Integrand size = 10, antiderivative size = 74 \[ \int \frac {\arccos (a x)^2}{x^2} \, dx=-\frac {\arccos (a x)^2}{x}-4 i a \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+2 i a \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i a \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right ) \]
-arccos(a*x)^2/x-4*I*a*arccos(a*x)*arctan(a*x+I*(-a^2*x^2+1)^(1/2))+2*I*a* polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))-2*I*a*polylog(2,I*(a*x+I*(-a^2*x^ 2+1)^(1/2)))
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.32 \[ \int \frac {\arccos (a x)^2}{x^2} \, dx=-\frac {\arccos (a x) \left (\arccos (a x)+2 a x \left (-\log \left (1-i e^{i \arccos (a x)}\right )+\log \left (1+i e^{i \arccos (a x)}\right )\right )\right )}{x}+2 i a \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i a \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right ) \]
-((ArcCos[a*x]*(ArcCos[a*x] + 2*a*x*(-Log[1 - I*E^(I*ArcCos[a*x])] + Log[1 + I*E^(I*ArcCos[a*x])])))/x) + (2*I)*a*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (2*I)*a*PolyLog[2, I*E^(I*ArcCos[a*x])]
Time = 0.39 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5139, 5219, 3042, 4669, 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.
\(\displaystyle \int \frac {\arccos (a x)^2}{x^2} \, dx\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle -2 a \int \frac {\arccos (a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\arccos (a x)^2}{x}\) |
\(\Big \downarrow \) 5219 |
\(\displaystyle 2 a \int \frac {\arccos (a x)}{a x}d\arccos (a x)-\frac {\arccos (a x)^2}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \int \arccos (a x) \csc \left (\arccos (a x)+\frac {\pi }{2}\right )d\arccos (a x)-\frac {\arccos (a x)^2}{x}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle -\frac {\arccos (a x)^2}{x}+2 a \left (-\int \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+\int \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\arccos (a x)^2}{x}+2 a \left (i \int e^{-i \arccos (a x)} \log \left (1-i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \int e^{-i \arccos (a x)} \log \left (1+i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\) |
-(ArcCos[a*x]^2/x) + 2*a*((-2*I)*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] + I *PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - I*PolyLog[2, I*E^(I*ArcCos[a*x])])
3.1.18.3.1 Defintions of rubi rules used
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[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(-(c^(m + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[ d + e*x^2]] Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; Fr eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.43 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.84
method | result | size |
derivativedivides | \(a \left (-\frac {\arccos \left (a x \right )^{2}}{a x}-2 \arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 \arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\right )\) | \(136\) |
default | \(a \left (-\frac {\arccos \left (a x \right )^{2}}{a x}-2 \arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 \arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\right )\) | \(136\) |
a*(-arccos(a*x)^2/a/x-2*arccos(a*x)*ln(1+I*(I*(-a^2*x^2+1)^(1/2)+a*x))+2*a rccos(a*x)*ln(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x))+2*I*dilog(1+I*(I*(-a^2*x^2+1 )^(1/2)+a*x))-2*I*dilog(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x)))
\[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x^{2}} \,d x } \]
\[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int \frac {\operatorname {acos}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x^{2}} \,d x } \]
(2*a*x*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x + 1)*sqrt(- a*x + 1), a*x)/(a^2*x^3 - x), x) - arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a *x)^2)/x
\[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^2}{x^2} \,d x \]