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*(cos(1/2*b*x+1/2*a)^2)^(1/2)/cos(1/2*b*x+1/2*a)*EllipticE(sin(1/2*b*x+1/2*a),2^(1/2))/b/cos(b*x+a)^(1/2)/(c*
sec(b*x+a))^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, size = 38, normalized size = 1.00 \[ \frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}} \]

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]])

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fricas [F]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \sec \left (b x + a\right )}}{c \sec \left (b x + a\right )}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \sec \left (b x + a\right )}}\,{d x} \]

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)

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maple [C]  time = 1.06, size = 306, normalized size = 8.05 \[ \frac {2 \left (i \cos \left (b x +a \right ) \sin \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right )-i \cos \left (b x +a \right ) \sin \left (b x +a \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+i \sin \left (b x +a \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-i \sin \left (b x +a \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-\left (\cos ^{2}\left (b x +a \right )\right )+\cos \left (b x +a \right )\right ) \sqrt {\frac {c}{\cos \left (b x +a \right )}}}{b \sin \left (b x +a \right ) c} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \sec \left (b x + a\right )}}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c/cos(a + b*x))^(1/2),x)

[Out]

int(1/(c/cos(a + b*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \sec {\left (a + b x \right )}}}\, dx \]

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)

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