\(\int \text {sech}^3(a+b \log (c x^n)) \, dx\) [184]

Optimal result

Integrand size = 13, antiderivative size = 71 \[ \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {8 e^{-3 a} x \left (c x^n\right )^{-3 b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {1}{b n}\right ),\frac {1}{2} \left (5-\frac {1}{b n}\right ),-e^{-2 a} \left (c x^n\right )^{-2 b}\right )}{1-3 b n} \] Output:

8*x*hypergeom([3, 3/2-1/2/b/n],[5/2-1/2/b/n],-1/exp(2*a)/((c*x^n)^(2*b)))/ 
exp(3*a)/(-3*b*n+1)/((c*x^n)^(3*b))
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.42 \[ \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (2 e^a (-1+b n) \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {1}{b n}\right ),\frac {1}{2} \left (3+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )+\text {sech}\left (a+b \log \left (c x^n\right )\right ) \left (1+b n \tanh \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b^2 n^2} \] Input:

Integrate[Sech[a + b*Log[c*x^n]]^3,x]
 

Output:

(x*(2*E^a*(-1 + b*n)*(c*x^n)^b*Hypergeometric2F1[1, (1 + 1/(b*n))/2, (3 + 
1/(b*n))/2, -(E^(2*a)*(c*x^n)^(2*b))] + Sech[a + b*Log[c*x^n]]*(1 + b*n*Ta 
nh[a + b*Log[c*x^n]])))/(2*b^2*n^2)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6079, 6081, 795, 888}

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.

\(\displaystyle \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 6079

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

\(\Big \downarrow \) 6081

\(\displaystyle \frac {8 e^{-3 a} x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{-3 b+\frac {1}{n}-1}}{\left (e^{-2 a} \left (c x^n\right )^{-2 b}+1\right )^3}d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 795

\(\displaystyle \frac {8 e^{-3 a} x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{3 b+\frac {1}{n}-1}}{\left (\left (c x^n\right )^{2 b}+e^{-2 a}\right )^3}d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {8 e^{3 a} x \left (c x^n\right )^{3 b} \operatorname {Hypergeometric2F1}\left (3,\frac {3 b+\frac {1}{n}}{2 b},\frac {1}{2} \left (5+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{3 b n+1}\)

Input:

Int[Sech[a + b*Log[c*x^n]]^3,x]
 

Output:

(8*E^(3*a)*x*(c*x^n)^(3*b)*Hypergeometric2F1[3, (3*b + n^(-1))/(2*b), (5 + 
 1/(b*n))/2, -(E^(2*a)*(c*x^n)^(2*b))])/(1 + 3*b*n)
 

Defintions of rubi rules used

rule 795

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* 
(b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 6079

Int[Sech[((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)*Sech[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 6081

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

Maple [F]

\[\int {\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}d x\]

Input:

int(sech(a+b*ln(c*x^n))^3,x)
 

Output:

int(sech(a+b*ln(c*x^n))^3,x)
 

Fricas [F]

\[ \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{3} \,d x } \] Input:

integrate(sech(a+b*log(c*x^n))^3,x, algorithm="fricas")
 

Output:

integral(sech(b*log(c*x^n) + a)^3, x)
 

Sympy [F]

\[ \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \operatorname {sech}^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \] Input:

integrate(sech(a+b*ln(c*x**n))**3,x)
 

Output:

Integral(sech(a + b*log(c*x**n))**3, x)
 

Maxima [F]

\[ \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{3} \,d x } \] Input:

integrate(sech(a+b*log(c*x^n))^3,x, algorithm="maxima")
 

Output:

8*(b^2*c^b*n^2 - c^b)*integrate(1/8*e^(b*log(x^n) + a)/(b^2*c^(2*b)*n^2*e^ 
(2*b*log(x^n) + 2*a) + b^2*n^2), x) + ((b*c^(3*b)*n + c^(3*b))*x*e^(3*b*lo 
g(x^n) + 3*a) - (b*c^b*n - c^b)*x*e^(b*log(x^n) + a))/(b^2*c^(4*b)*n^2*e^( 
4*b*log(x^n) + 4*a) + 2*b^2*c^(2*b)*n^2*e^(2*b*log(x^n) + 2*a) + b^2*n^2)
 

Giac [F]

\[ \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{3} \,d x } \] Input:

integrate(sech(a+b*log(c*x^n))^3,x, algorithm="giac")
 

Output:

integrate(sech(b*log(c*x^n) + a)^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx=8 e^{3 a} c^{3 b} \left (\int \frac {x^{3 b n}}{x^{6 b n} e^{6 a} c^{6 b}+3 x^{4 b n} e^{4 a} c^{4 b}+3 x^{2 b n} e^{2 a} c^{2 b}+1}d x \right ) \] Input:

int(sech(a+b*log(c*x^n))^3,x)
 

Output:

8*e**(3*a)*c**(3*b)*int(x**(3*b*n)/(x**(6*b*n)*e**(6*a)*c**(6*b) + 3*x**(4 
*b*n)*e**(4*a)*c**(4*b) + 3*x**(2*b*n)*e**(2*a)*c**(2*b) + 1),x)