3.267 \(\int \frac{\sqrt{\sec (a+b \log (c x^n))}}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0431732, antiderivative size = 54, normalized size of antiderivative = 1., 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 = {3771, 2641} \[ \frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sec[a + b*Log[c*x^n]]]/x,x]

[Out]

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

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 2641

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

Rubi steps

\begin{align*} \int \frac{\sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \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 ) \operatorname{Subst}\left (\int \frac{1}{\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 )} F\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*}

Mathematica [A]  time = 0.115889, size = 54, normalized size = 1. \[ \frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 1.984, size = 181, normalized size = 3.4 \begin{align*} -2\,{\frac{\sqrt{ \left ( 2\, \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) ,\sqrt{2} \right ) }{n\sqrt{-2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}+ \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \sqrt{2\, \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)/(-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)*EllipticF(cos(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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sec \left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sec(a + b*log(c*x**n)))/x, x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out