3.7.57 \(\int \frac {(1-x^2)^{3/2} \arccos (x)}{x} \, dx\) [657]

3.7.57.1 Optimal result

Integrand size = 17, antiderivative size = 95 \[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+2 i \arccos (x) \arctan \left (e^{i \arccos (x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (x)}\right ) \]

output

4/3*x-1/9*x^3+1/3*(-x^2+1)^(3/2)*arccos(x)+2*I*arccos(x)*arctan(x+I*(-x^2+ 
1)^(1/2))-I*polylog(2,-I*(x+I*(-x^2+1)^(1/2)))+I*polylog(2,I*(x+I*(-x^2+1) 
^(1/2)))+arccos(x)*(-x^2+1)^(1/2)
 

3.7.57.2 Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=x+\sqrt {1-x^2} \arccos (x)+\frac {1}{36} \left (9 x+12 \left (1-x^2\right )^{3/2} \arccos (x)-\cos (3 \arccos (x))\right )-\arccos (x) \log \left (1-i e^{i \arccos (x)}\right )+\arccos (x) \log \left (1+i e^{i \arccos (x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (x)}\right ) \]

input

Integrate[((1 - x^2)^(3/2)*ArcCos[x])/x,x]
 

output

x + Sqrt[1 - x^2]*ArcCos[x] + (9*x + 12*(1 - x^2)^(3/2)*ArcCos[x] - Cos[3* 
ArcCos[x]])/36 - ArcCos[x]*Log[1 - I*E^(I*ArcCos[x])] + ArcCos[x]*Log[1 + 
I*E^(I*ArcCos[x])] - I*PolyLog[2, (-I)*E^(I*ArcCos[x])] + I*PolyLog[2, I*E 
^(I*ArcCos[x])]
 

3.7.57.3 Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {5203, 2009, 5199, 24, 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.

\(\displaystyle \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx\)

\(\Big \downarrow \) 5203

\(\displaystyle \int \frac {\sqrt {1-x^2} \arccos (x)}{x}dx+\frac {1}{3} \int \left (1-x^2\right )dx+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {\sqrt {1-x^2} \arccos (x)}{x}dx+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\frac {1}{3} \left (x-\frac {x^3}{3}\right )\)

\(\Big \downarrow \) 5199

\(\displaystyle \int \frac {\arccos (x)}{x \sqrt {1-x^2}}dx+\int 1dx+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (x-\frac {x^3}{3}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \int \frac {\arccos (x)}{x \sqrt {1-x^2}}dx+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (x-\frac {x^3}{3}\right )+x\)

\(\Big \downarrow \) 5219

\(\displaystyle -\int \frac {\arccos (x)}{x}d\arccos (x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (x-\frac {x^3}{3}\right )+x\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int \arccos (x) \csc \left (\arccos (x)+\frac {\pi }{2}\right )d\arccos (x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (x-\frac {x^3}{3}\right )+x\)

\(\Big \downarrow \) 4669

\(\displaystyle \int \log \left (1-i e^{i \arccos (x)}\right )d\arccos (x)-\int \log \left (1+i e^{i \arccos (x)}\right )d\arccos (x)+2 i \arccos (x) \arctan \left (e^{i \arccos (x)}\right )+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (x-\frac {x^3}{3}\right )+x\)

\(\Big \downarrow \) 2715

\(\displaystyle -i \int e^{-i \arccos (x)} \log \left (1-i e^{i \arccos (x)}\right )de^{i \arccos (x)}+i \int e^{-i \arccos (x)} \log \left (1+i e^{i \arccos (x)}\right )de^{i \arccos (x)}+2 i \arccos (x) \arctan \left (e^{i \arccos (x)}\right )+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (x-\frac {x^3}{3}\right )+x\)

\(\Big \downarrow \) 2838

\(\displaystyle 2 i \arccos (x) \arctan \left (e^{i \arccos (x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (x)}\right )+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (x-\frac {x^3}{3}\right )+x\)

input

Int[((1 - x^2)^(3/2)*ArcCos[x])/x,x]
 

output

x + (x - x^3/3)/3 + Sqrt[1 - x^2]*ArcCos[x] + ((1 - x^2)^(3/2)*ArcCos[x])/ 
3 + (2*I)*ArcCos[x]*ArcTan[E^(I*ArcCos[x])] - I*PolyLog[2, (-I)*E^(I*ArcCo 
s[x])] + I*PolyLog[2, I*E^(I*ArcCos[x])]
 

3.7.57.3.1 Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 4669

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]
 

rule 5199

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
(e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcC 
os[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2]), x], x 
] + Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, 
 f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 5203

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcC 
os[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(f*(m + 
2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 - c^2 
*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, 
e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
 

rule 5219

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]
 

3.7.57.4 Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.63

method	result	size
default	\(-\frac {5 \left (-\sqrt {-x^{2}+1}+i x \right ) \left (\arccos \left (x \right )+i\right )}{8}+\frac {5 \left (i x +\sqrt {-x^{2}+1}\right ) \left (\arccos \left (x \right )-i\right )}{8}+\arccos \left (x \right ) \ln \left (1+i \left (i \sqrt {-x^{2}+1}+x \right )\right )-\arccos \left (x \right ) \ln \left (1-i \left (i \sqrt {-x^{2}+1}+x \right )\right )-i \operatorname {dilog}\left (1+i \left (i \sqrt {-x^{2}+1}+x \right )\right )+i \operatorname {dilog}\left (1-i \left (i \sqrt {-x^{2}+1}+x \right )\right )-\frac {\cos \left (3 \arccos \left (x \right )\right )}{36}-\frac {\arccos \left (x \right ) \sin \left (3 \arccos \left (x \right )\right )}{12}\)	\(155\)

input

int((-x^2+1)^(3/2)*arccos(x)/x,x,method=_RETURNVERBOSE)
 

output

-5/8*(-(-x^2+1)^(1/2)+I*x)*(arccos(x)+I)+5/8*(I*x+(-x^2+1)^(1/2))*(arccos( 
x)-I)+arccos(x)*ln(1+I*(I*(-x^2+1)^(1/2)+x))-arccos(x)*ln(1-I*(I*(-x^2+1)^ 
(1/2)+x))-I*dilog(1+I*(I*(-x^2+1)^(1/2)+x))+I*dilog(1-I*(I*(-x^2+1)^(1/2)+ 
x))-1/36*cos(3*arccos(x))-1/12*arccos(x)*sin(3*arccos(x))
 

3.7.57.5 Fricas [F]

\[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\int { \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )}{x} \,d x } \]

input

integrate((-x^2+1)^(3/2)*arccos(x)/x,x, algorithm="fricas")
 

output

integral(-(x^2 - 1)*sqrt(-x^2 + 1)*arccos(x)/x, x)
 

3.7.57.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\text {Timed out} \]

input

integrate((-x**2+1)**(3/2)*acos(x)/x,x)
 

output

Timed out
 

3.7.57.7 Maxima [F]

\[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\int { \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )}{x} \,d x } \]

input

integrate((-x^2+1)^(3/2)*arccos(x)/x,x, algorithm="maxima")
 

output

integrate((-x^2 + 1)^(3/2)*arccos(x)/x, x)
 

3.7.57.8 Giac [F]

\[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\int { \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )}{x} \,d x } \]

input

integrate((-x^2+1)^(3/2)*arccos(x)/x,x, algorithm="giac")
 

output

integrate((-x^2 + 1)^(3/2)*arccos(x)/x, x)
 

3.7.57.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\int \frac {\mathrm {acos}\left (x\right )\,{\left (1-x^2\right )}^{3/2}}{x} \,d x \]

input

int((acos(x)*(1 - x^2)^(3/2))/x,x)
 

output

int((acos(x)*(1 - x^2)^(3/2))/x, x)
 

3.7.57.10 Reduce [F]

\[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\int \frac {\sqrt {-x^{2}+1}\, \mathit {acos} \left (x \right )}{x}d x -\left (\int \sqrt {-x^{2}+1}\, \mathit {acos} \left (x \right ) x d x \right ) \]

input

int((sqrt( - x**2 + 1)*acos(x)*( - x**2 + 1))/x,x)
 

output

int((sqrt( - x**2 + 1)*acos(x))/x,x) - int(sqrt( - x**2 + 1)*acos(x)*x,x)