\(\int \frac {\sqrt {\csc (a+b \log (c x^n))}}{x} \, dx\) [309]

Optimal result

Integrand size = 19, antiderivative size = 59 \[ \int \frac {\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 59, 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} \[ \int \frac {\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{b n} \]

[In]

Int[Sqrt[Csc[a + b*Log[c*x^n]]]/x,x]

[Out]

(2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticF[(a - Pi/2 + b*Log[c*x^n])/2, 2]*Sqrt[Sin[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 {\csc (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\left (\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=-\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{4} \left (-2 a+\pi -2 b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[In]

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

[Out]

(-2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticF[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(b*n
)

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.73

method	result	size
derivativedivides	\(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(102\)
default	\(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(102\)

[In]

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

[Out]

1/n*(sin(a+b*ln(c*x^n))+1)^(1/2)*(-2*sin(a+b*ln(c*x^n))+2)^(1/2)*(-sin(a+b*ln(c*x^n)))^(1/2)*EllipticF((sin(a+
b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))/cos(a+b*ln(c*x^n))/sin(a+b*ln(c*x^n))^(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 = 78, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {-i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{b n} \]

[In]

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

[Out]

(-I*sqrt(2*I)*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*I)*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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 27.53 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=-\frac {2\,\sqrt {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}}{2}\right )\middle |2\right )\,\sqrt {{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2}\,\sqrt {\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}}}{b\,n\,\cos \left (a+b\,\ln \left (c\,x^n\right )\right )} \]

[In]

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

[Out]

-(2*sin(a + b*log(c*x^n))^(1/2)*ellipticF(asin((2^(1/2)*(1 - sin(a + b*log(c*x^n)))^(1/2))/2), 2)*(cos(a + b*l
og(c*x^n))^2)^(1/2)*(1/sin(a + b*log(c*x^n)))^(1/2))/(b*n*cos(a + b*log(c*x^n)))