3.21 \(\int \frac{1}{\sqrt{c \sec (a+b x)}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{\cos (a+b x)} \sqrt{c \sec (a+b x)}} \]

[Out]

(2*EllipticE[(a + b*x)/2, 2])/(b*Sqrt[Cos[a + b*x]]*Sqrt[c*Sec[a + b*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0272559, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3771, 2639} \[ \frac{2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{\cos (a+b x)} \sqrt{c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c*Sec[a + b*x]],x]

[Out]

(2*EllipticE[(a + b*x)/2, 2])/(b*Sqrt[Cos[a + b*x]]*Sqrt[c*Sec[a + b*x]])

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{c \sec (a+b x)}} \, dx &=\frac{\int \sqrt{\cos (a+b x)} \, dx}{\sqrt{\cos (a+b x)} \sqrt{c \sec (a+b x)}}\\ &=\frac{2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{\cos (a+b x)} \sqrt{c \sec (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0290382, size = 38, normalized size = 1. \[ \frac{2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{\cos (a+b x)} \sqrt{c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c*Sec[a + b*x]],x]

[Out]

(2*EllipticE[(a + b*x)/2, 2])/(b*Sqrt[Cos[a + b*x]]*Sqrt[c*Sec[a + b*x]])

________________________________________________________________________________________

Maple [C]  time = 0.164, size = 306, normalized size = 8.1 \begin{align*} 2\,{\frac{1}{b\sin \left ( bx+a \right ) c} \left ( i\sqrt{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( bx+a \right ) }{\cos \left ( bx+a \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }},i \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) -i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }},i \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) \sqrt{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( bx+a \right ) }{\cos \left ( bx+a \right ) +1}}}+i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }},i \right ) \sqrt{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( bx+a \right ) }{\cos \left ( bx+a \right ) +1}}}\sin \left ( bx+a \right ) -i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }},i \right ) \sin \left ( bx+a \right ) \sqrt{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( bx+a \right ) }{\cos \left ( bx+a \right ) +1}}}- \left ( \cos \left ( bx+a \right ) \right ) ^{2}+\cos \left ( bx+a \right ) \right ) \sqrt{{\frac{c}{\cos \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*sec(b*x+a))^(1/2),x)

[Out]

2/b*(I*(1/(cos(b*x+a)+1))^(1/2)*(cos(b*x+a)/(cos(b*x+a)+1))^(1/2)*EllipticF(I*(-1+cos(b*x+a))/sin(b*x+a),I)*co
s(b*x+a)*sin(b*x+a)-I*EllipticE(I*(-1+cos(b*x+a))/sin(b*x+a),I)*cos(b*x+a)*sin(b*x+a)*(1/(cos(b*x+a)+1))^(1/2)
*(cos(b*x+a)/(cos(b*x+a)+1))^(1/2)+I*EllipticF(I*(-1+cos(b*x+a))/sin(b*x+a),I)*(1/(cos(b*x+a)+1))^(1/2)*(cos(b
*x+a)/(cos(b*x+a)+1))^(1/2)*sin(b*x+a)-I*EllipticE(I*(-1+cos(b*x+a))/sin(b*x+a),I)*sin(b*x+a)*(1/(cos(b*x+a)+1
))^(1/2)*(cos(b*x+a)/(cos(b*x+a)+1))^(1/2)-cos(b*x+a)^2+cos(b*x+a))*(c/cos(b*x+a))^(1/2)/sin(b*x+a)/c

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sec(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(c*sec(b*x + a)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sec(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*sec(b*x + a))/(c*sec(b*x + a)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sec(b*x+a))**(1/2),x)

[Out]

Integral(1/sqrt(c*sec(a + b*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sec(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(c*sec(b*x + a)), x)