\(\int \frac {\sqrt {\cos (a+b \log (c x^n))}}{x} \, dx\) [111]

Optimal result

Integrand size = 19, antiderivative size = 24 \[ \int \frac {\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

[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)
)/b/n

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2719} \[ \int \frac {\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

[In]

Int[Sqrt[Cos[a + b*Log[c*x^n]]]/x,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(180\) vs. \(2(60)=120\).

Time = 2.87 (sec) , antiderivative size = 181, normalized size of antiderivative = 7.54

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

[In]

int(cos(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)*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

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.50 \[ \int \frac {\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) - i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )}{b n} \]

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 26.58 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2\,\mathrm {E}\left (\frac {a}{2}+\frac {b\,\ln \left (c\,x^n\right )}{2}\middle |2\right )}{b\,n} \]

[In]

int(cos(a + b*log(c*x^n))^(1/2)/x,x)

[Out]

(2*ellipticE(a/2 + (b*log(c*x^n))/2, 2))/(b*n)