3.264 \(\int \frac {x^4 \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\)

Optimal. Leaf size=157 \[ \frac {\sin ^{-1}(a x)^3}{8 a^5}-\frac {15 \sin ^{-1}(a x)}{64 a^5}+\frac {3 x^2 \sin ^{-1}(a x)}{8 a^3}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a^2}+\frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{8 a^4}+\frac {x^4 \sin ^{-1}(a x)}{8 a} \]

[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]  time = 0.27, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4707, 4641, 4627, 321, 216} \[ \frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}+\frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a^2}+\frac {3 x^2 \sin ^{-1}(a x)}{8 a^3}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{8 a^4}+\frac {\sin ^{-1}(a x)^3}{8 a^5}-\frac {15 \sin ^{-1}(a x)}{64 a^5}+\frac {x^4 \sin ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[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 216

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 321

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 4627

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 4641

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

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^4 \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a^2}+\frac {3 \int \frac {x^2 \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}+\frac {\int x^3 \sin ^{-1}(a x) \, dx}{2 a}\\ &=\frac {x^4 \sin ^{-1}(a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a^2}-\frac {1}{8} \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx+\frac {3 \int \frac {\sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4}+\frac {3 \int x \sin ^{-1}(a x) \, dx}{4 a^3}\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}+\frac {3 x^2 \sin ^{-1}(a x)}{8 a^3}+\frac {x^4 \sin ^{-1}(a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a^2}+\frac {\sin ^{-1}(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 \sin ^{-1}(a x)}{8 a^3}+\frac {x^4 \sin ^{-1}(a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a^2}+\frac {\sin ^{-1}(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 \sin ^{-1}(a x)}{64 a^5}+\frac {3 x^2 \sin ^{-1}(a x)}{8 a^3}+\frac {x^4 \sin ^{-1}(a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a^2}+\frac {\sin ^{-1}(a x)^3}{8 a^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 100, normalized size = 0.64 \[ \frac {a x \sqrt {1-a^2 x^2} \left (2 a^2 x^2+15\right )-8 a x \sqrt {1-a^2 x^2} \left (2 a^2 x^2+3\right ) \sin ^{-1}(a x)^2+\left (8 a^4 x^4+24 a^2 x^2-15\right ) \sin ^{-1}(a x)+8 \sin ^{-1}(a x)^3}{64 a^5} \]

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 84, normalized size = 0.54 \[ \frac {8 \, \arcsin \left (a x\right )^{3} + {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right ) + {\left (2 \, a^{3} x^{3} - 8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{2} + 15 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{64 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [A]  time = 0.60, size = 143, normalized size = 0.91 \[ \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{2}}{4 \, a^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{8 \, a^{4}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{32 \, a^{4}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )}{8 \, a^{5}} + \frac {\arcsin \left (a x\right )^{3}}{8 \, a^{5}} + \frac {17 \, \sqrt {-a^{2} x^{2} + 1} x}{64 \, a^{4}} + \frac {5 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{8 \, a^{5}} + \frac {17 \, \arcsin \left (a x\right )}{64 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

maple [A]  time = 0.12, size = 129, normalized size = 0.82 \[ \frac {-16 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, x^{3} a^{3}+8 a^{4} x^{4} \arcsin \left (a x \right )+2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-24 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, x a +24 a^{2} x^{2} \arcsin \left (a x \right )+8 \arcsin \left (a x \right )^{3}+15 a x \sqrt {-a^{2} x^{2}+1}-15 \arcsin \left (a x \right )}{64 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(-16*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*x^3*a^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)*x*a+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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4\,{\mathrm {asin}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 3.50, size = 146, normalized size = 0.93 \[ \begin {cases} \frac {x^{4} \operatorname {asin}{\left (a x \right )}}{8 a} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{4 a^{2}} + \frac {x^{3} \sqrt {- a^{2} x^{2} + 1}}{32 a^{2}} + \frac {3 x^{2} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{8 a^{4}} + \frac {15 x \sqrt {- a^{2} x^{2} + 1}}{64 a^{4}} + \frac {\operatorname {asin}^{3}{\left (a x \right )}}{8 a^{5}} - \frac {15 \operatorname {asin}{\left (a x \right )}}{64 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________