\(\int \frac {(a+b \arccos (c x))^3}{x} \, dx\) [156]

Optimal result

Integrand size = 14, antiderivative size = 127 \[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=-\frac {i (a+b \arccos (c x))^4}{4 b}+(a+b \arccos (c x))^3 \log \left (1+e^{2 i \arccos (c x)}\right )-\frac {3}{2} i b (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+\frac {3}{2} b^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )+\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arccos (c x)}\right ) \] Output:

-1/4*I*(a+b*arccos(c*x))^4/b+(a+b*arccos(c*x))^3*ln(1+(c*x+I*(-c^2*x^2+1)^ 
(1/2))^2)-3/2*I*b*(a+b*arccos(c*x))^2*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2) 
)^2)+3/2*b^2*(a+b*arccos(c*x))*polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+3/ 
4*I*b^3*polylog(4,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=\frac {1}{4} \left (-6 i a^2 b \arccos (c x)^2-4 i a b^2 \arccos (c x)^3-i b^3 \arccos (c x)^4+12 a^2 b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+12 a b^2 \arccos (c x)^2 \log \left (1+e^{2 i \arccos (c x)}\right )+4 b^3 \arccos (c x)^3 \log \left (1+e^{2 i \arccos (c x)}\right )+4 a^3 \log (c x)-6 i b (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+6 b^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )+3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arccos (c x)}\right )\right ) \] Input:

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

Output:

((-6*I)*a^2*b*ArcCos[c*x]^2 - (4*I)*a*b^2*ArcCos[c*x]^3 - I*b^3*ArcCos[c*x 
]^4 + 12*a^2*b*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] + 12*a*b^2*ArcCo 
s[c*x]^2*Log[1 + E^((2*I)*ArcCos[c*x])] + 4*b^3*ArcCos[c*x]^3*Log[1 + E^(( 
2*I)*ArcCos[c*x])] + 4*a^3*Log[c*x] - (6*I)*b*(a + b*ArcCos[c*x])^2*PolyLo 
g[2, -E^((2*I)*ArcCos[c*x])] + 6*b^2*(a + b*ArcCos[c*x])*PolyLog[3, -E^((2 
*I)*ArcCos[c*x])] + (3*I)*b^3*PolyLog[4, -E^((2*I)*ArcCos[c*x])])/4
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5137, 3042, 4202, 2620, 3011, 7163, 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.

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

\(\displaystyle -\int (a+b \arccos (c x))^3 \tan (\arccos (c x))d\arccos (c x)\)

\(\Big \downarrow \) 4202

\(\displaystyle 2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))^3}{1+e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^4}{4 b}\)

\(\Big \downarrow \) 2620

\(\displaystyle 2 i \left (\frac {3}{2} i b \int (a+b \arccos (c x))^2 \log \left (1+e^{2 i \arccos (c x)}\right )d\arccos (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^3\right )-\frac {i (a+b \arccos (c x))^4}{4 b}\)

\(\Big \downarrow \) 3011

\(\displaystyle 2 i \left (\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2-i b \int (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^3\right )-\frac {i (a+b \arccos (c x))^4}{4 b}\)

\(\Big \downarrow \) 7163

\(\displaystyle 2 i \left (\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2-i b \left (\frac {1}{2} i b \int \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )d\arccos (c x)-\frac {1}{2} i \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^3\right )-\frac {i (a+b \arccos (c x))^4}{4 b}\)

\(\Big \downarrow \) 2720

\(\displaystyle 2 i \left (\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2-i b \left (\frac {1}{4} b \int e^{-2 i \arccos (c x)} \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}-\frac {1}{2} i \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^3\right )-\frac {i (a+b \arccos (c x))^4}{4 b}\)

\(\Big \downarrow \) 7143

\(\displaystyle 2 i \left (\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2-i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (4,-e^{2 i \arccos (c x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^3\right )-\frac {i (a+b \arccos (c x))^4}{4 b}\)

Input:

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

Output:

((-1/4*I)*(a + b*ArcCos[c*x])^4)/b + (2*I)*((-1/2*I)*(a + b*ArcCos[c*x])^3 
*Log[1 + E^((2*I)*ArcCos[c*x])] + ((3*I)/2)*b*((I/2)*(a + b*ArcCos[c*x])^2 
*PolyLog[2, -E^((2*I)*ArcCos[c*x])] - I*b*((-1/2*I)*(a + b*ArcCos[c*x])*Po 
lyLog[3, -E^((2*I)*ArcCos[c*x])] + (b*PolyLog[4, -E^((2*I)*ArcCos[c*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 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 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 5137

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[ 
(a + b*x)^n*Tan[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0 
]
 

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]
 

rule 7163

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (160 ) = 320\).

Time = 0.40 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.56

method	result	size
parts	\(a^{3} \ln \left (x \right )+b^{3} \left (-\frac {i \arccos \left (c x \right )^{4}}{4}+\arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \arccos \left (c x \right )^{2} \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 \arccos \left (c x \right ) \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i \operatorname {polylog}\left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}\right )+3 a \,b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )+3 a^{2} b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\)	\(325\)
derivativedivides	\(a^{3} \ln \left (c x \right )+b^{3} \left (-\frac {i \arccos \left (c x \right )^{4}}{4}+\arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \arccos \left (c x \right )^{2} \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 \arccos \left (c x \right ) \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i \operatorname {polylog}\left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}\right )+3 a \,b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )+3 a^{2} b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\)	\(327\)
default	\(a^{3} \ln \left (c x \right )+b^{3} \left (-\frac {i \arccos \left (c x \right )^{4}}{4}+\arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \arccos \left (c x \right )^{2} \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 \arccos \left (c x \right ) \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i \operatorname {polylog}\left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}\right )+3 a \,b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )+3 a^{2} b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\)	\(327\)

Input:

int((a+b*arccos(c*x))^3/x,x,method=_RETURNVERBOSE)
 

Output:

a^3*ln(x)+b^3*(-1/4*I*arccos(c*x)^4+arccos(c*x)^3*ln(1+(c*x+I*(-c^2*x^2+1) 
^(1/2))^2)-3/2*I*arccos(c*x)^2*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+3/ 
2*arccos(c*x)*polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+3/4*I*polylog(4,-(c 
*x+I*(-c^2*x^2+1)^(1/2))^2))+3*a*b^2*(-1/3*I*arccos(c*x)^3+arccos(c*x)^2*l 
n(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-I*arccos(c*x)*polylog(2,-(c*x+I*(-c^2*x^ 
2+1)^(1/2))^2)+1/2*polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))^2))+3*a^2*b*(-1/2 
*I*arccos(c*x)^2+arccos(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-1/2*I*poly 
log(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2))
                                                                                    
                                                                                    
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

a^3*log(x) + integrate((b^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 + 
 3*a*b^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 3*a^2*b*arctan2(sq 
rt(c*x + 1)*sqrt(-c*x + 1), c*x))/x, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arccos (c x))^3}{x} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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