\(\int x^2 \arccos (a+b x) \, dx\) [2]

Optimal result

Integrand size = 10, antiderivative size = 114 \[ \int x^2 \arccos (a+b x) \, dx=-\frac {\left (1+3 a^2\right ) \sqrt {1-(a+b x)^2}}{3 b^3}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{2 b^3}+\frac {\left (1-(a+b x)^2\right )^{3/2}}{9 b^3}+\frac {1}{3} x^3 \arccos (a+b x)-\frac {a \left (3+2 a^2\right ) \arcsin (a+b x)}{6 b^3} \] Output:

-1/3*(3*a^2+1)*(1-(b*x+a)^2)^(1/2)/b^3+1/2*a*(b*x+a)*(1-(b*x+a)^2)^(1/2)/b 
^3+1/9*(1-(b*x+a)^2)^(3/2)/b^3+1/3*x^3*arccos(b*x+a)-1/6*a*(2*a^2+3)*arcsi 
n(b*x+a)/b^3
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73 \[ \int x^2 \arccos (a+b x) \, dx=-\frac {\sqrt {1-a^2-2 a b x-b^2 x^2} \left (4+11 a^2-5 a b x+2 b^2 x^2\right )-6 b^3 x^3 \arccos (a+b x)+3 a \left (3+2 a^2\right ) \arcsin (a+b x)}{18 b^3} \] Input:

Integrate[x^2*ArcCos[a + b*x],x]
 

Output:

-1/18*(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(4 + 11*a^2 - 5*a*b*x + 2*b^2*x^2 
) - 6*b^3*x^3*ArcCos[a + b*x] + 3*a*(3 + 2*a^2)*ArcSin[a + b*x])/b^3
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5305, 27, 5243, 497, 25, 676, 223}

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.

\(\displaystyle \int x^2 \arccos (a+b x) \, dx\)

\(\Big \downarrow \) 5305

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5243

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

\(\Big \downarrow \) 497

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{3} \int \frac {b x \left (3 a^2-5 (a+b x) a+2\right )}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {1}{3} b^2 x^2 \sqrt {1-(a+b x)^2}\right )+\frac {1}{3} b^3 x^3 \arccos (a+b x)}{b^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {1}{3} \int -\frac {b x \left (3 a^2-5 (a+b x) a+2\right )}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {1}{3} b^2 x^2 \sqrt {1-(a+b x)^2}\right )+\frac {1}{3} b^3 x^3 \arccos (a+b x)}{b^3}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{3} \left (-\frac {3}{2} a \left (2 a^2+3\right ) \int \frac {1}{\sqrt {1-(a+b x)^2}}d(a+b x)-2 \left (4 a^2+1\right ) \sqrt {1-(a+b x)^2}+\frac {5}{2} a (a+b x) \sqrt {1-(a+b x)^2}\right )-\frac {1}{3} b^2 x^2 \sqrt {1-(a+b x)^2}\right )+\frac {1}{3} b^3 x^3 \arccos (a+b x)}{b^3}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{3} \left (-\frac {3}{2} a \left (2 a^2+3\right ) \arcsin (a+b x)-2 \left (4 a^2+1\right ) \sqrt {1-(a+b x)^2}+\frac {5}{2} a (a+b x) \sqrt {1-(a+b x)^2}\right )-\frac {1}{3} b^2 x^2 \sqrt {1-(a+b x)^2}\right )+\frac {1}{3} b^3 x^3 \arccos (a+b x)}{b^3}\)

Input:

Int[x^2*ArcCos[a + b*x],x]
 

Output:

((b^3*x^3*ArcCos[a + b*x])/3 + (-1/3*(b^2*x^2*Sqrt[1 - (a + b*x)^2]) + (-2 
*(1 + 4*a^2)*Sqrt[1 - (a + b*x)^2] + (5*a*(a + b*x)*Sqrt[1 - (a + b*x)^2]) 
/2 - (3*a*(3 + 2*a^2)*ArcSin[a + b*x])/2)/3)/3)/b^3
 

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 497

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b 
*(n + 2*p + 1))   Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 
 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n 
, p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p 
+ 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
 

rule 676

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim 
p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p 
+ 3))/(c*(2*p + 3))   Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g 
, p}, x] &&  !LeQ[p, -1]
 

rule 5243

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

rule 5305

Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.27

method	result	size
orering	\(\frac {\left (10 b^{4} x^{4}-2 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+40 a^{3} b x +22 a^{4}+4 b^{2} x^{2}+35 a b x -14 a^{2}-8\right ) \arccos \left (b x +a \right )}{18 b^{4} x}-\frac {\left (2 b^{2} x^{2}-5 a b x +11 a^{2}+4\right ) \left (b x +a -1\right ) \left (b x +a +1\right ) \left (2 x \arccos \left (b x +a \right )-\frac {x^{2} b}{\sqrt {1-\left (b x +a \right )^{2}}}\right )}{18 b^{4} x^{2}}\)	\(145\)
derivativedivides	\(\frac {-\frac {\arccos \left (b x +a \right ) a^{3}}{3}+\arccos \left (b x +a \right ) a^{2} \left (b x +a \right )-\arccos \left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\frac {a^{3} \arcsin \left (b x +a \right )}{3}-\frac {\left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}}{9}-\frac {2 \sqrt {1-\left (b x +a \right )^{2}}}{9}-a \left (-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{2}+\frac {\arcsin \left (b x +a \right )}{2}\right )-a^{2} \sqrt {1-\left (b x +a \right )^{2}}}{b^{3}}\)	\(161\)
default	\(\frac {-\frac {\arccos \left (b x +a \right ) a^{3}}{3}+\arccos \left (b x +a \right ) a^{2} \left (b x +a \right )-\arccos \left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\frac {a^{3} \arcsin \left (b x +a \right )}{3}-\frac {\left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}}{9}-\frac {2 \sqrt {1-\left (b x +a \right )^{2}}}{9}-a \left (-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{2}+\frac {\arcsin \left (b x +a \right )}{2}\right )-a^{2} \sqrt {1-\left (b x +a \right )^{2}}}{b^{3}}\)	\(161\)
parts	\(\frac {x^{3} \arccos \left (b x +a \right )}{3}+\frac {b \left (-\frac {x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 b^{2}}-\frac {5 a \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}+\frac {2 \left (-a^{2}+1\right ) \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{3 b^{2}}\right )}{3}\)	\(303\)

Input:

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

Output:

1/18*(10*b^4*x^4-2*a*b^3*x^3+6*a^2*b^2*x^2+40*a^3*b*x+22*a^4+4*b^2*x^2+35* 
a*b*x-14*a^2-8)/b^4/x*arccos(b*x+a)-1/18*(2*b^2*x^2-5*a*b*x+11*a^2+4)/b^4/ 
x^2*(b*x+a-1)*(b*x+a+1)*(2*x*arccos(b*x+a)-x^2*b/(1-(b*x+a)^2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int x^2 \arccos (a+b x) \, dx=\frac {3 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \arccos \left (b x + a\right ) - {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{18 \, b^{3}} \] Input:

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

Output:

1/18*(3*(2*b^3*x^3 + 2*a^3 + 3*a)*arccos(b*x + a) - (2*b^2*x^2 - 5*a*b*x + 
 11*a^2 + 4)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/b^3
 

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.49 \[ \int x^2 \arccos (a+b x) \, dx=\begin {cases} \frac {a^{3} \operatorname {acos}{\left (a + b x \right )}}{3 b^{3}} - \frac {11 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{18 b^{3}} + \frac {5 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{18 b^{2}} + \frac {a \operatorname {acos}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {acos}{\left (a + b x \right )}}{3} - \frac {x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{9 b} - \frac {2 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {acos}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \] Input:

integrate(x**2*acos(b*x+a),x)
 

Output:

Piecewise((a**3*acos(a + b*x)/(3*b**3) - 11*a**2*sqrt(-a**2 - 2*a*b*x - b* 
*2*x**2 + 1)/(18*b**3) + 5*a*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(18*b 
**2) + a*acos(a + b*x)/(2*b**3) + x**3*acos(a + b*x)/3 - x**2*sqrt(-a**2 - 
 2*a*b*x - b**2*x**2 + 1)/(9*b) - 2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/ 
(9*b**3), Ne(b, 0)), (x**3*acos(a)/3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (98) = 196\).

Time = 0.11 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.93 \[ \int x^2 \arccos (a+b x) \, dx=\frac {1}{3} \, x^{3} \arccos \left (b x + a\right ) - \frac {1}{18} \, b {\left (\frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x^{2}}{b^{2}} - \frac {15 \, a^{3} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{4}} - \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x}{b^{3}} + \frac {9 \, {\left (a^{2} - 1\right )} a \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{4}} + \frac {15 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{4}} - \frac {4 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{b^{4}}\right )} \] Input:

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

Output:

1/3*x^3*arccos(b*x + a) - 1/18*b*(2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*x^2 
/b^2 - 15*a^3*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^4 - 5 
*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a*x/b^3 + 9*(a^2 - 1)*a*arcsin(-(b^2*x 
 + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^4 + 15*sqrt(-b^2*x^2 - 2*a*b*x - 
a^2 + 1)*a^2/b^4 - 4*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1)/b^4)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.37 \[ \int x^2 \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )}^{3} \arccos \left (b x + a\right )}{3 \, b^{3}} - \frac {{\left (b x + a\right )}^{2} a \arccos \left (b x + a\right )}{b^{3}} + \frac {{\left (b x + a\right )} a^{2} \arccos \left (b x + a\right )}{b^{3}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}^{2}}{9 \, b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{2 \, b^{3}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{b^{3}} + \frac {a \arccos \left (b x + a\right )}{2 \, b^{3}} - \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1}}{9 \, b^{3}} \] Input:

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

Output:

1/3*(b*x + a)^3*arccos(b*x + a)/b^3 - (b*x + a)^2*a*arccos(b*x + a)/b^3 + 
(b*x + a)*a^2*arccos(b*x + a)/b^3 - 1/9*sqrt(-(b*x + a)^2 + 1)*(b*x + a)^2 
/b^3 + 1/2*sqrt(-(b*x + a)^2 + 1)*(b*x + a)*a/b^3 - sqrt(-(b*x + a)^2 + 1) 
*a^2/b^3 + 1/2*a*arccos(b*x + a)/b^3 - 2/9*sqrt(-(b*x + a)^2 + 1)/b^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.26 \[ \int x^2 \arccos (a+b x) \, dx=\frac {6 \mathit {acos} \left (b x +a \right ) b^{3} x^{3}-6 \mathit {asin} \left (b x +a \right ) a^{3}-9 \mathit {asin} \left (b x +a \right ) a -11 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}+5 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x -2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{2} x^{2}-4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{18 b^{3}} \] Input:

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

Output:

(6*acos(a + b*x)*b**3*x**3 - 6*asin(a + b*x)*a**3 - 9*asin(a + b*x)*a - 11 
*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**2 + 5*sqrt( - a**2 - 2*a*b*x - 
 b**2*x**2 + 1)*a*b*x - 2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*b**2*x** 
2 - 4*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1))/(18*b**3)