3.674 \(\int \frac{1}{\sqrt{3-2 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx\)

Optimal. Leaf size=113 \[ \frac{4 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),6\right )}{3 d \sqrt{3-2 \sec (c+d x)}}+\frac{2 \sqrt{3-2 \sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |6\right )}{3 d \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}} \]

[Out]

(2*EllipticE[(c + d*x)/2, 6]*Sqrt[3 - 2*Sec[c + d*x]])/(3*d*Sqrt[-2 + 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4
*Sqrt[-2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[3 - 2*Sec[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.176649, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3862, 3856, 2653, 3858, 2661} \[ \frac{4 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |6\right )}{3 d \sqrt{3-2 \sec (c+d x)}}+\frac{2 \sqrt{3-2 \sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |6\right )}{3 d \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticE[(c + d*x)/2, 6]*Sqrt[3 - 2*Sec[c + d*x]])/(3*d*Sqrt[-2 + 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4
*Sqrt[-2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[3 - 2*Sec[c + d*x]])

Rule 3862

Int[1/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Dist[1/a,
Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Dist[b/(a*d), Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b
*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3856

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Dist[Sqrt[a +
b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; Free
Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 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]

Rule 3858

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(Sqrt[d*
Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && 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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{3-2 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx &=\frac{1}{3} \int \frac{\sqrt{3-2 \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx+\frac{2}{3} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{3-2 \sec (c+d x)}} \, dx\\ &=\frac{\sqrt{3-2 \sec (c+d x)} \int \sqrt{-2+3 \cos (c+d x)} \, dx}{3 \sqrt{-2+3 \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left (2 \sqrt{-2+3 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{-2+3 \cos (c+d x)}} \, dx}{3 \sqrt{3-2 \sec (c+d x)}}\\ &=\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |6\right ) \sqrt{3-2 \sec (c+d x)}}{3 d \sqrt{-2+3 \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{4 \sqrt{-2+3 \cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |6\right ) \sqrt{\sec (c+d x)}}{3 d \sqrt{3-2 \sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.113527, size = 72, normalized size = 0.64 \[ \frac{\sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} \left (4 \text{EllipticF}\left (\frac{1}{2} (c+d x),6\right )+2 E\left (\left .\frac{1}{2} (c+d x)\right |6\right )\right )}{3 d \sqrt{3-2 \sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-2 + 3*Cos[c + d*x]]*(2*EllipticE[(c + d*x)/2, 6] + 4*EllipticF[(c + d*x)/2, 6])*Sqrt[Sec[c + d*x]])/(3*
d*Sqrt[3 - 2*Sec[c + d*x]])

________________________________________________________________________________________

Maple [C]  time = 0.283, size = 381, normalized size = 3.4 \begin{align*}{\frac{2}{15\,d\sin \left ( dx+c \right ) \left ( -2+3\,\cos \left ( dx+c \right ) \right ) } \left ( 3\,\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \sqrt{5}\sqrt{{\frac{-2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}{\it EllipticF} \left ({\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},i/5\sqrt{5} \right ) -5\,\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \sqrt{5}\sqrt{{\frac{-2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}{\it EllipticE} \left ({\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},i/5\sqrt{5} \right ) +3\,\sqrt{5}{\it EllipticF} \left ({\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},i/5\sqrt{5} \right ) \sqrt{{\frac{-2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) -5\,\sqrt{5}{\it EllipticE} \left ({\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},i/5\sqrt{5} \right ) \sqrt{{\frac{-2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) -15\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+25\,\cos \left ( dx+c \right ) -10 \right ) \sqrt{{\frac{-2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15/d*(3*sin(d*x+c)*cos(d*x+c)*5^(1/2)*((-2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*Elli
pticF((-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2))-5*sin(d*x+c)*cos(d*x+c)*5^(1/2)*((-2+3*cos(d*x+c))/(co
s(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2))+3*5^(1
/2)*EllipticF((-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2))*((-2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(c
os(d*x+c)+1))^(1/2)*sin(d*x+c)-5*5^(1/2)*EllipticE((-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2))*((-2+3*co
s(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-15*cos(d*x+c)^2+25*cos(d*x+c)-10)*((-2+3*c
os(d*x+c))/cos(d*x+c))^(1/2)/(1/cos(d*x+c))^(1/2)/sin(d*x+c)/(-2+3*cos(d*x+c))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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