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

3.4.17.1 Optimal result

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

output

-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*l 
n(c*x^n))^(1/2)/b/n
 

3.4.17.2 Mathematica [A] (verified)

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

input

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

output

-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)
 

3.4.17.3 Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3039, 3042, 4256, 3042, 4258, 3042, 3120}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \frac {1}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {1}{\csc \left (a+b \log \left (c x^n\right )\right )^{3/2}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {\frac {1}{3} \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}}{n}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {\frac {1}{3} \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \int \frac {1}{\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \int \frac {1}{\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}}{n}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{3 b}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}}{n}\)

input

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

output

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

3.4.17.3.1 Defintions of rubi rules used

rule 3039

Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst 
[[3]]   Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /;  !FalseQ[lst]] /; 
NonsumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3120

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 4256

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] + Simp[(n + 1)/(b^2*n)   Int[(b*Csc[c 
+ d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* 
n]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
 

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

input

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

output

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)*(-si 
n(a+b*ln(c*x^n)))^(1/2)*EllipticF((sin(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
 

3.4.17.5 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 \[ \int \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sqrt {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} + i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) - i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{3 \, b n} \]

input

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

output

-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 
)
 

3.4.17.6 Sympy [F]

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

input

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

output

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

3.4.17.7 Maxima [F]

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

input

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

output

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

3.4.17.8 Giac [F]

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

input

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

output

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

3.4.17.9 Mupad [F(-1)]

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

input

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

output

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