Integrand size = 8, antiderivative size = 81 \[ \int x \arccos (a+b x) \, dx=\frac {a \sqrt {1-(a+b x)^2}}{b^2}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{4 b^2}+\frac {1}{2} x^2 \arccos (a+b x)+\frac {\left (1+2 a^2\right ) \arcsin (a+b x)}{4 b^2} \] Output:
a*(1-(b*x+a)^2)^(1/2)/b^2-1/4*(b*x+a)*(1-(b*x+a)^2)^(1/2)/b^2+1/2*x^2*arcc os(b*x+a)+1/4*(2*a^2+1)*arcsin(b*x+a)/b^2
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85 \[ \int x \arccos (a+b x) \, dx=\frac {(3 a-b x) \sqrt {1-a^2-2 a b x-b^2 x^2}+2 b^2 x^2 \arccos (a+b x)+\left (1+2 a^2\right ) \arcsin (a+b x)}{4 b^2} \] Input:
Integrate[x*ArcCos[a + b*x],x]
Output:
((3*a - b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + 2*b^2*x^2*ArcCos[a + b*x] + (1 + 2*a^2)*ArcSin[a + b*x])/(4*b^2)
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5305, 25, 27, 5243, 497, 25, 455, 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.
\(\displaystyle \int x \arccos (a+b x) \, dx\) |
\(\Big \downarrow \) 5305 |
\(\displaystyle \frac {\int x \arccos (a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -x \arccos (a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int -b x \arccos (a+b x)d(a+b x)}{b^2}\) |
\(\Big \downarrow \) 5243 |
\(\displaystyle -\frac {-\frac {1}{2} \int \frac {b^2 x^2}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {1}{2} b^2 x^2 \arccos (a+b x)}{b^2}\) |
\(\Big \downarrow \) 497 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \int -\frac {2 a^2-3 (a+b x) a+1}{\sqrt {1-(a+b x)^2}}d(a+b x)+\frac {1}{2} b x \sqrt {1-(a+b x)^2}\right )-\frac {1}{2} b^2 x^2 \arccos (a+b x)}{b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} b x \sqrt {1-(a+b x)^2}-\frac {1}{2} \int \frac {2 a^2-3 (a+b x) a+1}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )-\frac {1}{2} b^2 x^2 \arccos (a+b x)}{b^2}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \left (-\left (2 a^2+1\right ) \int \frac {1}{\sqrt {1-(a+b x)^2}}d(a+b x)-3 a \sqrt {1-(a+b x)^2}\right )+\frac {1}{2} b x \sqrt {1-(a+b x)^2}\right )-\frac {1}{2} b^2 x^2 \arccos (a+b x)}{b^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \left (-\left (2 a^2+1\right ) \arcsin (a+b x)-3 a \sqrt {1-(a+b x)^2}\right )+\frac {1}{2} b x \sqrt {1-(a+b x)^2}\right )-\frac {1}{2} b^2 x^2 \arccos (a+b x)}{b^2}\) |
Input:
Int[x*ArcCos[a + b*x],x]
Output:
-((-1/2*(b^2*x^2*ArcCos[a + b*x]) + ((b*x*Sqrt[1 - (a + b*x)^2])/2 + (-3*a *Sqrt[1 - (a + b*x)^2] - (1 + 2*a^2)*ArcSin[a + b*x])/2)/2)/b^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
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]
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]
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]
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\arccos \left (b x +a \right ) a \left (b x +a \right )-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{4}+\frac {\arcsin \left (b x +a \right )}{4}+a \sqrt {1-\left (b x +a \right )^{2}}}{b^{2}}\) | \(78\) |
default | \(\frac {\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\arccos \left (b x +a \right ) a \left (b x +a \right )-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{4}+\frac {\arcsin \left (b x +a \right )}{4}+a \sqrt {1-\left (b x +a \right )^{2}}}{b^{2}}\) | \(78\) |
orering | \(-\frac {\left (-3 b^{3} x^{3}+a \,b^{2} x^{2}+7 a^{2} b x +3 a^{3}+2 b x -3 a \right ) \arccos \left (b x +a \right )}{4 b^{3} x}+\frac {\left (-b x +3 a \right ) \left (b x +a -1\right ) \left (b x +a +1\right ) \left (\arccos \left (b x +a \right )-\frac {x b}{\sqrt {1-\left (b x +a \right )^{2}}}\right )}{4 b^{3} x}\) | \(104\) |
parts | \(\frac {x^{2} \arccos \left (b x +a \right )}{2}+\frac {b \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{2}\) | \(179\) |
Input:
int(x*arccos(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/b^2*(1/2*arccos(b*x+a)*(b*x+a)^2-arccos(b*x+a)*a*(b*x+a)-1/4*(b*x+a)*(1- (b*x+a)^2)^(1/2)+1/4*arcsin(b*x+a)+a*(1-(b*x+a)^2)^(1/2))
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.73 \[ \int x \arccos (a+b x) \, dx=\frac {{\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arccos \left (b x + a\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x - 3 \, a\right )}}{4 \, b^{2}} \] Input:
integrate(x*arccos(b*x+a),x, algorithm="fricas")
Output:
1/4*((2*b^2*x^2 - 2*a^2 - 1)*arccos(b*x + a) - sqrt(-b^2*x^2 - 2*a*b*x - a ^2 + 1)*(b*x - 3*a))/b^2
Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.28 \[ \int x \arccos (a+b x) \, dx=\begin {cases} - \frac {a^{2} \operatorname {acos}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{4 b^{2}} + \frac {x^{2} \operatorname {acos}{\left (a + b x \right )}}{2} - \frac {x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{4 b} - \frac {\operatorname {acos}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {acos}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*acos(b*x+a),x)
Output:
Piecewise((-a**2*acos(a + b*x)/(2*b**2) + 3*a*sqrt(-a**2 - 2*a*b*x - b**2* x**2 + 1)/(4*b**2) + x**2*acos(a + b*x)/2 - x*sqrt(-a**2 - 2*a*b*x - b**2* x**2 + 1)/(4*b) - acos(a + b*x)/(4*b**2), Ne(b, 0)), (x**2*acos(a)/2, True ))
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (71) = 142\).
Time = 0.12 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.89 \[ \int x \arccos (a+b x) \, dx=\frac {1}{2} \, x^{2} \arccos \left (b x + a\right ) - \frac {1}{4} \, b {\left (\frac {3 \, a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{b^{2}} - \frac {{\left (a^{2} - 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3}}\right )} \] Input:
integrate(x*arccos(b*x+a),x, algorithm="maxima")
Output:
1/2*x^2*arccos(b*x + a) - 1/4*b*(3*a^2*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^3 + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*x/b^2 - (a^2 - 1)*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^3 - 3*sqrt(-b^2* x^2 - 2*a*b*x - a^2 + 1)*a/b^3)
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09 \[ \int x \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )}^{2} \arccos \left (b x + a\right )}{2 \, b^{2}} - \frac {{\left (b x + a\right )} a \arccos \left (b x + a\right )}{b^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}}{4 \, b^{2}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a}{b^{2}} - \frac {\arccos \left (b x + a\right )}{4 \, b^{2}} \] Input:
integrate(x*arccos(b*x+a),x, algorithm="giac")
Output:
1/2*(b*x + a)^2*arccos(b*x + a)/b^2 - (b*x + a)*a*arccos(b*x + a)/b^2 - 1/ 4*sqrt(-(b*x + a)^2 + 1)*(b*x + a)/b^2 + sqrt(-(b*x + a)^2 + 1)*a/b^2 - 1/ 4*arccos(b*x + a)/b^2
Timed out. \[ \int x \arccos (a+b x) \, dx=\int x\,\mathrm {acos}\left (a+b\,x\right ) \,d x \] Input:
int(x*acos(a + b*x),x)
Output:
int(x*acos(a + b*x), x)
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.10 \[ \int x \arccos (a+b x) \, dx=\frac {2 \mathit {acos} \left (b x +a \right ) b^{2} x^{2}+2 \mathit {asin} \left (b x +a \right ) a^{2}+\mathit {asin} \left (b x +a \right )+3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -4 a}{4 b^{2}} \] Input:
int(x*acos(b*x+a),x)
Output:
(2*acos(a + b*x)*b**2*x**2 + 2*asin(a + b*x)*a**2 + asin(a + b*x) + 3*sqrt ( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a - sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*b*x - 4*a)/(4*b**2)