Integrand size = 10, antiderivative size = 68 \[ \int \frac {\arccos \left (a x^n\right )}{x} \, dx=-\frac {i \arccos \left (a x^n\right )^2}{2 n}+\frac {\arccos \left (a x^n\right ) \log \left (1+e^{2 i \arccos \left (a x^n\right )}\right )}{n}-\frac {i \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (a x^n\right )}\right )}{2 n} \] Output:
-1/2*I*arccos(a*x^n)^2/n+arccos(a*x^n)*ln(1+(a*x^n+I*(1-a^2*(x^n)^2)^(1/2) )^2)/n-1/2*I*polylog(2,-(a*x^n+I*(1-a^2*(x^n)^2)^(1/2))^2)/n
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(68)=136\).
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.07 \[ \int \frac {\arccos \left (a x^n\right )}{x} \, dx=\arccos \left (a x^n\right ) \log (x)+\frac {a \left (-\text {arcsinh}\left (\sqrt {-a^2} x^n\right )^2-2 \text {arcsinh}\left (\sqrt {-a^2} x^n\right ) \log \left (1-e^{-2 \text {arcsinh}\left (\sqrt {-a^2} x^n\right )}\right )+2 n \log (x) \log \left (\sqrt {-a^2} x^n+\sqrt {1-a^2 x^{2 n}}\right )+\operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\sqrt {-a^2} x^n\right )}\right )\right )}{2 \sqrt {-a^2} n} \] Input:
Integrate[ArcCos[a*x^n]/x,x]
Output:
ArcCos[a*x^n]*Log[x] + (a*(-ArcSinh[Sqrt[-a^2]*x^n]^2 - 2*ArcSinh[Sqrt[-a^ 2]*x^n]*Log[1 - E^(-2*ArcSinh[Sqrt[-a^2]*x^n])] + 2*n*Log[x]*Log[Sqrt[-a^2 ]*x^n + Sqrt[1 - a^2*x^(2*n)]] + PolyLog[2, E^(-2*ArcSinh[Sqrt[-a^2]*x^n]) ]))/(2*Sqrt[-a^2]*n)
Time = 0.35 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5330, 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.
\(\displaystyle \int \frac {\arccos \left (a x^n\right )}{x} \, dx\) |
\(\Big \downarrow \) 5330 |
\(\displaystyle -\frac {\int \frac {x^{-n} \sqrt {1-a^2 x^{2 n}} \arccos \left (a x^n\right )}{a}d\arccos \left (a x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \arccos \left (a x^n\right ) \tan \left (\arccos \left (a x^n\right )\right )d\arccos \left (a x^n\right )}{n}\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle -\frac {\frac {1}{2} i \arccos \left (a x^n\right )^2-2 i \int \frac {e^{2 i \arccos \left (a x^n\right )} \arccos \left (a x^n\right )}{1+e^{2 i \arccos \left (a x^n\right )}}d\arccos \left (a x^n\right )}{n}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {\frac {1}{2} i \arccos \left (a x^n\right )^2-2 i \left (\frac {1}{2} i \int \log \left (1+e^{2 i \arccos \left (a x^n\right )}\right )d\arccos \left (a x^n\right )-\frac {1}{2} i \arccos \left (a x^n\right ) \log \left (1+e^{2 i \arccos \left (a x^n\right )}\right )\right )}{n}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\frac {1}{2} i \arccos \left (a x^n\right )^2-2 i \left (\frac {1}{4} \int e^{-2 i \arccos \left (a x^n\right )} \log \left (1+e^{2 i \arccos \left (a x^n\right )}\right )de^{2 i \arccos \left (a x^n\right )}-\frac {1}{2} i \arccos \left (a x^n\right ) \log \left (1+e^{2 i \arccos \left (a x^n\right )}\right )\right )}{n}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\frac {1}{2} i \arccos \left (a x^n\right )^2-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (a x^n\right )}\right )-\frac {1}{2} i \arccos \left (a x^n\right ) \log \left (1+e^{2 i \arccos \left (a x^n\right )}\right )\right )}{n}\) |
Input:
Int[ArcCos[a*x^n]/x,x]
Output:
-(((I/2)*ArcCos[a*x^n]^2 - (2*I)*((-1/2*I)*ArcCos[a*x^n]*Log[1 + E^((2*I)* ArcCos[a*x^n])] - PolyLog[2, -E^((2*I)*ArcCos[a*x^n])]/4))/n)
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[ArcCos[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> Simp[-p^(-1) Subst[I nt[x^n*Tan[x], x], x, ArcCos[a*x^p]], x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(\frac {-\frac {i \arccos \left (a \,x^{n}\right )^{2}}{2}+\arccos \left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )}{2}}{n}\) | \(84\) |
default | \(\frac {-\frac {i \arccos \left (a \,x^{n}\right )^{2}}{2}+\arccos \left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )}{2}}{n}\) | \(84\) |
Input:
int(arccos(a*x^n)/x,x,method=_RETURNVERBOSE)
Output:
1/n*(-1/2*I*arccos(a*x^n)^2+arccos(a*x^n)*ln(1+(a*x^n+I*(1-a^2*(x^n)^2)^(1 /2))^2)-1/2*I*polylog(2,-(a*x^n+I*(1-a^2*(x^n)^2)^(1/2))^2))
Exception generated. \[ \int \frac {\arccos \left (a x^n\right )}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(arccos(a*x^n)/x,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\arccos \left (a x^n\right )}{x} \, dx=\int \frac {\operatorname {acos}{\left (a x^{n} \right )}}{x}\, dx \] Input:
integrate(acos(a*x**n)/x,x)
Output:
Integral(acos(a*x**n)/x, x)
\[ \int \frac {\arccos \left (a x^n\right )}{x} \, dx=\int { \frac {\arccos \left (a x^{n}\right )}{x} \,d x } \] Input:
integrate(arccos(a*x^n)/x,x, algorithm="maxima")
Output:
-a*n*integrate(sqrt(a*x^n + 1)*sqrt(-a*x^n + 1)*x^n*log(x)/(a^2*x*x^(2*n) - x), x) + arctan(sqrt(a*x^n + 1)*sqrt(-a*x^n + 1)/(a*x^n))*log(x)
\[ \int \frac {\arccos \left (a x^n\right )}{x} \, dx=\int { \frac {\arccos \left (a x^{n}\right )}{x} \,d x } \] Input:
integrate(arccos(a*x^n)/x,x, algorithm="giac")
Output:
integrate(arccos(a*x^n)/x, x)
Timed out. \[ \int \frac {\arccos \left (a x^n\right )}{x} \, dx=\int \frac {\mathrm {acos}\left (a\,x^n\right )}{x} \,d x \] Input:
int(acos(a*x^n)/x,x)
Output:
int(acos(a*x^n)/x, x)
\[ \int \frac {\arccos \left (a x^n\right )}{x} \, dx=\int \frac {\mathit {acos} \left (x^{n} a \right )}{x}d x \] Input:
int(acos(a*x^n)/x,x)
Output:
int(acos(x**n*a)/x,x)