\(\int \frac {\sqrt {\sec (a+b \log (c x^n))}}{x} \, dx\) [267]

Optimal result

Integrand size = 19, antiderivative size = 54 \[ \int \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[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))*EllipticF(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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3856, 2720} \[ \int \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]

[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 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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*} \text {integral}& = \frac {\text {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 ) \text {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 )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(180\) vs. \(2(86)=172\).

Time = 1.44 (sec) , antiderivative size = 181, normalized size of antiderivative = 3.35

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

[In]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {-i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{b n} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int \frac {\sqrt {\sec {\left (a + b \log {\left (c x^{n} \right )} \right )}}}{x}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int { \frac {\sqrt {\sec \left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int { \frac {\sqrt {\sec \left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 26.66 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2\,\sqrt {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\,\sqrt {\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}}\,\mathrm {F}\left (\frac {a}{2}+\frac {b\,\ln \left (c\,x^n\right )}{2}\middle |2\right )}{b\,n} \]

[In]

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

[Out]

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