\(\int \frac {\text {sech}^5(a+b \log (c x^n))}{x} \, dx\) [195]

Optimal result

Integrand size = 17, antiderivative size = 89 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]

[Out]

3/8*arctan(sinh(a+b*ln(c*x^n)))/b/n+3/8*sech(a+b*ln(c*x^n))*tanh(a+b*ln(c*x^n))/b/n+1/4*sech(a+b*ln(c*x^n))^3*
tanh(a+b*ln(c*x^n))/b/n

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3853, 3855} \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]

[In]

Int[Sech[a + b*Log[c*x^n]]^5/x,x]

[Out]

(3*ArcTan[Sinh[a + b*Log[c*x^n]]])/(8*b*n) + (3*Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]])/(8*b*n) + (Sech
[a + b*Log[c*x^n]]^3*Tanh[a + b*Log[c*x^n]])/(4*b*n)

Rule 3853

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] + Dist[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 3855

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {sech}^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int \text {sech}^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n} \\ & = \frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int \text {sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{8 n} \\ & = \frac {3 \arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]

[In]

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

[Out]

(3*ArcTan[Sinh[a + b*Log[c*x^n]]])/(8*b*n) + (3*Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]])/(8*b*n) + (Sech
[a + b*Log[c*x^n]]^3*Tanh[a + b*Log[c*x^n]])/(4*b*n)

Maple [A] (verified)

Time = 233.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {\left (\frac {{\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}+\frac {3 \,\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+\frac {3 \arctan \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{4}}{n b}\)	\(64\)
default	\(\frac {\left (\frac {{\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}+\frac {3 \,\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+\frac {3 \arctan \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{4}}{n b}\)	\(64\)
parallelrisch	\(\frac {3 i \left (-\cosh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )-4 \cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )-3\right ) \ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )-i\right )+3 i \left (\cosh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )+4 \cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+3\right ) \ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )+i\right )+22 \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+6 \sinh \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )}{8 b n \left (\cosh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )+4 \cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+3\right )}\)	\(183\)
risch	\(\text {Expression too large to display}\)	\(748\)

[In]

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

[Out]

1/n/b*((1/4*sech(a+b*ln(c*x^n))^3+3/8*sech(a+b*ln(c*x^n)))*tanh(a+b*ln(c*x^n))+3/4*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. 1326 vs. \(2 (83) = 166\).

Time = 0.28 (sec) , antiderivative size = 1326, normalized size of antiderivative = 14.90 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(3*cosh(b*n*log(x) + b*log(c) + a)^7 + 21*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^
6 + 3*sinh(b*n*log(x) + b*log(c) + a)^7 + (63*cosh(b*n*log(x) + b*log(c) + a)^2 + 11)*sinh(b*n*log(x) + b*log(
c) + a)^5 + 11*cosh(b*n*log(x) + b*log(c) + a)^5 + 5*(21*cosh(b*n*log(x) + b*log(c) + a)^3 + 11*cosh(b*n*log(x
) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^4 + (105*cosh(b*n*log(x) + b*log(c) + a)^4 + 110*cosh(b*n*l
og(x) + b*log(c) + a)^2 - 11)*sinh(b*n*log(x) + b*log(c) + a)^3 - 11*cosh(b*n*log(x) + b*log(c) + a)^3 + (63*c
osh(b*n*log(x) + b*log(c) + a)^5 + 110*cosh(b*n*log(x) + b*log(c) + a)^3 - 33*cosh(b*n*log(x) + b*log(c) + a))
*sinh(b*n*log(x) + b*log(c) + a)^2 + 3*(cosh(b*n*log(x) + b*log(c) + a)^8 + 8*cosh(b*n*log(x) + b*log(c) + a)*
sinh(b*n*log(x) + b*log(c) + a)^7 + sinh(b*n*log(x) + b*log(c) + a)^8 + 4*(7*cosh(b*n*log(x) + b*log(c) + a)^2
+ 1)*sinh(b*n*log(x) + b*log(c) + a)^6 + 4*cosh(b*n*log(x) + b*log(c) + a)^6 + 8*(7*cosh(b*n*log(x) + b*log(c
) + a)^3 + 3*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(35*cosh(b*n*log(x) + b*lo
g(c) + a)^4 + 30*cosh(b*n*log(x) + b*log(c) + a)^2 + 3)*sinh(b*n*log(x) + b*log(c) + a)^4 + 6*cosh(b*n*log(x)
+ b*log(c) + a)^4 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^5 + 10*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*cosh(b*n
*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*cosh(b*n*log(x) + b*log(c) + a)^6 + 15*cosh(
b*n*log(x) + b*log(c) + a)^4 + 9*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log(c) + a)^2 + 4*
cosh(b*n*log(x) + b*log(c) + a)^2 + 8*(cosh(b*n*log(x) + b*log(c) + a)^7 + 3*cosh(b*n*log(x) + b*log(c) + a)^5
+ 3*cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)
*arctan(cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)) + (21*cosh(b*n*log(x) + b*log(c) +
a)^6 + 55*cosh(b*n*log(x) + b*log(c) + a)^4 - 33*cosh(b*n*log(x) + b*log(c) + a)^2 - 3)*sinh(b*n*log(x) + b*lo
g(c) + a) - 3*cosh(b*n*log(x) + b*log(c) + a))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^8 + 8*b*n*cosh(b*n*log(x)
+ b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + b*n*sinh(b*n*log(x) + b*log(c) + a)^8 + 4*b*n*cosh(b*n*log
(x) + b*log(c) + a)^6 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n)*sinh(b*n*log(x) + b*log(c) + a)^6 +
6*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n*cosh(b*n*log(x) +
b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(35*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 30*b*n*cosh(
b*n*log(x) + b*log(c) + a)^2 + 3*b*n)*sinh(b*n*log(x) + b*log(c) + a)^4 + 4*b*n*cosh(b*n*log(x) + b*log(c) + a
)^2 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5 + 10*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n*cosh(b*n*l
og(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^6 + 15*b*n
*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*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 + 8*(b*n*cosh(b*n*log(x) + b*log(c) + a)^7 + 3*b*n*cosh(b*n*log(x) + b*log(c) + a)^5 + 3*b*n*cos
h(b*n*log(x) + b*log(c) + a)^3 + b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))

Sympy [F]

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

[In]

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

[Out]

Integral(sech(a + b*log(c*x**n))**5/x, x)

Maxima [F]

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

[In]

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

[Out]

96*c^b*integrate(1/128*e^(b*log(x^n) + a)/(c^(2*b)*x*e^(2*b*log(x^n) + 2*a) + x), x) + 1/4*(3*c^(7*b)*e^(7*b*l
og(x^n) + 7*a) + 11*c^(5*b)*e^(5*b*log(x^n) + 5*a) - 11*c^(3*b)*e^(3*b*log(x^n) + 3*a) - 3*c^b*e^(b*log(x^n) +
a))/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n
) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.71 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {1}{4} \, c^{5 \, b} {\left (\frac {3 \, \arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-5 \, a\right )}}{b c^{4 \, b} c^{b} n} + \frac {{\left (3 \, c^{6 \, b} x^{7 \, b n} e^{\left (6 \, a\right )} + 11 \, c^{4 \, b} x^{5 \, b n} e^{\left (4 \, a\right )} - 11 \, c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - 3 \, x^{b n}\right )} e^{\left (-4 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{4} b c^{4 \, b} n}\right )} e^{\left (5 \, a\right )} \]

[In]

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

[Out]

1/4*c^(5*b)*(3*arctan(c^(2*b)*x^(b*n)*e^a/c^b)*e^(-5*a)/(b*c^(4*b)*c^b*n) + (3*c^(6*b)*x^(7*b*n)*e^(6*a) + 11*
c^(4*b)*x^(5*b*n)*e^(4*a) - 11*c^(2*b)*x^(3*b*n)*e^(2*a) - 3*x^(b*n))*e^(-4*a)/((c^(2*b)*x^(2*b*n)*e^(2*a) + 1
)^4*b*c^(4*b)*n))*e^(5*a)

Mupad [B] (verification not implemented)

Time = 2.03 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.53 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {3\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {3\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}+\frac {b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{4\,\sqrt {b^2\,n^2}}-\frac {3\,{\mathrm {e}}^{-a}}{4\,{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}+\frac {4\,{\mathrm {e}}^{-3\,a}}{{\left (c\,x^n\right )}^{3\,b}\,\left (b\,n+\frac {4\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {6\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}+\frac {4\,b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}+\frac {b\,n\,{\mathrm {e}}^{-8\,a}}{{\left (c\,x^n\right )}^{8\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{2\,{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \]

[In]

int(1/(x*cosh(a + b*log(c*x^n))^5),x)

[Out]

(2*exp(-a))/((c*x^n)^b*(b*n + (3*b*n*exp(-2*a))/(c*x^n)^(2*b) + (3*b*n*exp(-4*a))/(c*x^n)^(4*b) + (b*n*exp(-6*
a))/(c*x^n)^(6*b))) - (3*atan((exp(-a)*(b^2*n^2)^(1/2))/(b*n*(c*x^n)^b)))/(4*(b^2*n^2)^(1/2)) - (3*exp(-a))/(4
*(c*x^n)^b*(b*n + (b*n*exp(-2*a))/(c*x^n)^(2*b))) + (4*exp(-3*a))/((c*x^n)^(3*b)*(b*n + (4*b*n*exp(-2*a))/(c*x
^n)^(2*b) + (6*b*n*exp(-4*a))/(c*x^n)^(4*b) + (4*b*n*exp(-6*a))/(c*x^n)^(6*b) + (b*n*exp(-8*a))/(c*x^n)^(8*b))
) - exp(-a)/(2*(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)))