\(\int \frac {1}{x \cosh ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\) [256]

Optimal result

Integrand size = 19, antiderivative size = 63 \[ \int \frac {1}{x \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n}+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}} \] Output:

2*I*EllipticE(I*sinh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))/b/n+2*sinh(a+b*ln(c*x 
^n))/b/n/cosh(a+b*ln(c*x^n))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \left (i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\frac {\sinh \left (a+b \log \left (c x^n\right )\right )}{\sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}\right )}{b n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3039, 3042, 3116, 3042, 3119}

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.

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

\(\Big \downarrow \) 3039

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

\(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{b \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}-\int \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{b \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}-\int \sqrt {\sin \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{b \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b}}{n}\)

Input:

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

Output:

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

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 3116

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1))   I 
nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && 
 IntegerQ[2*n]
 

rule 3119

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(140\) vs. \(2(63)=126\).

Time = 1.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.24

method	result	size
derivativedivides	\(\frac {4 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+2 \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\)	\(141\)
default	\(\frac {4 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+2 \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\)	\(141\)

Input:

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

Output:

2/n*(2*cosh(1/2*a+1/2*b*ln(c*x^n))*sinh(1/2*a+1/2*b*ln(c*x^n))^2+(-sinh(1/ 
2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-2*sinh(1/2*a+1/2*b*ln(c*x^n))^2-1)^(1/2)*E 
llipticE(cosh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2)))/sinh(1/2*a+1/2*b*ln(c*x^n)) 
/(2*cosh(1/2*a+1/2*b*ln(c*x^n))^2-1)^(1/2)/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (61) = 122\).

Time = 0.09 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.86 \[ \int \frac {1}{x \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + \sqrt {2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + 2 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} \sqrt {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}\right )}}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n} \] Input:

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

Output:

2*((sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*sqrt(2)*cosh(b*n*log(x) 
+ b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sqrt(2)*sinh(b*n*log(x) 
+ b*log(c) + a)^2 + sqrt(2))*weierstrassZeta(-4, 0, weierstrassPInverse(-4 
, 0, cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a))) + 
 2*(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)* 
sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2)*sqrt( 
cosh(b*n*log(x) + b*log(c) + a)))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 
 2*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + b 
*n*sinh(b*n*log(x) + b*log(c) + a)^2 + b*n)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(cosh(log(x**n*c)*b + a))/(cosh(log(x**n*c)*b + a)**2*x),x)