\(\int \frac {\text {sech}^3(a+b \log (c x^n))}{x} \, dx\) [193]

Optimal result

Integrand size = 17, antiderivative size = 55 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n} \] Output:

1/2*arctan(sinh(a+b*ln(c*x^n)))/b/n+1/2*sech(a+b*ln(c*x^n))*tanh(a+b*ln(c* 
x^n))/b/n
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n} \] Input:

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

Output:

ArcTan[Sinh[a + b*Log[c*x^n]]]/(2*b*n) + (Sech[a + b*Log[c*x^n]]*Tanh[a + 
b*Log[c*x^n]])/(2*b*n)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3039, 3042, 4255, 3042, 4257}

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.

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

\(\Big \downarrow \) 3039

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4255

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\)

Input:

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

Output:

(ArcTan[Sinh[a + b*Log[c*x^n]]]/(2*b) + (Sech[a + b*Log[c*x^n]]*Tanh[a + b 
*Log[c*x^n]])/(2*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 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 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 17.40 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right ) \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\arctan \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{n b}\)	\(45\)
default	\(\frac {\frac {\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right ) \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\arctan \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{n b}\)	\(45\)
parallelrisch	\(\frac {i \left (-1-\cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )\right ) \ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )-i\right )+i \left (\cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+1\right ) \ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )+i\right )+2 \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n \left (\cosh \left (4 b \ln \left (\sqrt {c \,x^{n}}\right )+2 a \right )+1\right )}\)	\(121\)
risch	\(\frac {c^{b} \left (x^{n}\right )^{b} \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}}-{\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}}\right )}{b n {\left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}+1\right )}^{2}}+\frac {i \ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}}+i\right )}{2 b n}-\frac {i \ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}}-i\right )}{2 b n}\)	\(538\)

Input:

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

Output:

1/n/b*(1/2*sech(a+b*ln(c*x^n))*tanh(a+b*ln(c*x^n))+arctan(exp(a+b*ln(c*x^n 
))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (51) = 102\).

Time = 0.09 (sec) , antiderivative size = 452, normalized size of antiderivative = 8.22 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

(cosh(b*n*log(x) + b*log(c) + a)^3 + 3*cosh(b*n*log(x) + b*log(c) + a)*sin 
h(b*n*log(x) + b*log(c) + a)^2 + sinh(b*n*log(x) + b*log(c) + a)^3 + (cosh 
(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n 
*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh( 
b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log(c) + a)^2 + 2*co 
sh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 + c 
osh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*arcta 
n(cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)) + (3* 
cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a) - c 
osh(b*n*log(x) + b*log(c) + a))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 4 
*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + b 
*n*sinh(b*n*log(x) + b*log(c) + a)^4 + 2*b*n*cosh(b*n*log(x) + b*log(c) + 
a)^2 + 2*(3*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n)*sinh(b*n*log(x) + 
 b*log(c) + a)^2 + b*n + 4*(b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + b*n*co 
sh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (51) = 102\).

Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.09 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=c^{3 \, b} {\left (\frac {\arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-3 \, a\right )}}{b c^{2 \, b} c^{b} n} + \frac {{\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \] Input:

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

Output:

c^(3*b)*(arctan(c^(2*b)*x^(b*n)*e^a/c^b)*e^(-3*a)/(b*c^(2*b)*c^b*n) + (c^( 
2*b)*x^(3*b*n)*e^(2*a) - x^(b*n))*e^(-2*a)/((c^(2*b)*x^(2*b*n)*e^(2*a) + 1 
)^2*b*c^(2*b)*n))*e^(3*a)
 

Mupad [B] (verification not implemented)

Time = 2.49 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.53 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {b^2\,n^2}} \] Input:

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

Output:

(2*exp(-a))/((c*x^n)^b*(b*n + (2*b*n*exp(-2*a))/(c*x^n)^(2*b) + (b*n*exp(- 
4*a))/(c*x^n)^(4*b))) - exp(-a)/((c*x^n)^b*(b*n + (b*n*exp(-2*a))/(c*x^n)^ 
(2*b))) - atan((exp(-a)*(b^2*n^2)^(1/2))/(b*n*(c*x^n)^b))/(b^2*n^2)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.75 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {x^{4 b n} e^{4 a} c^{4 b} \mathit {atan} \left (x^{b n} e^{a} c^{b}\right )+2 x^{2 b n} e^{2 a} c^{2 b} \mathit {atan} \left (x^{b n} e^{a} c^{b}\right )+\mathit {atan} \left (x^{b n} e^{a} c^{b}\right )+x^{3 b n} e^{3 a} c^{3 b}-x^{b n} e^{a} c^{b}}{b n \left (x^{4 b n} e^{4 a} c^{4 b}+2 x^{2 b n} e^{2 a} c^{2 b}+1\right )} \] Input:

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

Output:

(x**(4*b*n)*e**(4*a)*c**(4*b)*atan(x**(b*n)*e**a*c**b) + 2*x**(2*b*n)*e**( 
2*a)*c**(2*b)*atan(x**(b*n)*e**a*c**b) + atan(x**(b*n)*e**a*c**b) + x**(3* 
b*n)*e**(3*a)*c**(3*b) - x**(b*n)*e**a*c**b)/(b*n*(x**(4*b*n)*e**(4*a)*c** 
(4*b) + 2*x**(2*b*n)*e**(2*a)*c**(2*b) + 1))