Integrand size = 10, antiderivative size = 69 \[ \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx=\frac {i \sec ^{-1}\left (a x^n\right )^2}{2 n}-\frac {\sec ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \sec ^{-1}\left (a x^n\right )}\right )}{n}+\frac {i \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (a x^n\right )}\right )}{2 n} \]
1/2*I*arcsec(a*x^n)^2/n-arcsec(a*x^n)*ln(1+(1/a/(x^n)+I*(1-1/a^2/(x^n)^2)^ (1/2))^2)/n+1/2*I*polylog(2,-(1/a/(x^n)+I*(1-1/a^2/(x^n)^2)^(1/2))^2)/n
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx=\frac {x^{-n} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {x^{-2 n}}{a^2}\right )}{a n}+\left (\sec ^{-1}\left (a x^n\right )+\arcsin \left (\frac {x^{-n}}{a}\right )\right ) \log (x) \]
HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, 1/(a^2*x^(2*n))]/(a*n*x^n) + (ArcSec[a*x^n] + ArcSin[1/(a*x^n)])*Log[x]
Time = 0.51 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {7282, 5741, 5137, 3042, 4202, 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.
\(\displaystyle \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx\) |
\(\Big \downarrow \) 7282 |
\(\displaystyle \frac {\int x^{-n} \sec ^{-1}\left (a x^n\right )dx^n}{n}\) |
\(\Big \downarrow \) 5741 |
\(\displaystyle -\frac {\int x^{-n} \arccos \left (\frac {x^{-n}}{a}\right )dx^{-n}}{n}\) |
\(\Big \downarrow \) 5137 |
\(\displaystyle \frac {\int a x^n \sqrt {1-\frac {x^{-2 n}}{a^2}} \arccos \left (\frac {x^{-n}}{a}\right )d\arccos \left (\frac {x^{-n}}{a}\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \arccos \left (\frac {x^{-n}}{a}\right ) \tan \left (\arccos \left (\frac {x^{-n}}{a}\right )\right )d\arccos \left (\frac {x^{-n}}{a}\right )}{n}\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle \frac {\frac {1}{2} i x^{2 n}-2 i \int \frac {e^{2 i \arccos \left (\frac {x^{-n}}{a}\right )} \arccos \left (\frac {x^{-n}}{a}\right )}{1+e^{2 i \arccos \left (\frac {x^{-n}}{a}\right )}}d\arccos \left (\frac {x^{-n}}{a}\right )}{n}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {1}{2} i x^{2 n}-2 i \left (\frac {1}{2} i \int \log \left (1+e^{2 i \arccos \left (\frac {x^{-n}}{a}\right )}\right )d\arccos \left (\frac {x^{-n}}{a}\right )-\frac {1}{2} i \arccos \left (\frac {x^{-n}}{a}\right ) \log \left (1+e^{2 i \arccos \left (\frac {x^{-n}}{a}\right )}\right )\right )}{n}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\frac {1}{2} i x^{2 n}-2 i \left (\frac {1}{4} \int e^{2 i \arccos \left (\frac {x^{-n}}{a}\right )} \log \left (1+e^{2 i \arccos \left (\frac {x^{-n}}{a}\right )}\right )de^{2 i \arccos \left (\frac {x^{-n}}{a}\right )}-\frac {1}{2} i \arccos \left (\frac {x^{-n}}{a}\right ) \log \left (1+e^{2 i \arccos \left (\frac {x^{-n}}{a}\right )}\right )\right )}{n}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\frac {1}{2} i x^{2 n}-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {x^{-n}}{a}\right )}\right )-\frac {1}{2} i \arccos \left (\frac {x^{-n}}{a}\right ) \log \left (1+e^{2 i \arccos \left (\frac {x^{-n}}{a}\right )}\right )\right )}{n}\) |
((I/2)*x^(2*n) - (2*I)*((-1/2*I)*ArcCos[1/(a*x^n)]*Log[1 + E^((2*I)*ArcCos [1/(a*x^n)])] - PolyLog[2, -E^((2*I)*ArcCos[1/(a*x^n)])]/4))/n
3.1.17.3.1 Defintions of rubi rules used
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_.) + (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*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[ (a + b*x)^n*Tan[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0 ]
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + b *ArcCos[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 = 1.44 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(\frac {\frac {i \operatorname {arcsec}\left (a \,x^{n}\right )^{2}}{2}-\operatorname {arcsec}\left (a \,x^{n}\right ) \ln \left (1+\left (\frac {x^{-n}}{a}+i \sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (\frac {x^{-n}}{a}+i \sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )^{2}\right )}{2}}{n}\) | \(93\) |
default | \(\frac {\frac {i \operatorname {arcsec}\left (a \,x^{n}\right )^{2}}{2}-\operatorname {arcsec}\left (a \,x^{n}\right ) \ln \left (1+\left (\frac {x^{-n}}{a}+i \sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (\frac {x^{-n}}{a}+i \sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )^{2}\right )}{2}}{n}\) | \(93\) |
1/n*(1/2*I*arcsec(a*x^n)^2-arcsec(a*x^n)*ln(1+(1/a/(x^n)+I*(1-1/a^2/(x^n)^ 2)^(1/2))^2)+1/2*I*polylog(2,-(1/a/(x^n)+I*(1-1/a^2/(x^n)^2)^(1/2))^2))
Exception generated. \[ \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx=\int \frac {\operatorname {asec}{\left (a x^{n} \right )}}{x}\, dx \]
\[ \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arcsec}\left (a x^{n}\right )}{x} \,d x } \]
-a^2*n*integrate(sqrt(a*x^n + 1)*sqrt(a*x^n - 1)*log(x)/(a^4*x*x^(2*n) - a ^2*x), x) + arctan(sqrt(a*x^n + 1)*sqrt(a*x^n - 1))*log(x)
\[ \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arcsec}\left (a x^{n}\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx=\int \frac {\mathrm {acos}\left (\frac {1}{a\,x^n}\right )}{x} \,d x \]