -\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} \end {gather*}

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 2732

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

Rule 2740

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

Rule 2831

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 2832

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)*Sim
p[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 2870

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 \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*}

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Mathematica [A]
time = 0.15, size = 81, normalized size = 0.77 \begin {gather*} \frac {21 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )-\sqrt {7} F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )+2 \sqrt {3+4 \cos (c+d x)} (\sin (c+d x)+2 \sin (2 (c+d x)))}{20 d} \end {gather*}

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 = 0.16, size = 253, normalized size = 2.41

method	result	size
default	\(-\frac {\sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-256 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+384 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-140 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )\right )}{20 \sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(253\)

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,method=_RETURNVERBOSE)

[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*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+38
4*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-140*sin(1/2*d*x+1/2*c)^2*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)*EllipticF(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)))/(-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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 138, normalized size = 1.31 \begin {gather*} \frac {4 \, \sqrt {4 \, \cos \left (d x + c\right ) + 3} {\left (4 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 7 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 7 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 42 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 42 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{40 \, d} \end {gather*}

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]

1/40*(4*sqrt(4*cos(d*x + c) + 3)*(4*cos(d*x + c) + 1)*sin(d*x + c) - 7*I*sqrt(2)*weierstrassPInverse(-1, 1, co
s(d*x + c) + I*sin(d*x + c) + 1/2) + 7*I*sqrt(2)*weierstrassPInverse(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/
2) + 42*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, cos(d*x + c) + I*sin(d*x + c) + 1/2)) - 42
*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2)))/d

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

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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]

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

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