\(\int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [264]

Optimal result

Integrand size = 24, antiderivative size = 157 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}-\frac {15 \arcsin (a x)}{64 a^5}+\frac {3 x^2 \arcsin (a x)}{8 a^3}+\frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {\arcsin (a x)^3}{8 a^5} \]

[Out]

-15/64*arcsin(a*x)/a^5+3/8*x^2*arcsin(a*x)/a^3+1/8*x^4*arcsin(a*x)/a+1/8*arcsin(a*x)^3/a^5+15/64*x*(-a^2*x^2+1
)^(1/2)/a^4+1/32*x^3*(-a^2*x^2+1)^(1/2)/a^2-3/8*x*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a^4-1/4*x^3*arcsin(a*x)^2*(
-a^2*x^2+1)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4795, 4737, 4723, 327, 222} \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {\arcsin (a x)^3}{8 a^5}-\frac {15 \arcsin (a x)}{64 a^5}+\frac {3 x^2 \arcsin (a x)}{8 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}+\frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}+\frac {x^4 \arcsin (a x)}{8 a} \]

[In]

Int[(x^4*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2],x]

[Out]

(15*x*Sqrt[1 - a^2*x^2])/(64*a^4) + (x^3*Sqrt[1 - a^2*x^2])/(32*a^2) - (15*ArcSin[a*x])/(64*a^5) + (3*x^2*ArcS
in[a*x])/(8*a^3) + (x^4*ArcSin[a*x])/(8*a) - (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(8*a^4) - (x^3*Sqrt[1 - a^2
*x^2]*ArcSin[a*x]^2)/(4*a^2) + ArcSin[a*x]^3/(8*a^5)

Rule 222

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 4723

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

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[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]

Rule 4795

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] + (Dist[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] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {3 \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}+\frac {\int x^3 \arcsin (a x) \, dx}{2 a} \\ & = \frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}-\frac {1}{8} \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx+\frac {3 \int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4}+\frac {3 \int x \arcsin (a x) \, dx}{4 a^3} \\ & = \frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}+\frac {3 x^2 \arcsin (a x)}{8 a^3}+\frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {\arcsin (a x)^3}{8 a^5}-\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{32 a^2}-\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2} \\ & = \frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}+\frac {3 x^2 \arcsin (a x)}{8 a^3}+\frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {\arcsin (a x)^3}{8 a^5}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{64 a^4}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{16 a^4} \\ & = \frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}-\frac {15 \arcsin (a x)}{64 a^5}+\frac {3 x^2 \arcsin (a x)}{8 a^3}+\frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {\arcsin (a x)^3}{8 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right )+\left (-15+24 a^2 x^2+8 a^4 x^4\right ) \arcsin (a x)-8 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arcsin (a x)^2+8 \arcsin (a x)^3}{64 a^5} \]

[In]

Integrate[(x^4*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2],x]

[Out]

(a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2) + (-15 + 24*a^2*x^2 + 8*a^4*x^4)*ArcSin[a*x] - 8*a*x*Sqrt[1 - a^2*x^2]
*(3 + 2*a^2*x^2)*ArcSin[a*x]^2 + 8*ArcSin[a*x]^3)/(64*a^5)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {-16 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+8 a^{4} x^{4} \arcsin \left (a x \right )+2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-24 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +24 a^{2} x^{2} \arcsin \left (a x \right )+8 \arcsin \left (a x \right )^{3}+15 a x \sqrt {-a^{2} x^{2}+1}-15 \arcsin \left (a x \right )}{64 a^{5}}\)	\(129\)

[In]

int(x^4*arcsin(a*x)^2/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/64*(-16*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^3*x^3+8*a^4*x^4*arcsin(a*x)+2*a^3*x^3*(-a^2*x^2+1)^(1/2)-24*arcsi
n(a*x)^2*(-a^2*x^2+1)^(1/2)*a*x+24*a^2*x^2*arcsin(a*x)+8*arcsin(a*x)^3+15*a*x*(-a^2*x^2+1)^(1/2)-15*arcsin(a*x
))/a^5

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {8 \, \arcsin \left (a x\right )^{3} + {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right ) + {\left (2 \, a^{3} x^{3} - 8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{2} + 15 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{64 \, a^{5}} \]

[In]

integrate(x^4*arcsin(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/64*(8*arcsin(a*x)^3 + (8*a^4*x^4 + 24*a^2*x^2 - 15)*arcsin(a*x) + (2*a^3*x^3 - 8*(2*a^3*x^3 + 3*a*x)*arcsin(
a*x)^2 + 15*a*x)*sqrt(-a^2*x^2 + 1))/a^5

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {x^{4} \operatorname {asin}{\left (a x \right )}}{8 a} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{4 a^{2}} + \frac {x^{3} \sqrt {- a^{2} x^{2} + 1}}{32 a^{2}} + \frac {3 x^{2} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{8 a^{4}} + \frac {15 x \sqrt {- a^{2} x^{2} + 1}}{64 a^{4}} + \frac {\operatorname {asin}^{3}{\left (a x \right )}}{8 a^{5}} - \frac {15 \operatorname {asin}{\left (a x \right )}}{64 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**4*asin(a*x)**2/(-a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((x**4*asin(a*x)/(8*a) - x**3*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(4*a**2) + x**3*sqrt(-a**2*x**2 + 1)/
(32*a**2) + 3*x**2*asin(a*x)/(8*a**3) - 3*x*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(8*a**4) + 15*x*sqrt(-a**2*x**2
+ 1)/(64*a**4) + asin(a*x)**3/(8*a**5) - 15*asin(a*x)/(64*a**5), Ne(a, 0)), (0, True))

Maxima [F]

\[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]

[In]

integrate(x^4*arcsin(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4*arcsin(a*x)^2/sqrt(-a^2*x^2 + 1), x)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.91 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{2}}{4 \, a^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{8 \, a^{4}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{32 \, a^{4}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )}{8 \, a^{5}} + \frac {\arcsin \left (a x\right )^{3}}{8 \, a^{5}} + \frac {17 \, \sqrt {-a^{2} x^{2} + 1} x}{64 \, a^{4}} + \frac {5 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{8 \, a^{5}} + \frac {17 \, \arcsin \left (a x\right )}{64 \, a^{5}} \]

[In]

integrate(x^4*arcsin(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/4*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^2/a^4 - 5/8*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^2/a^4 - 1/32*(-a^2*x^2 + 1
)^(3/2)*x/a^4 + 1/8*(a^2*x^2 - 1)^2*arcsin(a*x)/a^5 + 1/8*arcsin(a*x)^3/a^5 + 17/64*sqrt(-a^2*x^2 + 1)*x/a^4 +
5/8*(a^2*x^2 - 1)*arcsin(a*x)/a^5 + 17/64*arcsin(a*x)/a^5

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {asin}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]

[In]

int((x^4*asin(a*x)^2)/(1 - a^2*x^2)^(1/2),x)

[Out]

int((x^4*asin(a*x)^2)/(1 - a^2*x^2)^(1/2), x)