3.1.21 \(\int \frac {1}{\sqrt {c \sec (a+b x)}} \, dx\) [21]

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

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Rubi [A]
time = 0.01, 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 = {3856, 2719} \begin {gather*} \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 {gather*}

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 2719

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

Rule 3856

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 \begin {gather*} \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 {gather*}

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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Maple [C] Result contains complex when optimal does not.
time = 29.64, size = 306, normalized size = 8.05

method	result	size
risch	\(-\frac {i \sqrt {2}}{b \sqrt {\frac {c \,{\mathrm e}^{i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}+1}}}-\frac {i \left (-\frac {2 \left ({\mathrm e}^{2 i \left (b x +a \right )} c +c \right )}{c \sqrt {{\mathrm e}^{i \left (b x +a \right )} \left ({\mathrm e}^{2 i \left (b x +a \right )} c +c \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (b x +a \right )}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 i \left (b x +a \right )} c +c \,{\mathrm e}^{i \left (b x +a \right )}}}\right ) \sqrt {2}\, \sqrt {c \,{\mathrm e}^{i \left (b x +a \right )} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}}{b \sqrt {\frac {c \,{\mathrm e}^{i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}+1}}\, \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}\)	\(299\)
default	\(\frac {2 \left (i \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right )-i \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+i \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sin \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-i \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sin \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-\left (\cos ^{2}\left (b x +a \right )\right )+\cos \left (b x +a \right )\right ) \sqrt {\frac {c}{\cos \left (b x +a \right )}}}{b \sin \left (b x +a \right ) c}\)	\(306\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.91, size = 66, normalized size = 1.74 \begin {gather*} \frac {i \, \sqrt {2} \sqrt {c} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - i \, \sqrt {2} \sqrt {c} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*sqrt(2)*sqrt(c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a))) - I*sqrt(
2)*sqrt(c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a))))/(b*c)

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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