Integrand size = 10, antiderivative size = 61 \[ \int \frac {\text {sech}^{-1}\left (a x^n\right )}{x} \, dx=\frac {\text {sech}^{-1}\left (a x^n\right )^2}{2 n}-\frac {\text {sech}^{-1}\left (a x^n\right ) \log \left (1+e^{2 \text {sech}^{-1}\left (a x^n\right )}\right )}{n}-\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}\left (a x^n\right )}\right )}{2 n} \] Output:
1/2*arcsech(a*x^n)^2/n-arcsech(a*x^n)*ln(1+(1/a/(x^n)+(1/a/(x^n)-1)^(1/2)* (1/a/(x^n)+1)^(1/2))^2)/n-1/2*polylog(2,-(1/a/(x^n)+(1/a/(x^n)-1)^(1/2)*(1 /a/(x^n)+1)^(1/2))^2)/n
Leaf count is larger than twice the leaf count of optimal. \(219\) vs. \(2(61)=122\).
Time = 0.64 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.59 \[ \int \frac {\text {sech}^{-1}\left (a x^n\right )}{x} \, dx=\text {sech}^{-1}\left (a x^n\right ) \log (x)+\frac {\sqrt {\frac {1-a x^n}{1+a x^n}} \left (4 \sqrt {-1+a^2 x^{2 n}} \arctan \left (\sqrt {-1+a^2 x^{2 n}}\right ) \left (2 n \log (x)-\log \left (a^2 x^{2 n}\right )\right )+\sqrt {1-a^2 x^{2 n}} \left (\log ^2\left (a^2 x^{2 n}\right )-4 \log \left (a^2 x^{2 n}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {1-a^2 x^{2 n}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1-a^2 x^{2 n}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {1-a^2 x^{2 n}}\right )\right )\right )}{8 \left (n-a n x^n\right )} \] Input:
Integrate[ArcSech[a*x^n]/x,x]
Output:
ArcSech[a*x^n]*Log[x] + (Sqrt[(1 - a*x^n)/(1 + a*x^n)]*(4*Sqrt[-1 + a^2*x^ (2*n)]*ArcTan[Sqrt[-1 + a^2*x^(2*n)]]*(2*n*Log[x] - Log[a^2*x^(2*n)]) + Sq rt[1 - a^2*x^(2*n)]*(Log[a^2*x^(2*n)]^2 - 4*Log[a^2*x^(2*n)]*Log[(1 + Sqrt [1 - a^2*x^(2*n)])/2] + 2*Log[(1 + Sqrt[1 - a^2*x^(2*n)])/2]^2 - 4*PolyLog [2, 1/2 - Sqrt[1 - a^2*x^(2*n)]/2])))/(8*(n - a*n*x^n))
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {7282, 6835, 6297, 3042, 26, 4201, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {\text {sech}^{-1}\left (a x^n\right )}{x} \, dx\) |
| \(\Big \downarrow \) 7282 |
| \(\displaystyle \frac {\int x^{-n} \text {sech}^{-1}\left (a x^n\right )dx^n}{n}\) |
| \(\Big \downarrow \) 6835 |
| \(\displaystyle -\frac {\int x^{-n} \text {arccosh}\left (\frac {x^{-n}}{a}\right )dx^{-n}}{n}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle -\frac {\int a x^n \sqrt {\frac {\frac {x^{-n}}{a}-1}{\frac {x^{-n}}{a}+1}} \left (\frac {x^{-n}}{a}+1\right ) \text {arccosh}\left (\frac {x^{-n}}{a}\right )d\text {arccosh}\left (\frac {x^{-n}}{a}\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int -i \text {arccosh}\left (\frac {x^{-n}}{a}\right ) \tan \left (i \text {arccosh}\left (\frac {x^{-n}}{a}\right )\right )d\text {arccosh}\left (\frac {x^{-n}}{a}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \int \text {arccosh}\left (\frac {x^{-n}}{a}\right ) \tan \left (i \text {arccosh}\left (\frac {x^{-n}}{a}\right )\right )d\text {arccosh}\left (\frac {x^{-n}}{a}\right )}{n}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {i \left (2 i \int \frac {e^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )} \text {arccosh}\left (\frac {x^{-n}}{a}\right )}{1+e^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )}}d\text {arccosh}\left (\frac {x^{-n}}{a}\right )-\frac {1}{2} i x^{2 n}\right )}{n}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (\frac {x^{-n}}{a}\right ) \log \left (e^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )}\right )d\text {arccosh}\left (\frac {x^{-n}}{a}\right )\right )-\frac {1}{2} i x^{2 n}\right )}{n}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (\frac {x^{-n}}{a}\right ) \log \left (e^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )}+1\right )-\frac {1}{4} \int e^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )} \log \left (1+e^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )}\right )de^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )}\right )-\frac {1}{2} i x^{2 n}\right )}{n}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )}\right )+\frac {1}{2} \text {arccosh}\left (\frac {x^{-n}}{a}\right ) \log \left (e^{2 \text {arccosh}\left (\frac {x^{-n}}{a}\right )}+1\right )\right )-\frac {1}{2} i x^{2 n}\right )}{n}\) |
Input:
Int[ArcSech[a*x^n]/x,x]
Output:
(I*((-1/2*I)*x^(2*n) + (2*I)*((ArcCosh[1/(a*x^n)]*Log[1 + E^(2*ArcCosh[1/( a*x^n)])])/2 + PolyLog[2, -E^(2*ArcCosh[1/(a*x^n)])]/4)))/n
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + b*ArcCosh[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 /lst[[2]] Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ u] && !RationalFunctionQ[u, x]
Time = 0.66 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.82
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arcsech}\left (a \,x^{n}\right )^{2}}{2}-\operatorname {arcsech}\left (a \,x^{n}\right ) \ln \left (1+\left (\frac {x^{-n}}{a}+\sqrt {\frac {x^{-n}}{a}-1}\, \sqrt {\frac {x^{-n}}{a}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {x^{-n}}{a}+\sqrt {\frac {x^{-n}}{a}-1}\, \sqrt {\frac {x^{-n}}{a}+1}\right )^{2}\right )}{2}}{n}\) | \(111\) |
| default | \(\frac {\frac {\operatorname {arcsech}\left (a \,x^{n}\right )^{2}}{2}-\operatorname {arcsech}\left (a \,x^{n}\right ) \ln \left (1+\left (\frac {x^{-n}}{a}+\sqrt {\frac {x^{-n}}{a}-1}\, \sqrt {\frac {x^{-n}}{a}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {x^{-n}}{a}+\sqrt {\frac {x^{-n}}{a}-1}\, \sqrt {\frac {x^{-n}}{a}+1}\right )^{2}\right )}{2}}{n}\) | \(111\) |
Input:
int(arcsech(a*x^n)/x,x,method=_RETURNVERBOSE)
Output:
1/n*(1/2*arcsech(a*x^n)^2-arcsech(a*x^n)*ln(1+(1/a/(x^n)+(1/a/(x^n)-1)^(1/ 2)*(1/a/(x^n)+1)^(1/2))^2)-1/2*polylog(2,-(1/a/(x^n)+(1/a/(x^n)-1)^(1/2)*( 1/a/(x^n)+1)^(1/2))^2))
Exception generated. \[ \int \frac {\text {sech}^{-1}\left (a x^n\right )}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(arcsech(a*x^n)/x,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\text {sech}^{-1}\left (a x^n\right )}{x} \, dx=\int \frac {\operatorname {asech}{\left (a x^{n} \right )}}{x}\, dx \] Input:
integrate(asech(a*x**n)/x,x)
Output:
Integral(asech(a*x**n)/x, x)
\[ \int \frac {\text {sech}^{-1}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arsech}\left (a x^{n}\right )}{x} \,d x } \] Input:
integrate(arcsech(a*x^n)/x,x, algorithm="maxima")
Output:
a^2*n*integrate(x^(2*n)*log(x)/(a^2*x*x^(2*n) + (a^2*x*x^(2*n) - x)*sqrt(a *x^n + 1)*sqrt(-a*x^n + 1) - x), x) + n*integrate(1/2*log(x)/(a*x*x^n + x) , x) - n*integrate(1/2*log(x)/(a*x*x^n - x), x) + log(sqrt(a*x^n + 1)*sqrt (-a*x^n + 1) + 1)*log(x) - log(a)*log(x) - log(x)*log(x^n)
\[ \int \frac {\text {sech}^{-1}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arsech}\left (a x^{n}\right )}{x} \,d x } \] Input:
integrate(arcsech(a*x^n)/x,x, algorithm="giac")
Output:
integrate(arcsech(a*x^n)/x, x)
Timed out. \[ \int \frac {\text {sech}^{-1}\left (a x^n\right )}{x} \, dx=\int \frac {\mathrm {acosh}\left (\frac {1}{a\,x^n}\right )}{x} \,d x \] Input:
int(acosh(1/(a*x^n))/x,x)
Output:
int(acosh(1/(a*x^n))/x, x)
\[ \int \frac {\text {sech}^{-1}\left (a x^n\right )}{x} \, dx=\int \frac {\mathit {asech} \left (x^{n} a \right )}{x}d x \] Input:
int(asech(a*x^n)/x,x)
Output:
int(asech(x**n*a)/x,x)