\(\int \frac {\arccos (a x^2)}{x} \, dx\) [28]

Optimal result

Integrand size = 10, antiderivative size = 62 \[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=-\frac {1}{4} i \arccos \left (a x^2\right )^2+\frac {1}{2} \arccos \left (a x^2\right ) \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (a x^2\right )}\right ) \] Output:

-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)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=-\frac {1}{4} i \left (\arccos \left (a x^2\right ) \left (\arccos \left (a x^2\right )+2 i \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (a x^2\right )}\right )\right ) \] Input:

Integrate[ArcCos[a*x^2]/x,x]
 

Output:

(-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])])
 

Rubi [A] (verified)

Time = 0.36 (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

\(\displaystyle \frac {1}{2} \left (2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (a x^2\right )}\right )-\frac {1}{2} i \arccos \left (a x^2\right ) \log \left (1+e^{2 i \arccos \left (a x^2\right )}\right )\right )-\frac {1}{2} i \arccos \left (a x^2\right )^2\right )\)

Input:

Int[ArcCos[a*x^2]/x,x]
 

Output:

((-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
 

Defintions of rubi rules used

rule 2620

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]
 

rule 2715

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]
 

rule 2838

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4202

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]
 

rule 5330

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]
 

Maple [F]

\[\int \frac {\arccos \left (a \,x^{2}\right )}{x}d x\]

Input:

int(arccos(a*x^2)/x,x)
 

Output:

int(arccos(a*x^2)/x,x)
 

Fricas [F]

\[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int { \frac {\arccos \left (a x^{2}\right )}{x} \,d x } \] Input:

integrate(arccos(a*x^2)/x,x, algorithm="fricas")
 

Output:

integral(arccos(a*x^2)/x, x)
 

Sympy [F]

\[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int \frac {\operatorname {acos}{\left (a x^{2} \right )}}{x}\, dx \] Input:

integrate(acos(a*x**2)/x,x)
 

Output:

Integral(acos(a*x**2)/x, x)
 

Maxima [F]

\[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int { \frac {\arccos \left (a x^{2}\right )}{x} \,d x } \] Input:

integrate(arccos(a*x^2)/x,x, algorithm="maxima")
 

Output:

integrate(arccos(a*x^2)/x, x)
 

Giac [F]

\[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int { \frac {\arccos \left (a x^{2}\right )}{x} \,d x } \] Input:

integrate(arccos(a*x^2)/x,x, algorithm="giac")
 

Output:

integrate(arccos(a*x^2)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int \frac {\mathrm {acos}\left (a\,x^2\right )}{x} \,d x \] Input:

int(acos(a*x^2)/x,x)
 

Output:

int(acos(a*x^2)/x, x)
 

Reduce [F]

\[ \int \frac {\arccos \left (a x^2\right )}{x} \, dx=\int \frac {\mathit {acos} \left (a \,x^{2}\right )}{x}d x \] Input:

int(acos(a*x^2)/x,x)
                                                                                    
                                                                                    
 

Output:

int(acos(a*x**2)/x,x)