3.445 \(\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{3+2 \cos (c+d x)}} \, dx\)

Optimal. Leaf size=72 \[ \frac{2 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{3 d} \]

[Out]

(2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c +
d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)

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Rubi [A]  time = 0.0825297, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {2994} \[ \frac{2 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]),x]

[Out]

(2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c +
d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)

Rule 2994

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*A*(c - d)*Tan[e + f*x]*Rt[(c + d)/b, 2]*Sqrt[(c*(1 + Csc[e + f*x]))/(c
- d)]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/(Sqrt[b*Sin[e + f*x]]*Rt[
(c + d)/b, 2])], -((c + d)/(c - d))])/(f*b*c^2), x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] &&
EqQ[A, B] && PosQ[(c + d)/b]

Rubi steps

\begin{align*} \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{3+2 \cos (c+d x)}} \, dx &=\frac{2 \cot (c+d x) E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3+2 \cos (c+d x)}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right ) \sqrt{1-\sec (c+d x)} \sqrt{1+\sec (c+d x)}}{3 d}\\ \end{align*}

Mathematica [F]  time = 37.7417, size = 0, normalized size = 0. \[ \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{3+2 \cos (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]),x]

[Out]

Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]), x]

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Maple [B]  time = 0.448, size = 665, normalized size = 9.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15/d/(3+2*cos(d*x+c))^(1/2)*(3*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*10^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))
^(1/2)*cos(d*x+c)^2*sin(d*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*I*5^(1/2))*2^(1/2)+6*(cos(d*x+c)/(1+co
s(d*x+c)))^(3/2)*10^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)*sin(d*x+c)*EllipticF((-1+cos(d*x+
c))/sin(d*x+c),1/5*I*5^(1/2))*2^(1/2)+3*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*10^(1/2)*((3+2*cos(d*x+c))/(
1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*I*5^(1/2))*sin(d*x+c)+3*2^(1/2)*(cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*10^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*I*
5^(1/2))*cos(d*x+c)^2*sin(d*x+c)-5*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((3+2*cos(d*x+c))/(1+cos
(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),1/5*I*5^(1/2))*cos(d*x+c)^2*sin(d*x+c)+3*2^(1/2)*(cos(d*x
+c)/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c
),1/5*I*5^(1/2))*cos(d*x+c)*sin(d*x+c)-5*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((3+2*cos(d*x+c))/
(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),1/5*I*5^(1/2))*cos(d*x+c)*sin(d*x+c)+20*cos(d*x+c)^
3+10*cos(d*x+c)^2-30*cos(d*x+c))/cos(d*x+c)^(3/2)/sin(d*x+c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right ) + 1}{\sqrt{2 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((cos(d*x + c) + 1)/(sqrt(2*cos(d*x + c) + 3)*cos(d*x + c)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{2 \, \cos \left (d x + c\right ) + 3}{\left (\cos \left (d x + c\right ) + 1\right )} \sqrt{\cos \left (d x + c\right )}}{2 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(2*cos(d*x + c) + 3)*(cos(d*x + c) + 1)*sqrt(cos(d*x + c))/(2*cos(d*x + c)^3 + 3*cos(d*x + c)^2),
x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (c + d x \right )} + 1}{\sqrt{2 \cos{\left (c + d x \right )} + 3} \cos ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right ) + 1}{\sqrt{2 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((cos(d*x + c) + 1)/(sqrt(2*cos(d*x + c) + 3)*cos(d*x + c)^(3/2)), x)