\(\int \frac {\arccos (\sqrt {x})}{x} \, dx\) [40]

Optimal result

Integrand size = 10, antiderivative size = 56 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x} \, dx=-i \arccos \left (\sqrt {x}\right )^2+2 \arccos \left (\sqrt {x}\right ) \log \left (1+e^{2 i \arccos \left (\sqrt {x}\right )}\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\sqrt {x}\right )}\right ) \] Output:

-I*arccos(x^(1/2))^2+2*arccos(x^(1/2))*ln(1+(x^(1/2)+I*(1-x)^(1/2))^2)-I*p 
olylog(2,-(x^(1/2)+I*(1-x)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x} \, dx=-i \left (\arccos \left (\sqrt {x}\right ) \left (\arccos \left (\sqrt {x}\right )+2 i \log \left (1+e^{2 i \arccos \left (\sqrt {x}\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\sqrt {x}\right )}\right )\right ) \] Input:

Integrate[ArcCos[Sqrt[x]]/x,x]
 

Output:

(-I)*(ArcCos[Sqrt[x]]*(ArcCos[Sqrt[x]] + (2*I)*Log[1 + E^((2*I)*ArcCos[Sqr 
t[x]])]) + PolyLog[2, -E^((2*I)*ArcCos[Sqrt[x]])])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.23, 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.

\(\displaystyle \int \frac {\arccos \left (\sqrt {x}\right )}{x} \, dx\)

\(\Big \downarrow \) 5330

\(\displaystyle -2 \int \frac {\sqrt {1-x} \arccos \left (\sqrt {x}\right )}{\sqrt {x}}d\arccos \left (\sqrt {x}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -2 \int \arccos \left (\sqrt {x}\right ) \tan \left (\arccos \left (\sqrt {x}\right )\right )d\arccos \left (\sqrt {x}\right )\)

\(\Big \downarrow \) 4202

\(\displaystyle -2 \left (\frac {1}{2} i \arccos \left (\sqrt {x}\right )^2-2 i \int \frac {e^{2 i \arccos \left (\sqrt {x}\right )} \arccos \left (\sqrt {x}\right )}{1+e^{2 i \arccos \left (\sqrt {x}\right )}}d\arccos \left (\sqrt {x}\right )\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle -2 \left (\frac {1}{2} i \arccos \left (\sqrt {x}\right )^2-2 i \left (\frac {1}{2} i \int \log \left (1+e^{2 i \arccos \left (\sqrt {x}\right )}\right )d\arccos \left (\sqrt {x}\right )-\frac {1}{2} i \arccos \left (\sqrt {x}\right ) \log \left (1+e^{2 i \arccos \left (\sqrt {x}\right )}\right )\right )\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle -2 \left (\frac {1}{2} i \arccos \left (\sqrt {x}\right )^2-2 i \left (\frac {1}{4} \int e^{-2 i \arccos \left (\sqrt {x}\right )} \log \left (1+e^{2 i \arccos \left (\sqrt {x}\right )}\right )de^{2 i \arccos \left (\sqrt {x}\right )}-\frac {1}{2} i \arccos \left (\sqrt {x}\right ) \log \left (1+e^{2 i \arccos \left (\sqrt {x}\right )}\right )\right )\right )\)

\(\Big \downarrow \) 2838

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

Input:

Int[ArcCos[Sqrt[x]]/x,x]
 

Output:

-2*((I/2)*ArcCos[Sqrt[x]]^2 - (2*I)*((-1/2*I)*ArcCos[Sqrt[x]]*Log[1 + E^(( 
2*I)*ArcCos[Sqrt[x]])] - PolyLog[2, -E^((2*I)*ArcCos[Sqrt[x]])]/4))
 

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 [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05

method	result	size
derivativedivides	\(-i \arccos \left (\sqrt {x}\right )^{2}+2 \arccos \left (\sqrt {x}\right ) \ln \left (1+\left (\sqrt {x}+i \sqrt {1-x}\right )^{2}\right )-i \operatorname {polylog}\left (2, -\left (\sqrt {x}+i \sqrt {1-x}\right )^{2}\right )\)	\(59\)
default	\(-i \arccos \left (\sqrt {x}\right )^{2}+2 \arccos \left (\sqrt {x}\right ) \ln \left (1+\left (\sqrt {x}+i \sqrt {1-x}\right )^{2}\right )-i \operatorname {polylog}\left (2, -\left (\sqrt {x}+i \sqrt {1-x}\right )^{2}\right )\)	\(59\)

Input:

int(arccos(x^(1/2))/x,x,method=_RETURNVERBOSE)
 

Output:

-I*arccos(x^(1/2))^2+2*arccos(x^(1/2))*ln(1+(x^(1/2)+I*(1-x)^(1/2))^2)-I*p 
olylog(2,-(x^(1/2)+I*(1-x)^(1/2))^2)
 

Fricas [F]

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

integrate(arccos(x^(1/2))/x,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(arccos(sqrt(x))/x, x)
 

Sympy [F]

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

integrate(acos(x**(1/2))/x,x)
 

Output:

Integral(acos(sqrt(x))/x, x)
 

Maxima [F]

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

integrate(arccos(x^(1/2))/x,x, algorithm="maxima")
 

Output:

integrate(arccos(sqrt(x))/x, x)
 

Giac [F]

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

integrate(arccos(x^(1/2))/x,x, algorithm="giac")
 

Output:

integrate(arccos(sqrt(x))/x, x)
 

Mupad [F(-1)]

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

int(acos(x^(1/2))/x,x)
 

Output:

int(acos(x^(1/2))/x, x)
 

Reduce [F]

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

int(acos(x^(1/2))/x,x)
 

Output:

int(acos(sqrt(x))/x,x)