Integrand size = 22, antiderivative size = 67 \[ \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {6 x}{a}+\frac {6 \sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}+\frac {3 x \arcsin (a x)^2}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2} \] Output:
-6*x/a+6*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/a^2+3*x*arcsin(a*x)^2/a-(-a^2*x^2+ 1)^(1/2)*arcsin(a*x)^3/a^2
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {-6 a x+6 \sqrt {1-a^2 x^2} \arcsin (a x)+3 a x \arcsin (a x)^2-\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2} \] Input:
Integrate[(x*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]
Output:
(-6*a*x + 6*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 3*a*x*ArcSin[a*x]^2 - Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2
Time = 0.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5182, 5130, 5182, 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.
\(\displaystyle \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle \frac {3 \int \arcsin (a x)^2dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle \frac {3 \left (x \arcsin (a x)^2-2 a \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle \frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\) |
Input:
Int[(x*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]
Output:
-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2) + (3*(x*ArcSin[a*x]^2 - 2*a*(x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2)))/a
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[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*ArcSin[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 = 0.38 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.60
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\arcsin \left (a x \right )^{3} a^{2} x^{2}-\arcsin \left (a x \right )^{3}+3 \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{2} a x -6 \arcsin \left (a x \right ) a^{2} x^{2}+6 \arcsin \left (a x \right )-6 \sqrt {-a^{2} x^{2}+1}\, x a \right )}{a^{2} \left (a^{2} x^{2}-1\right )}\) | \(107\) |
orering | \(\frac {\left (a^{4} x^{4}-8 a^{2} x^{2}+8\right ) \arcsin \left (a x \right )^{3}}{a^{4} x^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\left (a^{4} x^{4}-6 a^{2} x^{2}+8\right ) \left (\frac {\arcsin \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}}+\frac {3 x \arcsin \left (a x \right )^{2} a}{-a^{2} x^{2}+1}+\frac {x^{2} \arcsin \left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{x^{2} a^{4}}-\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}+2\right ) \left (\frac {6 \arcsin \left (a x \right )^{2} a}{-a^{2} x^{2}+1}+\frac {3 \arcsin \left (a x \right )^{3} x \,a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 x \arcsin \left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {9 x^{2} \arcsin \left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {3 x^{3} \arcsin \left (a x \right )^{3} a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{x \,a^{4}}-\frac {\left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (\frac {18 \arcsin \left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {39 \arcsin \left (a x \right )^{2} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {18 \arcsin \left (a x \right )^{3} x^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {3 \arcsin \left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 x \,a^{3}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {36 x^{2} \arcsin \left (a x \right ) a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {45 x^{3} \arcsin \left (a x \right )^{2} a^{5}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {15 x^{4} \arcsin \left (a x \right )^{3} a^{6}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{a^{4}}\) | \(491\) |
Input:
int(x*arcsin(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)*(arcsin(a*x)^3*a^2*x^2-arcsin(a*x)^3 +3*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^2*a*x-6*arcsin(a*x)*a^2*x^2+6*arcsin(a*x )-6*(-a^2*x^2+1)^(1/2)*x*a)
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.69 \[ \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 \, a x \arcsin \left (a x\right )^{2} - 6 \, a x - \sqrt {-a^{2} x^{2} + 1} {\left (\arcsin \left (a x\right )^{3} - 6 \, \arcsin \left (a x\right )\right )}}{a^{2}} \] Input:
integrate(x*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
(3*a*x*arcsin(a*x)^2 - 6*a*x - sqrt(-a^2*x^2 + 1)*(arcsin(a*x)^3 - 6*arcsi n(a*x)))/a^2
Time = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {3 x \operatorname {asin}^{2}{\left (a x \right )}}{a} - \frac {6 x}{a} - \frac {\sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{a^{2}} + \frac {6 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x*asin(a*x)**3/(-a**2*x**2+1)**(1/2),x)
Output:
Piecewise((3*x*asin(a*x)**2/a - 6*x/a - sqrt(-a**2*x**2 + 1)*asin(a*x)**3/ a**2 + 6*sqrt(-a**2*x**2 + 1)*asin(a*x)/a**2, Ne(a, 0)), (0, True))
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 \, x \arcsin \left (a x\right )^{2}}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2}} - \frac {6 \, {\left (x - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a} \] Input:
integrate(x*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
3*x*arcsin(a*x)^2/a - sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/a^2 - 6*(x - sqrt(- a^2*x^2 + 1)*arcsin(a*x)/a)/a
Time = 0.16 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2}} + \frac {3 \, {\left (x \arcsin \left (a x\right )^{2} - 2 \, x + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a} \] Input:
integrate(x*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
-sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/a^2 + 3*(x*arcsin(a*x)^2 - 2*x + 2*sqrt( -a^2*x^2 + 1)*arcsin(a*x)/a)/a
Timed out. \[ \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x\,{\mathrm {asin}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x*asin(a*x)^3)/(1 - a^2*x^2)^(1/2),x)
Output:
int((x*asin(a*x)^3)/(1 - a^2*x^2)^(1/2), x)
Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.82 \[ \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {-\sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )^{3}+3 \mathit {asin} \left (a x \right )^{2} a x +6 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-6 a x}{a^{2}} \] Input:
int(x*asin(a*x)^3/(-a^2*x^2+1)^(1/2),x)
Output:
( - sqrt( - a**2*x**2 + 1)*asin(a*x)**3 + 3*asin(a*x)**2*a*x + 6*sqrt( - a **2*x**2 + 1)*asin(a*x) - 6*a*x)/a**2