Integrand size = 17, antiderivative size = 55 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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)
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 |
\(\displaystyle \frac {\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
(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
3.2.93.3.1 Defintions of rubi rules used
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 = 19.64 (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 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}^{-\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}^{a} {\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}} {\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}}\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 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}} {\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}}+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 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}} {\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}}-i\right )}{2 b n}\) | \(538\) |
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (51) = 102\).
Time = 0.27 (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=\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right )} \arctan \left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n + 4 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]
(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))
\[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {sech}^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
\[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x} \,d x } \]
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 )
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (51) = 102\).
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.09 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=c^{3 \, b} {\left (\frac {\arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-3 \, a\right )}}{b c^{2 \, b} c^{b} n} + \frac {{\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \]
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)
Time = 2.06 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.53 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {b^2\,n^2}} \]