\(\int \frac {1}{x \csc ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\) [317]

Optimal result

Integrand size = 19, antiderivative size = 98 \[ \int \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{3 b n} \]

[Out]

-2/3*cos(a+b*ln(c*x^n))/b/n/csc(a+b*ln(c*x^n))^(1/2)-2/3*(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 = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3854, 3856, 2720} \[ \int \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \]

[In]

Int[1/(x*Csc[a + b*Log[c*x^n]]^(3/2)),x]

[Out]

(-2*Cos[a + b*Log[c*x^n]])/(3*b*n*Sqrt[Csc[a + b*Log[c*x^n]]]) + (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]]])/(3*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 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

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 \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}+\frac {\text {Subst}\left (\int \sqrt {\csc (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n} \\ & = -\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}+\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 )}{3 n} \\ & = -\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}+\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 )}}{3 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (2 \operatorname {EllipticF}\left (\frac {1}{4} \left (-2 a+\pi -2 b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}+\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{3 b n} \]

[In]

Integrate[1/(x*Csc[a + b*Log[c*x^n]]^(3/2)),x]

[Out]

-1/3*(Sqrt[Csc[a + b*Log[c*x^n]]]*(2*EllipticF[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]]
+ Sin[2*(a + b*Log[c*x^n])]))/(b*n)

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{3}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(131\)
default	\(\frac {\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{3}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(131\)

[In]

int(1/x/csc(a+b*ln(c*x^n))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/n*(1/3*(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((s
in(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))-2/3*cos(a+b*ln(c*x^n))^2*sin(a+b*ln(c*x^n)))/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 = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sqrt {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} + i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) - i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{3 \, b n} \]

[In]

integrate(1/x/csc(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")

[Out]

-1/3*(2*cos(b*n*log(x) + b*log(c) + a)*sqrt(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)) - 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]

\[ \int \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \csc ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

[In]

integrate(1/x/csc(a+b*ln(c*x**n))**(3/2),x)

[Out]

Integral(1/(x*csc(a + b*log(c*x**n))**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/x/csc(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/(x*csc(b*log(c*x^n) + a)^(3/2)), x)

Giac [F]

\[ \int \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/x/csc(a+b*log(c*x^n))^(3/2),x, algorithm="giac")

[Out]

integrate(1/(x*csc(b*log(c*x^n) + a)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x\,{\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}} \,d x \]

[In]

int(1/(x*(1/sin(a + b*log(c*x^n)))^(3/2)),x)

[Out]

int(1/(x*(1/sin(a + b*log(c*x^n)))^(3/2)), x)