Integrand size = 22, antiderivative size = 88 \[ \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 x^2}{16 a^3}+\frac {x^4}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {3 \arcsin (a x)^2}{16 a^5} \] Output:
3/16*x^2/a^3+1/16*x^4/a-3/8*x*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/a^4-1/4*x^3*( -a^2*x^2+1)^(1/2)*arcsin(a*x)/a^2+3/16*arcsin(a*x)^2/a^5
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {a^2 x^2 \left (3+a^2 x^2\right )-2 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arcsin (a x)+3 \arcsin (a x)^2}{16 a^5} \] Input:
Integrate[(x^4*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]
Output:
(a^2*x^2*(3 + a^2*x^2) - 2*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a* x] + 3*ArcSin[a*x]^2)/(16*a^5)
Time = 0.48 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5210, 15, 5210, 15, 5152}
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.
\(\displaystyle \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int x^3dx}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle -\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {3 \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}+\frac {x^4}{16 a}\) |
Input:
Int[(x^4*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]
Output:
x^4/(16*a) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(4*a^2) + (3*(x^2/(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*a^2) + ArcSin[a*x]^2/(4*a^3)))/(4*a^ 2)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 0.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {-16 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+4 a^{4} x^{4}-24 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +12 a^{2} x^{2}+12 \arcsin \left (a x \right )^{2}+9}{64 a^{5}}\) | \(76\) |
Input:
int(x^4*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/64*(-16*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a^3*x^3+4*a^4*x^4-24*arcsin(a*x)* (-a^2*x^2+1)^(1/2)*a*x+12*a^2*x^2+12*arcsin(a*x)^2+9)/a^5
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {a^{4} x^{4} + 3 \, a^{2} x^{2} - 2 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right ) + 3 \, \arcsin \left (a x\right )^{2}}{16 \, a^{5}} \] Input:
integrate(x^4*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
1/16*(a^4*x^4 + 3*a^2*x^2 - 2*(2*a^3*x^3 + 3*a*x)*sqrt(-a^2*x^2 + 1)*arcsi n(a*x) + 3*arcsin(a*x)^2)/a^5
Time = 0.44 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {x^{4}}{16 a} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{2}}{16 a^{3}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{8 a^{4}} + \frac {3 \operatorname {asin}^{2}{\left (a x \right )}}{16 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**4*asin(a*x)/(-a**2*x**2+1)**(1/2),x)
Output:
Piecewise((x**4/(16*a) - x**3*sqrt(-a**2*x**2 + 1)*asin(a*x)/(4*a**2) + 3* x**2/(16*a**3) - 3*x*sqrt(-a**2*x**2 + 1)*asin(a*x)/(8*a**4) + 3*asin(a*x) **2/(16*a**5), Ne(a, 0)), (0, True))
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {1}{16} \, {\left (\frac {x^{4}}{a^{2}} + \frac {3 \, x^{2}}{a^{4}} - \frac {3 \, \arcsin \left (a x\right )^{2}}{a^{6}}\right )} a - \frac {1}{8} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{4}} - \frac {3 \, \arcsin \left (a x\right )}{a^{5}}\right )} \arcsin \left (a x\right ) \] Input:
integrate(x^4*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
1/16*(x^4/a^2 + 3*x^2/a^4 - 3*arcsin(a*x)^2/a^6)*a - 1/8*(2*sqrt(-a^2*x^2 + 1)*x^3/a^2 + 3*sqrt(-a^2*x^2 + 1)*x/a^4 - 3*arcsin(a*x)/a^5)*arcsin(a*x)
Time = 0.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.03 \[ \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{4 \, a^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{8 \, a^{4}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2}}{16 \, a^{5}} + \frac {3 \, \arcsin \left (a x\right )^{2}}{16 \, a^{5}} + \frac {5 \, {\left (a^{2} x^{2} - 1\right )}}{16 \, a^{5}} + \frac {17}{128 \, a^{5}} \] Input:
integrate(x^4*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
1/4*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)/a^4 - 5/8*sqrt(-a^2*x^2 + 1)*x*arcs in(a*x)/a^4 + 1/16*(a^2*x^2 - 1)^2/a^5 + 3/16*arcsin(a*x)^2/a^5 + 5/16*(a^ 2*x^2 - 1)/a^5 + 17/128/a^5
Timed out. \[ \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,\mathrm {asin}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x^4*asin(a*x))/(1 - a^2*x^2)^(1/2),x)
Output:
int((x^4*asin(a*x))/(1 - a^2*x^2)^(1/2), x)
\[ \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {asin} \left (a x \right ) x^{4}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(x^4*asin(a*x)/(-a^2*x^2+1)^(1/2),x)
Output:
int((asin(a*x)*x**4)/sqrt( - a**2*x**2 + 1),x)