3.3.73 \(\int \frac {1}{x \sqrt {\sec (a+b \log (c x^n))}} \, dx\) [273]

Optimal. Leaf size=54 \[ \frac {2 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3856, 2719} \begin {gather*} \frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[Sec[a + b*Log[c*x^n]]]),x]

[Out]

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

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

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Mathematica [A]
time = 0.11, size = 54, normalized size = 1.00 \begin {gather*} \frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[Sec[a + b*Log[c*x^n]]]),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(180\) vs. \(2(86)=172\).
time = 0.30, size = 181, normalized size = 3.35

method	result	size
derivativedivides	\(\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sqrt {-2 \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b}\)	\(181\)
default	\(\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sqrt {-2 \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b}\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/n*((2*cos(1/2*a+1/2*b*ln(c*x^n))^2-1)*sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/
2)*(-2*cos(1/2*a+1/2*b*ln(c*x^n))^2+1)^(1/2)*EllipticE(cos(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))/(-2*sin(1/2*a+1/2*b
*ln(c*x^n))^4+sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/sin(1/2*a+1/2*b*ln(c*x^n))/(2*cos(1/2*a+1/2*b*ln(c*x^n))^2-1
)^(1/2)/b

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

[Out]

integrate(1/(x*sqrt(sec(b*log(c*x^n) + a))), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.51, size = 84, normalized size = 1.56 \begin {gather*} \frac {i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) - i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )}{b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(sec(a + b*log(c*x**n)))), 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/x/sec(a+b*log(c*x^n))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(x*sqrt(sec(b*log(c*x^n) + a))), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x\,\sqrt {\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(1/cos(a + b*log(c*x^n)))^(1/2)),x)

[Out]

int(1/(x*(1/cos(a + b*log(c*x^n)))^(1/2)), x)

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