3.2.69 \(\int \frac {x}{(a+b \arccos (c x))^3} \, dx\) [169]

3.2.69.1 Optimal result

Integrand size = 12, antiderivative size = 130 \[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x^2}{b^2 (a+b \arccos (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{b^3 c^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^3 c^2} \]

output

-1/2/b^2/c^2/(a+b*arccos(c*x))+x^2/b^2/(a+b*arccos(c*x))+cos(2*a/b)*Si(2*( 
a+b*arccos(c*x))/b)/b^3/c^2-Ci(2*(a+b*arccos(c*x))/b)*sin(2*a/b)/b^3/c^2+1 
/2*x*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))^2
 

3.2.69.2 Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\frac {\frac {b^2 c x \sqrt {1-c^2 x^2}}{(a+b \arccos (c x))^2}+\frac {b \left (-1+2 c^2 x^2\right )}{a+b \arccos (c x)}-2 \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )+2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{2 b^3 c^2} \]

input

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

output

((b^2*c*x*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x])^2 + (b*(-1 + 2*c^2*x^2))/ 
(a + b*ArcCos[c*x]) - 2*CosIntegral[2*(a/b + ArcCos[c*x])]*Sin[(2*a)/b] + 
2*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcCos[c*x])])/(2*b^3*c^2)
 

3.2.69.3 Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {5145, 5153, 5223, 5147, 25, 4906, 27, 3042, 3784, 25, 3042, 3780, 3783}

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.

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

\(\Big \downarrow \) 5145

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5223

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

\(\Big \downarrow \) 5147

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

\(\displaystyle \frac {c \left (\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}+\frac {\pi }{2}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^3}+\frac {x^2}{b c (a+b \arccos (c x))}\right )}{b}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}\)

\(\Big \downarrow \) 3783

\(\displaystyle \frac {c \left (\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )-\sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^2 c^3}+\frac {x^2}{b c (a+b \arccos (c x))}\right )}{b}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}\)

input

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

output

(x*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcCos[c*x])^2) - 1/(2*b^2*c^2*(a + b* 
ArcCos[c*x])) + (c*(x^2/(b*c*(a + b*ArcCos[c*x])) + (-(CosIntegral[(2*(a + 
 b*ArcCos[c*x]))/b]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*(a + b*Arc 
Cos[c*x]))/b])/(b^2*c^3)))/b
 

3.2.69.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5145

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[( 
-x^m)*Sqrt[1 - c^2*x^2]*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] + ( 
-Simp[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcCos[c*x])^(n + 1)/ 
Sqrt[1 - c^2*x^2]), x], x] + Simp[m/(b*c*(n + 1))   Int[x^(m - 1)*((a + b*A 
rcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && I 
GtQ[m, 0] && LtQ[n, -2]
 

rule 5147

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- 
(b*c^(m + 1))^(-1)   Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b], x], x 
, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
 

rule 5153

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] 
]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 
2*d + e, 0] && NeQ[n, -1]
 

rule 5223

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) 
+ (e_.)*(x_)^2], x_Symbol] :> Simp[(-(f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c 
^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] + Simp[f*(m/(b*c*( 
n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*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] && LtQ[n, -1]
 

3.2.69.4 Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {\frac {\sin \left (2 \arccos \left (c x \right )\right )}{4 \left (a +b \arccos \left (c x \right )\right )^{2} b}-\frac {2 \arccos \left (c x \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) b -2 \arccos \left (c x \right ) \operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +2 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) a -2 \,\operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -\cos \left (2 \arccos \left (c x \right )\right ) b}{2 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{2}}\)	\(158\)
default	\(\frac {\frac {\sin \left (2 \arccos \left (c x \right )\right )}{4 \left (a +b \arccos \left (c x \right )\right )^{2} b}-\frac {2 \arccos \left (c x \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) b -2 \arccos \left (c x \right ) \operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +2 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) a -2 \,\operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -\cos \left (2 \arccos \left (c x \right )\right ) b}{2 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{2}}\)	\(158\)

input

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

output

1/c^2*(1/4*sin(2*arccos(c*x))/(a+b*arccos(c*x))^2/b-1/2*(2*arccos(c*x)*sin 
(2*a/b)*Ci(2*arccos(c*x)+2*a/b)*b-2*arccos(c*x)*Si(2*arccos(c*x)+2*a/b)*co 
s(2*a/b)*b+2*sin(2*a/b)*Ci(2*arccos(c*x)+2*a/b)*a-2*Si(2*arccos(c*x)+2*a/b 
)*cos(2*a/b)*a-cos(2*arccos(c*x))*b)/(a+b*arccos(c*x))/b^3)
 

3.2.69.5 Fricas [F]

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

input

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

output

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

3.2.69.6 Sympy [F]

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

input

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

output

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

3.2.69.7 Maxima [F]

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

input

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

output

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

3.2.69.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (124) = 248\).

Time = 0.33 (sec) , antiderivative size = 860, normalized size of antiderivative = 6.62 \[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\text {Too large to display} \]

input

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

output

b^2*c^2*x^2*arccos(c*x)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + 
 a^2*b^3*c^2) - 2*b^2*arccos(c*x)^2*cos(a/b)*cos_integral(2*a/b + 2*arccos 
(c*x))*sin(a/b)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3 
*c^2) + 2*b^2*arccos(c*x)^2*cos(a/b)^2*sin_integral(2*a/b + 2*arccos(c*x)) 
/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) + a*b*c^2 
*x^2/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 4*a 
*b*arccos(c*x)*cos(a/b)*cos_integral(2*a/b + 2*arccos(c*x))*sin(a/b)/(b^5* 
c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) + 4*a*b*arccos( 
c*x)*cos(a/b)^2*sin_integral(2*a/b + 2*arccos(c*x))/(b^5*c^2*arccos(c*x)^2 
 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 2*a^2*cos(a/b)*cos_integral(2* 
a/b + 2*arccos(c*x))*sin(a/b)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos( 
c*x) + a^2*b^3*c^2) - b^2*arccos(c*x)^2*sin_integral(2*a/b + 2*arccos(c*x) 
)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) + 2*a^2* 
cos(a/b)^2*sin_integral(2*a/b + 2*arccos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2* 
a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) + 1/2*sqrt(-c^2*x^2 + 1)*b^2*c*x/(b^5 
*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 2*a*b*arccos 
(c*x)*sin_integral(2*a/b + 2*arccos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4 
*c^2*arccos(c*x) + a^2*b^3*c^2) - 1/2*b^2*arccos(c*x)/(b^5*c^2*arccos(c*x) 
^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - a^2*sin_integral(2*a/b + 2*a 
rccos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*...
 

3.2.69.9 Mupad [F(-1)]

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

input

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

output

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