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

Optimal. Leaf size=70 \[ -\frac{\sqrt{5} \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left (\sin ^{-1}\left (\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right )|\frac{1}{5}\right )}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.103997, antiderivative size = 70, 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} \[ -\frac{\sqrt{5} \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left (\sin ^{-1}\left (\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right )|\frac{1}{5}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[5]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[-2 + 3*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c +
d*x]]*Sqrt[1 + Sec[c + d*x]])/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{-2+3 \cos (c+d x)}} \, dx &=-\frac{\sqrt{5} \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{-2+3 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right )|\frac{1}{5}\right ) \sqrt{-1+\sec (c+d x)} \sqrt{1+\sec (c+d x)}}{d}\\ \end{align*}

Mathematica [F]  time = 38.3501, size = 0, normalized size = 0. \[ \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-2+3 \cos (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.461, size = 601, normalized size = 8.6 \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)/(-2+3*cos(d*x+c))^(1/2),x)

[Out]

1/d/(-2+3*cos(d*x+c))^(1/2)*(-2*cos(d*x+c)^2*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*((-2+3*cos(d*x+c))/(1+cos(d*x+c
)))^(1/2)*sin(d*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),5^(1/2))-4*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(3
/2)*((-2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),5^(1/2))-2*(cos(d
*x+c)/(1+cos(d*x+c)))^(1/2)*((-2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),5^(1
/2))*cos(d*x+c)^2*sin(d*x+c)-(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((-2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*Ellipt
icE((-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*cos(d*x+c)^2*sin(d*x+c)-2*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)
*((-2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),5^(1/2))-2*(cos(d*x+c)/(1+cos(d
*x+c)))^(1/2)*((-2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*cos(d*x+c
)*sin(d*x+c)-(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((-2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+
c))/sin(d*x+c),5^(1/2))*cos(d*x+c)*sin(d*x+c)+3*cos(d*x+c)^3-5*cos(d*x+c)^2+2*cos(d*x+c))/cos(d*x+c)^(3/2)/sin
(d*x+c)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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