\(\int \frac {x^2}{(a+b \arccos (c x))^3} \, dx\) [168]

Optimal result

Integrand size = 14, antiderivative size = 197 \[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {x}{b^2 c^2 (a+b \arccos (c x))}+\frac {3 x^3}{2 b^2 (a+b \arccos (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b^3 c^3}-\frac {9 \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{8 b^3 c^3}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{8 b^3 c^3} \] Output:

1/2*x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))^2-x/b^2/c^2/(a+b*arccos(c 
*x))+3/2*x^3/b^2/(a+b*arccos(c*x))-1/8*Ci((a+b*arccos(c*x))/b)*sin(a/b)/b^ 
3/c^3-9/8*Ci(3*(a+b*arccos(c*x))/b)*sin(3*a/b)/b^3/c^3+1/8*cos(a/b)*Si((a+ 
b*arccos(c*x))/b)/b^3/c^3+9/8*cos(3*a/b)*Si(3*(a+b*arccos(c*x))/b)/b^3/c^3
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\frac {\frac {4 b^2 x^2 \sqrt {1-c^2 x^2}}{c (a+b \arccos (c x))^2}-\frac {8 b x}{c^2 (a+b \arccos (c x))}+\frac {12 b x^3}{a+b \arccos (c x)}-\frac {\operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )}{c^3}-\frac {9 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )}{c^3}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )}{c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{c^3}}{8 b^3} \] Input:

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

Output:

((4*b^2*x^2*Sqrt[1 - c^2*x^2])/(c*(a + b*ArcCos[c*x])^2) - (8*b*x)/(c^2*(a 
 + b*ArcCos[c*x])) + (12*b*x^3)/(a + b*ArcCos[c*x]) - (CosIntegral[a/b + A 
rcCos[c*x]]*Sin[a/b])/c^3 - (9*CosIntegral[3*(a/b + ArcCos[c*x])]*Sin[(3*a 
)/b])/c^3 + (Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]])/c^3 + (9*Cos[(3*a)/b 
]*SinIntegral[3*(a/b + ArcCos[c*x])])/c^3)/(8*b^3)
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.26, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5145, 5223, 5135, 25, 3042, 3784, 25, 3042, 3780, 3783, 5147, 25, 4906, 2009}

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.

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

\(\Big \downarrow \) 5145

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

\(\Big \downarrow \) 5223

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

\(\Big \downarrow \) 5135

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 5147

\(\displaystyle \frac {3 c \left (\frac {3 \int -\frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^4}+\frac {x^3}{b c (a+b \arccos (c x))}\right )}{2 b}-\frac {\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )-\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c^2}+\frac {x}{b c (a+b \arccos (c x))}}{b c}+\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 c \left (\frac {x^3}{b c (a+b \arccos (c x))}-\frac {3 \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^4}\right )}{2 b}-\frac {\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )-\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c^2}+\frac {x}{b c (a+b \arccos (c x))}}{b c}+\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {3 c \left (\frac {x^3}{b c (a+b \arccos (c x))}-\frac {3 \int \left (\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{4 (a+b \arccos (c x))}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{4 (a+b \arccos (c x))}\right )d(a+b \arccos (c x))}{b^2 c^4}\right )}{2 b}-\frac {\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )-\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c^2}+\frac {x}{b c (a+b \arccos (c x))}}{b c}+\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 c \left (\frac {3 \left (-\frac {1}{4} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )-\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )+\frac {1}{4} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )+\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )\right )}{b^2 c^4}+\frac {x^3}{b c (a+b \arccos (c x))}\right )}{2 b}-\frac {\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )-\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c^2}+\frac {x}{b c (a+b \arccos (c x))}}{b c}+\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}\)

Input:

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

Output:

(x^2*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcCos[c*x])^2) - (x/(b*c*(a + b*Arc 
Cos[c*x])) + (-(CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b]) + Cos[a/b]*Si 
nIntegral[(a + b*ArcCos[c*x])/b])/(b^2*c^2))/(b*c) + (3*c*(x^3/(b*c*(a + b 
*ArcCos[c*x])) + (3*(-1/4*(CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b]) - 
(CosIntegral[(3*(a + b*ArcCos[c*x]))/b]*Sin[(3*a)/b])/4 + (Cos[a/b]*SinInt 
egral[(a + b*ArcCos[c*x])/b])/4 + (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcC 
os[c*x]))/b])/4))/(b^2*c^4)))/(2*b)
 

Defintions of rubi rules used

rule 25

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

rule 2009

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

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 5135

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

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 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]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {\frac {\sin \left (3 \arccos \left (c x \right )\right )}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arccos \left (c x \right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{8}-\frac {9 \arccos \left (c x \right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{8}-\frac {9 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{8}+\frac {3 \cos \left (3 \arccos \left (c x \right )\right ) b}{8}}{\left (a +b \arccos \left (c x \right )\right ) b^{3}}+\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +c x b}{8 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{3}}\)	\(290\)
default	\(\frac {\frac {\sin \left (3 \arccos \left (c x \right )\right )}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arccos \left (c x \right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{8}-\frac {9 \arccos \left (c x \right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{8}-\frac {9 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{8}+\frac {3 \cos \left (3 \arccos \left (c x \right )\right ) b}{8}}{\left (a +b \arccos \left (c x \right )\right ) b^{3}}+\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +c x b}{8 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{3}}\)	\(290\)

Input:

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

Output:

1/c^3*(1/8*sin(3*arccos(c*x))/(a+b*arccos(c*x))^2/b+3/8*(3*arccos(c*x)*Si( 
3*arccos(c*x)+3*a/b)*cos(3*a/b)*b-3*arccos(c*x)*Ci(3*arccos(c*x)+3*a/b)*si 
n(3*a/b)*b+3*Si(3*arccos(c*x)+3*a/b)*cos(3*a/b)*a-3*Ci(3*arccos(c*x)+3*a/b 
)*sin(3*a/b)*a+cos(3*arccos(c*x))*b)/(a+b*arccos(c*x))/b^3+1/8*(-c^2*x^2+1 
)^(1/2)/(a+b*arccos(c*x))^2/b+1/8*(arccos(c*x)*Si(arccos(c*x)+a/b)*cos(a/b 
)*b-arccos(c*x)*Ci(arccos(c*x)+a/b)*sin(a/b)*b+Si(arccos(c*x)+a/b)*cos(a/b 
)*a-Ci(arccos(c*x)+a/b)*sin(a/b)*a+c*x*b)/(a+b*arccos(c*x))/b^3)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*(3*a*c^2*x^3 + sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x^2 - 2*a*x + (3*b*c^2 
*x^3 - 2*b*x)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) - 2*(b^4*c^2*arct 
an2(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(1/2*(9*c^2*x^2 - 2)/(b^3* 
c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b^2*c^2), x))/(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)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (183) = 366\).

Time = 0.22 (sec) , antiderivative size = 1479, normalized size of antiderivative = 7.51 \[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

3/2*b^2*c^3*x^3*arccos(c*x)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c* 
x) + a^2*b^3*c^3) + 3/2*a*b*c^3*x^3/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*a 
rccos(c*x) + a^2*b^3*c^3) - 9/2*b^2*arccos(c*x)^2*cos(a/b)^2*cos_integral( 
3*a/b + 3*arccos(c*x))*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arcco 
s(c*x) + a^2*b^3*c^3) + 9/2*b^2*arccos(c*x)^2*cos(a/b)^3*sin_integral(3*a/ 
b + 3*arccos(c*x))/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2* 
b^3*c^3) - 9*a*b*arccos(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*arccos(c*x) 
)*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) 
 + 9*a*b*arccos(c*x)*cos(a/b)^3*sin_integral(3*a/b + 3*arccos(c*x))/(b^5*c 
^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 1/2*sqrt(-c^2* 
x^2 + 1)*b^2*c^2*x^2/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^ 
2*b^3*c^3) + 9/8*b^2*arccos(c*x)^2*cos_integral(3*a/b + 3*arccos(c*x))*sin 
(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - 9/ 
2*a^2*cos(a/b)^2*cos_integral(3*a/b + 3*arccos(c*x))*sin(a/b)/(b^5*c^3*arc 
cos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - 1/8*b^2*arccos(c*x)^ 
2*cos_integral(a/b + arccos(c*x))*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^ 
4*c^3*arccos(c*x) + a^2*b^3*c^3) - 27/8*b^2*arccos(c*x)^2*cos(a/b)*sin_int 
egral(3*a/b + 3*arccos(c*x))/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c 
*x) + a^2*b^3*c^3) + 9/2*a^2*cos(a/b)^3*sin_integral(3*a/b + 3*arccos(c*x) 
)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 1/8...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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