3.2.57 \(\int \frac {(a+b \arccos (c x))^3}{x^2} \, dx\) [157]

3.2.57.1 Optimal result

Integrand size = 14, antiderivative size = 151 \[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=-\frac {(a+b \arccos (c x))^3}{x}-6 i b c (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )+6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )-6 b^3 c \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )+6 b^3 c \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right ) \]

output

-(a+b*arccos(c*x))^3/x-6*I*b*c*(a+b*arccos(c*x))^2*arctan(c*x+I*(-c^2*x^2+ 
1)^(1/2))+6*I*b^2*c*(a+b*arccos(c*x))*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/ 
2)))-6*I*b^2*c*(a+b*arccos(c*x))*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))-6 
*b^3*c*polylog(3,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))+6*b^3*c*polylog(3,I*(c*x+I 
*(-c^2*x^2+1)^(1/2)))
 

3.2.57.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(308\) vs. \(2(151)=302\).

Time = 0.22 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=-\frac {a^3}{x}-\frac {3 a^2 b \arccos (c x)}{x}-3 a^2 b c \log (x)+3 a^2 b c \log \left (1+\sqrt {1-c^2 x^2}\right )+3 a b^2 c \left (-\frac {\arccos (c x)^2}{c x}+2 \left (\arccos (c x) \left (\log \left (1-i e^{i \arccos (c x)}\right )-\log \left (1+i e^{i \arccos (c x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )\right )\right )+b^3 c \left (-\frac {\arccos (c x)^3}{c x}+3 \left (\arccos (c x)^2 \left (\log \left (1-i e^{i \arccos (c x)}\right )-\log \left (1+i e^{i \arccos (c x)}\right )\right )+2 i \arccos (c x) \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )-2 \left (\operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )\right )\right ) \]

input

Integrate[(a + b*ArcCos[c*x])^3/x^2,x]
 

output

-(a^3/x) - (3*a^2*b*ArcCos[c*x])/x - 3*a^2*b*c*Log[x] + 3*a^2*b*c*Log[1 + 
Sqrt[1 - c^2*x^2]] + 3*a*b^2*c*(-(ArcCos[c*x]^2/(c*x)) + 2*(ArcCos[c*x]*(L 
og[1 - I*E^(I*ArcCos[c*x])] - Log[1 + I*E^(I*ArcCos[c*x])]) + I*(PolyLog[2 
, (-I)*E^(I*ArcCos[c*x])] - PolyLog[2, I*E^(I*ArcCos[c*x])]))) + b^3*c*(-( 
ArcCos[c*x]^3/(c*x)) + 3*(ArcCos[c*x]^2*(Log[1 - I*E^(I*ArcCos[c*x])] - Lo 
g[1 + I*E^(I*ArcCos[c*x])]) + (2*I)*ArcCos[c*x]*(PolyLog[2, (-I)*E^(I*ArcC 
os[c*x])] - PolyLog[2, I*E^(I*ArcCos[c*x])]) - 2*(PolyLog[3, (-I)*E^(I*Arc 
Cos[c*x])] - PolyLog[3, I*E^(I*ArcCos[c*x])])))
 

3.2.57.3 Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5139, 5219, 3042, 4669, 3011, 2720, 7143}

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.

\(\displaystyle \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx\)

\(\Big \downarrow \) 5139

\(\displaystyle -3 b c \int \frac {(a+b \arccos (c x))^2}{x \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arccos (c x))^3}{x}\)

\(\Big \downarrow \) 5219

\(\displaystyle 3 b c \int \frac {(a+b \arccos (c x))^2}{c x}d\arccos (c x)-\frac {(a+b \arccos (c x))^3}{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 3 b c \int (a+b \arccos (c x))^2 \csc \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)-\frac {(a+b \arccos (c x))^3}{x}\)

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {(a+b \arccos (c x))^3}{x}+3 b c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {(a+b \arccos (c x))^3}{x}+3 b c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {(a+b \arccos (c x))^3}{x}+3 b c \left (-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )\right )\)

input

Int[(a + b*ArcCos[c*x])^3/x^2,x]
 

output

-((a + b*ArcCos[c*x])^3/x) + 3*b*c*((-2*I)*(a + b*ArcCos[c*x])^2*ArcTan[E^ 
(I*ArcCos[c*x])] + 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, (-I)*E^(I*ArcCos[ 
c*x])] - b*PolyLog[3, (-I)*E^(I*ArcCos[c*x])]) - 2*b*(I*(a + b*ArcCos[c*x] 
)*PolyLog[2, I*E^(I*ArcCos[c*x])] - b*PolyLog[3, I*E^(I*ArcCos[c*x])]))
 

3.2.57.3.1 Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

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 5139

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]
 

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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.2.57.4 Maple [F]

\[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{3}}{x^{2}}d x\]

input

int((a+b*arccos(c*x))^3/x^2,x)
 

output

int((a+b*arccos(c*x))^3/x^2,x)
 

3.2.57.5 Fricas [F]

\[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

input

integrate((a+b*arccos(c*x))^3/x^2,x, algorithm="fricas")
 

output

integral((b^3*arccos(c*x)^3 + 3*a*b^2*arccos(c*x)^2 + 3*a^2*b*arccos(c*x) 
+ a^3)/x^2, x)
 

3.2.57.6 Sympy [F]

\[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]

input

integrate((a+b*acos(c*x))**3/x**2,x)
 

output

Integral((a + b*acos(c*x))**3/x**2, x)
 

3.2.57.7 Maxima [F]

\[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

input

integrate((a+b*arccos(c*x))^3/x^2,x, algorithm="maxima")
 

output

3*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) - arccos(c*x)/x)*a^2*b - 
a^3/x - (b^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 - x*integrate(3* 
(sqrt(c*x + 1)*sqrt(-c*x + 1)*b^3*c*x*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1) 
, c*x)^2 + (a*b^2*c^2*x^2 - a*b^2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c 
*x)^2)/(c^2*x^4 - x^2), x))/x
 

3.2.57.8 Giac [F]

\[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

input

integrate((a+b*arccos(c*x))^3/x^2,x, algorithm="giac")
 

output

integrate((b*arccos(c*x) + a)^3/x^2, x)
 

3.2.57.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3}{x^2} \,d x \]

input

int((a + b*acos(c*x))^3/x^2,x)
 

output

int((a + b*acos(c*x))^3/x^2, x)