\(\int \frac {\text {sech}^{\frac {5}{2}}(a+b \log (c x^n))}{x} \, dx\) [196]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-1)]
Maxima [F]
Giac [F(-1)]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 19, antiderivative size = 97 \[ \int \frac {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{3 b n}+\frac {2 \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n} \] Output:

-2/3*I*cosh(a+b*ln(c*x^n))^(1/2)*InverseJacobiAM(1/2*I*(a+b*ln(c*x^n)),2^( 
1/2))*sech(a+b*ln(c*x^n))^(1/2)/b/n+2/3*sech(a+b*ln(c*x^n))^(3/2)*sinh(a+b 
*ln(c*x^n))/b/n
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int \frac {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \left (-i \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )+\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n} \] Input:

Integrate[Sech[a + b*Log[c*x^n]]^(5/2)/x,x]
 

Output:

(2*Sech[a + b*Log[c*x^n]]^(3/2)*((-I)*Cosh[a + b*Log[c*x^n]]^(3/2)*Ellipti 
cF[(I/2)*(a + b*Log[c*x^n]), 2] + Sinh[a + b*Log[c*x^n]]))/(3*b*n)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 95, 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, 4255, 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.

\(\displaystyle \int \frac {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )^{5/2}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4255

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b}+\frac {1}{3} \int \sqrt {\csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b}-\frac {2 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{3 b}}{n}\)

Input:

Int[Sech[a + b*Log[c*x^n]]^(5/2)/x,x]
 

Output:

((((-2*I)/3)*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticF[(I/2)*(a + b*Log[c*x^n 
]), 2]*Sqrt[Sech[a + b*Log[c*x^n]]])/b + (2*Sech[a + b*Log[c*x^n]]^(3/2)*S 
inh[a + b*Log[c*x^n]])/(3*b))/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 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 4255
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) 
  Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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]
 
Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(257\) vs. \(2(85)=170\).

Time = 94.31 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.66

method result size
derivativedivides \(\frac {\sqrt {\left (2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \left (\frac {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}}{3 {\left ({\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-\frac {1}{2}\right )}^{2}}+\frac {2 \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{3 \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}}\right )}{n \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) \(258\)
default \(\frac {\sqrt {\left (2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \left (\frac {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}}{3 {\left ({\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-\frac {1}{2}\right )}^{2}}+\frac {2 \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{3 \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}}\right )}{n \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) \(258\)

Input:

int(sech(a+b*ln(c*x^n))^(5/2)/x,x,method=_RETURNVERBOSE)
 

Output:

1/n*((2*cosh(1/2*a+1/2*b*ln(c*x^n))^2-1)*sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1 
/2)*(1/3*cosh(1/2*a+1/2*b*ln(c*x^n))*(2*sinh(1/2*a+1/2*b*ln(c*x^n))^4+sinh 
(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/(cosh(1/2*a+1/2*b*ln(c*x^n))^2-1/2)^2+2/3 
*(-sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-2*cosh(1/2*a+1/2*b*ln(c*x^n))^2+ 
1)^(1/2)/(2*sinh(1/2*a+1/2*b*ln(c*x^n))^4+sinh(1/2*a+1/2*b*ln(c*x^n))^2)^( 
1/2)*EllipticF(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. 315 vs. \(2 (84) = 168\).

Time = 0.13 (sec) , antiderivative size = 315, normalized size of antiderivative = 3.25 \[ \int \frac {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1}} + {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )}}{3 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n\right )}} \] Input:

integrate(sech(a+b*log(c*x^n))^(5/2)/x,x, algorithm="fricas")
 

Output:

2/3*(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*lo 
g(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a 
)^2 - 1)*sqrt((cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c 
) + a))/(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 + 
 1)) + (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))*weierstrassPInverse(-4, 0, cosh(b*n*log(x 
) + b*log(c) + a) + sinh(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(-1)]

Timed out. \[ \int \frac {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \] Input:

integrate(sech(a+b*ln(c*x**n))**(5/2)/x,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x} \,d x } \] Input:

integrate(sech(a+b*log(c*x^n))^(5/2)/x,x, algorithm="maxima")
 

Output:

integrate(sech(b*log(c*x^n) + a)^(5/2)/x, x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \] Input:

integrate(sech(a+b*log(c*x^n))^(5/2)/x,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}}{x} \,d x \] Input:

int((1/cosh(a + b*log(c*x^n)))^(5/2)/x,x)
 

Output:

int((1/cosh(a + b*log(c*x^n)))^(5/2)/x, x)
 

Reduce [F]

\[ \int \frac {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}\, {\mathrm {sech}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2}}{x}d x \] Input:

int(sech(a+b*log(c*x^n))^(5/2)/x,x)
 

Output:

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