\(\int \frac {1}{\cosh ^{\frac {3}{2}}(a+\frac {2 \log (c x^n)}{n})} \, dx\) [261]

Optimal result

Integrand size = 18, antiderivative size = 42 \[ \int \frac {1}{\cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=-\frac {x \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{2 \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \] Output:

-1/2*x*(1+1/exp(2*a)/((c*x^n)^(4/n)))/cosh(a+2*ln(c*x^n)/n)^(3/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\frac {-\cosh \left (a-2 \log (x)+\frac {2 \log \left (c x^n\right )}{n}\right )+\sinh \left (a-2 \log (x)+\frac {2 \log \left (c x^n\right )}{n}\right )}{x \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}} \] Input:

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

Output:

(-Cosh[a - 2*Log[x] + (2*Log[c*x^n])/n] + Sinh[a - 2*Log[x] + (2*Log[c*x^n 
])/n])/(x*Sqrt[Cosh[a + (2*Log[c*x^n])/n]])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6052, 6060, 796}

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.

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

\(\Big \downarrow \) 6052

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

\(\Big \downarrow \) 6060

\(\displaystyle \frac {x \left (c x^n\right )^{2/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2} \int \frac {\left (c x^n\right )^{-1-\frac {2}{n}}}{\left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}d\left (c x^n\right )}{n \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}\)

\(\Big \downarrow \) 796

\(\displaystyle -\frac {x \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}{2 \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}\)

Input:

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

Output:

-1/2*(x*(1 + 1/(E^(2*a)*(c*x^n)^(4/n))))/Cosh[a + (2*Log[c*x^n])/n]^(3/2)
 

Defintions of rubi rules used

rule 796

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, 
 p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
 

rule 6052

Int[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> S 
imp[x/(n*(c*x^n)^(1/n))   Subst[Int[x^(1/n - 1)*Cosh[d*(a + b*Log[x])]^p, x 
], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1] 
)
 

rule 6060

Int[Cosh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] 
:> Simp[Cosh[d*(a + b*Log[x])]^p/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p 
)   Int[(e*x)^m*x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p, x], x] /; FreeQ[ 
{a, b, d, e, m, p}, x] &&  !IntegerQ[p]
 

Maple [F]

\[\int \frac {1}{{\cosh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )}^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=-\frac {2 \, \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )}}{x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1} \] Input:

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

Output:

-2*sqrt(1/2)*x*sqrt((x^4*e^(2*(a*n + 2*log(c))/n) + 1)/x^2)*e^(-1/2*(a*n + 
 2*log(c))/n)/(x^4*e^(2*(a*n + 2*log(c))/n) + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

sage0*x
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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