Integrand size = 8, antiderivative size = 51 \[ \int \frac {\arccos (a x)}{x} \, dx=-\frac {1}{2} i \arccos (a x)^2+\arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right ) \] Output:
-1/2*I*arccos(a*x)^2+arccos(a*x)*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)-1/2*I* polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos (a x)}{x} \, dx=-\frac {1}{2} i \arccos (a x)^2+\arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right ) \] Input:
Integrate[ArcCos[a*x]/x,x]
Output:
(-1/2*I)*ArcCos[a*x]^2 + ArcCos[a*x]*Log[1 + E^((2*I)*ArcCos[a*x])] - (I/2 )*PolyLog[2, -E^((2*I)*ArcCos[a*x])]
Time = 0.33 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {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.
\(\displaystyle \int \frac {\arccos (a x)}{x} \, dx\) |
\(\Big \downarrow \) 5137 |
\(\displaystyle -\int \frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a x}d\arccos (a x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \arccos (a x) \tan (\arccos (a x))d\arccos (a x)\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle 2 i \int \frac {e^{2 i \arccos (a x)} \arccos (a x)}{1+e^{2 i \arccos (a x)}}d\arccos (a x)-\frac {1}{2} i \arccos (a x)^2\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle 2 i \left (\frac {1}{2} i \int \log \left (1+e^{2 i \arccos (a x)}\right )d\arccos (a x)-\frac {1}{2} i \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle 2 i \left (\frac {1}{4} \int e^{-2 i \arccos (a x)} \log \left (1+e^{2 i \arccos (a x)}\right )de^{2 i \arccos (a x)}-\frac {1}{2} i \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle 2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2\) |
Input:
Int[ArcCos[a*x]/x,x]
Output:
(-1/2*I)*ArcCos[a*x]^2 + (2*I)*((-1/2*I)*ArcCos[a*x]*Log[1 + E^((2*I)*ArcC os[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[((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 ]
Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(-\frac {i \arccos \left (a x \right )^{2}}{2}+\arccos \left (a x \right ) \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(68\) |
default | \(-\frac {i \arccos \left (a x \right )^{2}}{2}+\arccos \left (a x \right ) \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(68\) |
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*(-a^2*x^2+1)^(1/2))^2)-1/2*I* polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)
\[ \int \frac {\arccos (a x)}{x} \, dx=\int { \frac {\arccos \left (a x\right )}{x} \,d x } \] Input:
integrate(arccos(a*x)/x,x, algorithm="fricas")
Output:
integral(arccos(a*x)/x, x)
\[ \int \frac {\arccos (a x)}{x} \, dx=\int \frac {\operatorname {acos}{\left (a x \right )}}{x}\, dx \] Input:
integrate(acos(a*x)/x,x)
Output:
Integral(acos(a*x)/x, x)
\[ \int \frac {\arccos (a x)}{x} \, dx=\int { \frac {\arccos \left (a x\right )}{x} \,d x } \] Input:
integrate(arccos(a*x)/x,x, algorithm="maxima")
Output:
integrate(arccos(a*x)/x, x)
\[ \int \frac {\arccos (a x)}{x} \, dx=\int { \frac {\arccos \left (a x\right )}{x} \,d x } \] Input:
integrate(arccos(a*x)/x,x, algorithm="giac")
Output:
integrate(arccos(a*x)/x, x)
Timed out. \[ \int \frac {\arccos (a x)}{x} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{x} \,d x \] Input:
int(acos(a*x)/x,x)
Output:
int(acos(a*x)/x, x)
\[ \int \frac {\arccos (a x)}{x} \, dx=\int \frac {\mathit {acos} \left (a x \right )}{x}d x \] Input:
int(acos(a*x)/x,x)
Output:
int(acos(a*x)/x,x)