Integrand size = 10, antiderivative size = 62 \[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=-\frac {1}{4} i \arccos \left (a x^2\right )^2+\frac {1}{2} \arccos \left (a x^2\right ) \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (a x^2\right )}\right ) \]
-1/4*I*arccos(a*x^2)^2+1/2*arccos(a*x^2)*ln(1+(a*x^2+I*(-a^2*x^4+1)^(1/2)) ^2)-1/4*I*polylog(2,-(a*x^2+I*(-a^2*x^4+1)^(1/2))^2)
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=-\frac {1}{4} i \left (\arccos \left (a x^2\right ) \left (\arccos \left (a x^2\right )+2 i \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (a x^2\right )}\right )\right ) \]
(-1/4*I)*(ArcCos[a*x^2]*(ArcCos[a*x^2] + (2*I)*Log[1 + E^((2*I)*ArcCos[a*x ^2])]) + PolyLog[2, -E^((2*I)*ArcCos[a*x^2])])
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.15, 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.
\(\displaystyle \int \frac {\arccos \left (a x^2\right )}{x} \, dx\) |
\(\Big \downarrow \) 5330 |
\(\displaystyle -\frac {1}{2} \int \frac {\sqrt {1-a^2 x^4} \arccos \left (a x^2\right )}{a x^2}d\arccos \left (a x^2\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} \int \arccos \left (a x^2\right ) \tan \left (\arccos \left (a x^2\right )\right )d\arccos \left (a x^2\right )\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle \frac {1}{2} \left (2 i \int \frac {e^{2 i \arccos \left (a x^2\right )} \arccos \left (a x^2\right )}{1+e^{2 i \arccos \left (a x^2\right )}}d\arccos \left (a x^2\right )-\frac {1}{2} i \arccos \left (a x^2\right )^2\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {1}{2} \left (2 i \left (\frac {1}{2} i \int \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )d\arccos \left (a x^2\right )-\frac {1}{2} i \arccos \left (a x^2\right ) \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )\right )-\frac {1}{2} i \arccos \left (a x^2\right )^2\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {1}{2} \left (2 i \left (\frac {1}{4} \int e^{-2 i \arccos \left (a x^2\right )} \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )de^{2 i \arccos \left (a x^2\right )}-\frac {1}{2} i \arccos \left (a x^2\right ) \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )\right )-\frac {1}{2} i \arccos \left (a x^2\right )^2\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {1}{2} \left (2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (a x^2\right )}\right )-\frac {1}{2} i \arccos \left (a x^2\right ) \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )\right )-\frac {1}{2} i \arccos \left (a x^2\right )^2\right )\) |
((-1/2*I)*ArcCos[a*x^2]^2 + (2*I)*((-1/2*I)*ArcCos[a*x^2]*Log[1 + E^((2*I) *ArcCos[a*x^2])] - PolyLog[2, -E^((2*I)*ArcCos[a*x^2])]/4))/2
3.1.51.3.1 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[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]
\[\int \frac {\arccos \left (a \,x^{2}\right )}{x}d x\]
\[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int { \frac {\arccos \left (a x^{2}\right )}{x} \,d x } \]
\[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int \frac {\operatorname {acos}{\left (a x^{2} \right )}}{x}\, dx \]
\[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int { \frac {\arccos \left (a x^{2}\right )}{x} \,d x } \]
\[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int { \frac {\arccos \left (a x^{2}\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int \frac {\mathrm {acos}\left (a\,x^2\right )}{x} \,d x \]