Integrand size = 17, antiderivative size = 89 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n} \] Output:
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
Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n} \] Input:
Integrate[Sech[a + b*Log[c*x^n]]^5/x,x]
Output:
(3*ArcTan[Sinh[a + b*Log[c*x^n]]])/(8*b*n) + (3*Sech[a + b*Log[c*x^n]]*Tan h[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)
Time = 0.39 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3039, 3042, 4255, 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.
| \(\displaystyle \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \text {sech}^5\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )^5d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {3}{4} \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}+\frac {3}{4} \int \csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )^3d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \int \text {sech}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}+\frac {3}{4} \left (\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{2 b}+\frac {1}{2} \int \csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )d\log \left (c x^n\right )\right )}{n}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
Input:
Int[Sech[a + b*Log[c*x^n]]^5/x,x]
Output:
((Sech[a + b*Log[c*x^n]]^3*Tanh[a + b*Log[c*x^n]])/(4*b) + (3*(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)))/4)/n
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]
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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 187.71 (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 | \(\text {Expression too large to display}\) | \(748\) |
Input:
int(sech(a+b*ln(c*x^n))^5/x,x,method=_RETURNVERBOSE)
Output:
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))))
Leaf count of result is larger than twice the leaf count of optimal. 1326 vs. \(2 (83) = 166\).
Time = 0.21 (sec) , antiderivative size = 1326, normalized size of antiderivative = 14.90 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:
integrate(sech(a+b*log(c*x^n))^5/x,x, algorithm="fricas")
Output:
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*l og(c) + a)^4 + (105*cosh(b*n*log(x) + b*log(c) + a)^4 + 110*cosh(b*n*log(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*cosh(b*n*log(x) + b*log(c) + a)^5 + 110*cos h(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*co sh(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)*si nh(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*log(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...
\[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {sech}^{5}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(sech(a+b*ln(c*x**n))**5/x,x)
Output:
Integral(sech(a + b*log(c*x**n))**5/x, x)
\[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{5}}{x} \,d x } \] Input:
integrate(sech(a+b*log(c*x^n))^5/x,x, algorithm="maxima")
Output:
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*log(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)
Time = 0.12 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.71 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {1}{4} \, c^{5 \, b} {\left (\frac {3 \, \arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-5 \, a\right )}}{b c^{4 \, b} c^{b} n} + \frac {{\left (3 \, c^{6 \, b} x^{7 \, b n} e^{\left (6 \, a\right )} + 11 \, c^{4 \, b} x^{5 \, b n} e^{\left (4 \, a\right )} - 11 \, c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - 3 \, x^{b n}\right )} e^{\left (-4 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{4} b c^{4 \, b} n}\right )} e^{\left (5 \, a\right )} \] Input:
integrate(sech(a+b*log(c*x^n))^5/x,x, algorithm="giac")
Output:
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)
Time = 2.36 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.53 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {3\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {3\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}+\frac {b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{4\,\sqrt {b^2\,n^2}}-\frac {3\,{\mathrm {e}}^{-a}}{4\,{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}+\frac {4\,{\mathrm {e}}^{-3\,a}}{{\left (c\,x^n\right )}^{3\,b}\,\left (b\,n+\frac {4\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {6\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}+\frac {4\,b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}+\frac {b\,n\,{\mathrm {e}}^{-8\,a}}{{\left (c\,x^n\right )}^{8\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{2\,{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \] Input:
int(1/(x*cosh(a + b*log(c*x^n))^5),x)
Output:
(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)))
Time = 0.25 (sec) , antiderivative size = 290, normalized size of antiderivative = 3.26 \[ \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 x^{8 b n} e^{8 a} c^{8 b} \mathit {atan} \left (x^{b n} e^{a} c^{b}\right )+12 x^{6 b n} e^{6 a} c^{6 b} \mathit {atan} \left (x^{b n} e^{a} c^{b}\right )+18 x^{4 b n} e^{4 a} c^{4 b} \mathit {atan} \left (x^{b n} e^{a} c^{b}\right )+12 x^{2 b n} e^{2 a} c^{2 b} \mathit {atan} \left (x^{b n} e^{a} c^{b}\right )+3 \mathit {atan} \left (x^{b n} e^{a} c^{b}\right )+3 x^{7 b n} e^{7 a} c^{7 b}+11 x^{5 b n} e^{5 a} c^{5 b}-11 x^{3 b n} e^{3 a} c^{3 b}-3 x^{b n} e^{a} c^{b}}{4 b n \left (x^{8 b n} e^{8 a} c^{8 b}+4 x^{6 b n} e^{6 a} c^{6 b}+6 x^{4 b n} e^{4 a} c^{4 b}+4 x^{2 b n} e^{2 a} c^{2 b}+1\right )} \] Input:
int(sech(a+b*log(c*x^n))^5/x,x)
Output:
(3*x**(8*b*n)*e**(8*a)*c**(8*b)*atan(x**(b*n)*e**a*c**b) + 12*x**(6*b*n)*e **(6*a)*c**(6*b)*atan(x**(b*n)*e**a*c**b) + 18*x**(4*b*n)*e**(4*a)*c**(4*b )*atan(x**(b*n)*e**a*c**b) + 12*x**(2*b*n)*e**(2*a)*c**(2*b)*atan(x**(b*n) *e**a*c**b) + 3*atan(x**(b*n)*e**a*c**b) + 3*x**(7*b*n)*e**(7*a)*c**(7*b) + 11*x**(5*b*n)*e**(5*a)*c**(5*b) - 11*x**(3*b*n)*e**(3*a)*c**(3*b) - 3*x* *(b*n)*e**a*c**b)/(4*b*n*(x**(8*b*n)*e**(8*a)*c**(8*b) + 4*x**(6*b*n)*e**( 6*a)*c**(6*b) + 6*x**(4*b*n)*e**(4*a)*c**(4*b) + 4*x**(2*b*n)*e**(2*a)*c** (2*b) + 1))