Integrand size = 22, antiderivative size = 67 \[ \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {6 x}{a}+\frac {6 \sqrt {1-a^2 x^2} \arccos (a x)}{a^2}+\frac {3 x \arccos (a x)^2}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2} \] Output:
-6*x/a+6*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a^2+3*x*arccos(a*x)^2/a-(-a^2*x^2+ 1)^(1/2)*arccos(a*x)^3/a^2
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {6 a x+6 \sqrt {1-a^2 x^2} \arccos (a x)-3 a x \arccos (a x)^2-\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2} \] Input:
Integrate[(x*ArcCos[a*x]^3)/Sqrt[1 - a^2*x^2],x]
Output:
(6*a*x + 6*Sqrt[1 - a^2*x^2]*ArcCos[a*x] - 3*a*x*ArcCos[a*x]^2 - Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a^2
Time = 0.40 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5183, 5131, 5183, 24}
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.
\(\displaystyle \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle -\frac {3 \int \arccos (a x)^2dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}\) |
\(\Big \downarrow \) 5131 |
\(\displaystyle -\frac {3 \left (2 a \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+x \arccos (a x)^2\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle -\frac {3 \left (2 a \left (-\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}\right )+x \arccos (a x)^2\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\) |
Input:
Int[(x*ArcCos[a*x]^3)/Sqrt[1 - a^2*x^2],x]
Output:
-((Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a^2) - (3*(x*ArcCos[a*x]^2 + 2*a*(-(x/ a) - (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a^2)))/a
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cCos[c*x])^n, x] + Simp[b*c*n Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.60
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\arccos \left (a x \right )^{3} a^{2} x^{2}-\arccos \left (a x \right )^{3}-3 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x -6 a^{2} x^{2} \arccos \left (a x \right )+6 \arccos \left (a x \right )+6 \sqrt {-a^{2} x^{2}+1}\, a x \right )}{a^{2} \left (a^{2} x^{2}-1\right )}\) | \(107\) |
orering | \(\frac {\left (a^{4} x^{4}-8 a^{2} x^{2}+8\right ) \arccos \left (a x \right )^{3}}{a^{4} x^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\left (a^{4} x^{4}-6 a^{2} x^{2}+8\right ) \left (\frac {\arccos \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}}-\frac {3 x \arccos \left (a x \right )^{2} a}{-a^{2} x^{2}+1}+\frac {x^{2} \arccos \left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{x^{2} a^{4}}-\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}+2\right ) \left (-\frac {6 \arccos \left (a x \right )^{2} a}{-a^{2} x^{2}+1}+\frac {3 \arccos \left (a x \right )^{3} x \,a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 x \arccos \left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {9 x^{2} \arccos \left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {3 x^{3} \arccos \left (a x \right )^{3} a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{x \,a^{4}}-\frac {\left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (\frac {18 \arccos \left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {39 \arccos \left (a x \right )^{2} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {18 \arccos \left (a x \right )^{3} x^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {3 \arccos \left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {6 x \,a^{3}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {36 x^{2} \arccos \left (a x \right ) a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {45 x^{3} \arccos \left (a x \right )^{2} a^{5}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {15 x^{4} \arccos \left (a x \right )^{3} a^{6}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{a^{4}}\) | \(491\) |
Input:
int(x*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/a^2*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)*(arccos(a*x)^3*a^2*x^2-arccos(a*x)^3 -3*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)*a*x-6*a^2*x^2*arccos(a*x)+6*arccos(a*x )+6*(-a^2*x^2+1)^(1/2)*a*x)
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.69 \[ \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 \, a x \arccos \left (a x\right )^{2} - 6 \, a x + \sqrt {-a^{2} x^{2} + 1} {\left (\arccos \left (a x\right )^{3} - 6 \, \arccos \left (a x\right )\right )}}{a^{2}} \] Input:
integrate(x*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
-(3*a*x*arccos(a*x)^2 - 6*a*x + sqrt(-a^2*x^2 + 1)*(arccos(a*x)^3 - 6*arcc os(a*x)))/a^2
Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.01 \[ \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} - \frac {3 x \operatorname {acos}^{2}{\left (a x \right )}}{a} + \frac {6 x}{a} - \frac {\sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{a^{2}} + \frac {6 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{2}}{16} & \text {otherwise} \end {cases} \] Input:
integrate(x*acos(a*x)**3/(-a**2*x**2+1)**(1/2),x)
Output:
Piecewise((-3*x*acos(a*x)**2/a + 6*x/a - sqrt(-a**2*x**2 + 1)*acos(a*x)**3 /a**2 + 6*sqrt(-a**2*x**2 + 1)*acos(a*x)/a**2, Ne(a, 0)), (pi**3*x**2/16, True))
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 \, x \arccos \left (a x\right )^{2}}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a^{2}} + \frac {6 \, {\left (x + \frac {\sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a}\right )}}{a} \] Input:
integrate(x*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
-3*x*arccos(a*x)^2/a - sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/a^2 + 6*(x + sqrt( -a^2*x^2 + 1)*arccos(a*x)/a)/a
Time = 0.16 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a^{2}} - \frac {3 \, {\left (x \arccos \left (a x\right )^{2} - 2 \, x - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a}\right )}}{a} \] Input:
integrate(x*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
-sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/a^2 - 3*(x*arccos(a*x)^2 - 2*x - 2*sqrt( -a^2*x^2 + 1)*arccos(a*x)/a)/a
Timed out. \[ \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x\,{\mathrm {acos}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x*acos(a*x)^3)/(1 - a^2*x^2)^(1/2),x)
Output:
int((x*acos(a*x)^3)/(1 - a^2*x^2)^(1/2), x)
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.82 \[ \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {-\sqrt {-a^{2} x^{2}+1}\, \mathit {acos} \left (a x \right )^{3}-3 \mathit {acos} \left (a x \right )^{2} a x +6 \sqrt {-a^{2} x^{2}+1}\, \mathit {acos} \left (a x \right )+6 a x}{a^{2}} \] Input:
int(x*acos(a*x)^3/(-a^2*x^2+1)^(1/2),x)
Output:
( - sqrt( - a**2*x**2 + 1)*acos(a*x)**3 - 3*acos(a*x)**2*a*x + 6*sqrt( - a **2*x**2 + 1)*acos(a*x) + 6*a*x)/a**2