Integrand size = 10, antiderivative size = 60 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x} \, dx=\frac {1}{2} i \arccos \left (\frac {a}{x}\right )^2-\arccos \left (\frac {a}{x}\right ) \log \left (1+e^{2 i \arccos \left (\frac {a}{x}\right )}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {a}{x}\right )}\right ) \] Output:
1/2*I*arccos(a/x)^2-arccos(a/x)*ln(1+(a/x+I*(1-a^2/x^2)^(1/2))^2)+1/2*I*po lylog(2,-(a/x+I*(1-a^2/x^2)^(1/2))^2)
Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x} \, dx=\frac {1}{2} i \arccos \left (\frac {a}{x}\right )^2-\arccos \left (\frac {a}{x}\right ) \log \left (1+e^{2 i \arccos \left (\frac {a}{x}\right )}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {a}{x}\right )}\right ) \] Input:
Integrate[ArcCos[a/x]/x,x]
Output:
(I/2)*ArcCos[a/x]^2 - ArcCos[a/x]*Log[1 + E^((2*I)*ArcCos[a/x])] + (I/2)*P olyLog[2, -E^((2*I)*ArcCos[a/x])]
Time = 0.34 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.12, 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.
\(\displaystyle \int \frac {\arccos \left (\frac {a}{x}\right )}{x} \, dx\) |
\(\Big \downarrow \) 5330 |
\(\displaystyle \int \frac {x \sqrt {1-\frac {a^2}{x^2}} \arccos \left (\frac {a}{x}\right )}{a}d\arccos \left (\frac {a}{x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \arccos \left (\frac {a}{x}\right ) \tan \left (\arccos \left (\frac {a}{x}\right )\right )d\arccos \left (\frac {a}{x}\right )\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle \frac {1}{2} i \arccos \left (\frac {a}{x}\right )^2-2 i \int \frac {e^{2 i \arccos \left (\frac {a}{x}\right )} \arccos \left (\frac {a}{x}\right )}{1+e^{2 i \arccos \left (\frac {a}{x}\right )}}d\arccos \left (\frac {a}{x}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {1}{2} i \arccos \left (\frac {a}{x}\right )^2-2 i \left (\frac {1}{2} i \int \log \left (1+e^{2 i \arccos \left (\frac {a}{x}\right )}\right )d\arccos \left (\frac {a}{x}\right )-\frac {1}{2} i \arccos \left (\frac {a}{x}\right ) \log \left (1+e^{2 i \arccos \left (\frac {a}{x}\right )}\right )\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {1}{2} i \arccos \left (\frac {a}{x}\right )^2-2 i \left (\frac {1}{4} \int e^{-2 i \arccos \left (\frac {a}{x}\right )} \log \left (1+e^{2 i \arccos \left (\frac {a}{x}\right )}\right )de^{2 i \arccos \left (\frac {a}{x}\right )}-\frac {1}{2} i \arccos \left (\frac {a}{x}\right ) \log \left (1+e^{2 i \arccos \left (\frac {a}{x}\right )}\right )\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {1}{2} i \arccos \left (\frac {a}{x}\right )^2-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {a}{x}\right )}\right )-\frac {1}{2} i \arccos \left (\frac {a}{x}\right ) \log \left (1+e^{2 i \arccos \left (\frac {a}{x}\right )}\right )\right )\) |
Input:
Int[ArcCos[a/x]/x,x]
Output:
(I/2)*ArcCos[a/x]^2 - (2*I)*((-1/2*I)*ArcCos[a/x]*Log[1 + E^((2*I)*ArcCos[ a/x])] - PolyLog[2, -E^((2*I)*ArcCos[a/x])]/4)
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 = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\frac {i \arccos \left (\frac {a}{x}\right )^{2}}{2}-\arccos \left (\frac {a}{x}\right ) \ln \left (1+{\left (\frac {a}{x}+i \sqrt {1-\frac {a^{2}}{x^{2}}}\right )}^{2}\right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {a}{x}+i \sqrt {1-\frac {a^{2}}{x^{2}}}\right )}^{2}\right )}{2}\) | \(77\) |
default | \(\frac {i \arccos \left (\frac {a}{x}\right )^{2}}{2}-\arccos \left (\frac {a}{x}\right ) \ln \left (1+{\left (\frac {a}{x}+i \sqrt {1-\frac {a^{2}}{x^{2}}}\right )}^{2}\right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {a}{x}+i \sqrt {1-\frac {a^{2}}{x^{2}}}\right )}^{2}\right )}{2}\) | \(77\) |
Input:
int(arccos(a/x)/x,x,method=_RETURNVERBOSE)
Output:
1/2*I*arccos(a/x)^2-arccos(a/x)*ln(1+(a/x+I*(1-a^2/x^2)^(1/2))^2)+1/2*I*po lylog(2,-(a/x+I*(1-a^2/x^2)^(1/2))^2)
\[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x} \, dx=\int { \frac {\arccos \left (\frac {a}{x}\right )}{x} \,d x } \] Input:
integrate(arccos(a/x)/x,x, algorithm="fricas")
Output:
integral(arccos(a/x)/x, x)
\[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x} \, dx=\int \frac {\operatorname {acos}{\left (\frac {a}{x} \right )}}{x}\, dx \] Input:
integrate(acos(a/x)/x,x)
Output:
Integral(acos(a/x)/x, x)
\[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x} \, dx=\int { \frac {\arccos \left (\frac {a}{x}\right )}{x} \,d x } \] Input:
integrate(arccos(a/x)/x,x, algorithm="maxima")
Output:
-I*a^2*integrate(-log(x)/(a^2*x - x^3), x) - a*integrate(-sqrt(a + x)*sqrt (-a + x)*log(x)/(a^2*x - x^3), x) + arctan(sqrt(a + x)*sqrt(-a + x)/a)*log (x) - 1/2*I*log(x^2)*log(x) + 1/2*I*log(x)^2 + 1/2*I*log(x)*log((a + x)/a) + 1/2*I*log(x)*log((a - x)/a) + 1/2*I*dilog(x/a) + 1/2*I*dilog(-x/a)
Exception generated. \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(arccos(a/x)/x,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Limit: Max order reached or unable to make series expan sion Error: Bad Argument Value
Timed out. \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x} \, dx=\int \frac {\mathrm {acos}\left (\frac {a}{x}\right )}{x} \,d x \] Input:
int(acos(a/x)/x,x)
Output:
int(acos(a/x)/x, x)
\[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x} \, dx=\int \frac {\mathit {acos} \left (\frac {a}{x}\right )}{x}d x \] Input:
int(acos(a/x)/x,x)
Output:
int(acos(a/x)/x,x)