3.4.3 \(\int \frac {x^4 \text {ArcSin}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [303]

Optimal. Leaf size=191 \[ -\frac {45 x^2}{128 a^3}-\frac {3 x^4}{128 a}+\frac {45 x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{64 a^4}+\frac {3 x^3 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{32 a^2}-\frac {45 \text {ArcSin}(a x)^2}{128 a^5}+\frac {9 x^2 \text {ArcSin}(a x)^2}{16 a^3}+\frac {3 x^4 \text {ArcSin}(a x)^2}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{4 a^2}+\frac {3 \text {ArcSin}(a x)^4}{32 a^5} \]

[Out]

-45/128*x^2/a^3-3/128*x^4/a-45/128*arcsin(a*x)^2/a^5+9/16*x^2*arcsin(a*x)^2/a^3+3/16*x^4*arcsin(a*x)^2/a+3/32*
arcsin(a*x)^4/a^5+45/64*x*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^4+3/32*x^3*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^2-3/8*x
*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/a^4-1/4*x^3*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/a^2

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Rubi [A]
time = 0.32, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4795, 4737, 4723, 30} \begin {gather*} \frac {3 \text {ArcSin}(a x)^4}{32 a^5}-\frac {45 \text {ArcSin}(a x)^2}{128 a^5}+\frac {9 x^2 \text {ArcSin}(a x)^2}{16 a^3}-\frac {45 x^2}{128 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{4 a^2}+\frac {3 x^3 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{32 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{8 a^4}+\frac {45 x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{64 a^4}+\frac {3 x^4 \text {ArcSin}(a x)^2}{16 a}-\frac {3 x^4}{128 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]

[Out]

(-45*x^2)/(128*a^3) - (3*x^4)/(128*a) + (45*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(64*a^4) + (3*x^3*Sqrt[1 - a^2*x^
2]*ArcSin[a*x])/(32*a^2) - (45*ArcSin[a*x]^2)/(128*a^5) + (9*x^2*ArcSin[a*x]^2)/(16*a^3) + (3*x^4*ArcSin[a*x]^
2)/(16*a) - (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(4*a^2) + (3
*ArcSin[a*x]^4)/(32*a^5)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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*} \int \frac {x^4 \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac {3 \int \frac {x^2 \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}+\frac {3 \int x^3 \sin ^{-1}(a x)^2 \, dx}{4 a}\\ &=\frac {3 x^4 \sin ^{-1}(a x)^2}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}-\frac {3}{8} \int \frac {x^4 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx+\frac {3 \int \frac {\sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4}+\frac {9 \int x \sin ^{-1}(a x)^2 \, dx}{8 a^3}\\ &=\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{32 a^2}+\frac {9 x^2 \sin ^{-1}(a x)^2}{16 a^3}+\frac {3 x^4 \sin ^{-1}(a x)^2}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac {3 \sin ^{-1}(a x)^4}{32 a^5}-\frac {9 \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{32 a^2}-\frac {9 \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2}-\frac {3 \int x^3 \, dx}{32 a}\\ &=-\frac {3 x^4}{128 a}+\frac {45 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{64 a^4}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{32 a^2}+\frac {9 x^2 \sin ^{-1}(a x)^2}{16 a^3}+\frac {3 x^4 \sin ^{-1}(a x)^2}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac {3 \sin ^{-1}(a x)^4}{32 a^5}-\frac {9 \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{64 a^4}-\frac {9 \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 a^4}-\frac {9 \int x \, dx}{64 a^3}-\frac {9 \int x \, dx}{16 a^3}\\ &=-\frac {45 x^2}{128 a^3}-\frac {3 x^4}{128 a}+\frac {45 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{64 a^4}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{32 a^2}-\frac {45 \sin ^{-1}(a x)^2}{128 a^5}+\frac {9 x^2 \sin ^{-1}(a x)^2}{16 a^3}+\frac {3 x^4 \sin ^{-1}(a x)^2}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac {3 \sin ^{-1}(a x)^4}{32 a^5}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 125, normalized size = 0.65 \begin {gather*} \frac {-3 a^2 x^2 \left (15+a^2 x^2\right )+6 a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right ) \text {ArcSin}(a x)+3 \left (-15+24 a^2 x^2+8 a^4 x^4\right ) \text {ArcSin}(a x)^2-16 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \text {ArcSin}(a x)^3+12 \text {ArcSin}(a x)^4}{128 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]

[Out]

(-3*a^2*x^2*(15 + a^2*x^2) + 6*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2)*ArcSin[a*x] + 3*(-15 + 24*a^2*x^2 + 8*a^
4*x^4)*ArcSin[a*x]^2 - 16*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x]^3 + 12*ArcSin[a*x]^4)/(128*a^5)

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Maple [A]
time = 0.12, size = 160, normalized size = 0.84

method	result	size
default	\(\frac {-128 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+96 a^{4} x^{4} \arcsin \left (a x \right )^{2}+48 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-12 a^{4} x^{4}-192 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a x +288 \arcsin \left (a x \right )^{2} a^{2} x^{2}+48 \arcsin \left (a x \right )^{4}+360 a x \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-180 a^{2} x^{2}-180 \arcsin \left (a x \right )^{2}-27}{512 a^{5}}\)	\(160\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/512*(-128*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)*a^3*x^3+96*a^4*x^4*arcsin(a*x)^2+48*arcsin(a*x)*(-a^2*x^2+1)^(1/2
)*a^3*x^3-12*a^4*x^4-192*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)*a*x+288*arcsin(a*x)^2*a^2*x^2+48*arcsin(a*x)^4+360*a
*x*arcsin(a*x)*(-a^2*x^2+1)^(1/2)-180*a^2*x^2-180*arcsin(a*x)^2-27)/a^5

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4*arcsin(a*x)^3/sqrt(-a^2*x^2 + 1), x)

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Fricas [A]
time = 1.73, size = 111, normalized size = 0.58 \begin {gather*} -\frac {3 \, a^{4} x^{4} + 45 \, a^{2} x^{2} - 12 \, \arcsin \left (a x\right )^{4} - 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right )^{2} + 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{3} - 3 \, {\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \arcsin \left (a x\right )\right )}}{128 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/128*(3*a^4*x^4 + 45*a^2*x^2 - 12*arcsin(a*x)^4 - 3*(8*a^4*x^4 + 24*a^2*x^2 - 15)*arcsin(a*x)^2 + 2*sqrt(-a^
2*x^2 + 1)*(8*(2*a^3*x^3 + 3*a*x)*arcsin(a*x)^3 - 3*(2*a^3*x^3 + 15*a*x)*arcsin(a*x)))/a^5

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Sympy [A]
time = 0.89, size = 185, normalized size = 0.97 \begin {gather*} \begin {cases} \frac {3 x^{4} \operatorname {asin}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{4}}{128 a} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{32 a^{2}} + \frac {9 x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{16 a^{3}} - \frac {45 x^{2}}{128 a^{3}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{8 a^{4}} + \frac {45 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{64 a^{4}} + \frac {3 \operatorname {asin}^{4}{\left (a x \right )}}{32 a^{5}} - \frac {45 \operatorname {asin}^{2}{\left (a x \right )}}{128 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*asin(a*x)**3/(-a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((3*x**4*asin(a*x)**2/(16*a) - 3*x**4/(128*a) - x**3*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(4*a**2) + 3*x
**3*sqrt(-a**2*x**2 + 1)*asin(a*x)/(32*a**2) + 9*x**2*asin(a*x)**2/(16*a**3) - 45*x**2/(128*a**3) - 3*x*sqrt(-
a**2*x**2 + 1)*asin(a*x)**3/(8*a**4) + 45*x*sqrt(-a**2*x**2 + 1)*asin(a*x)/(64*a**4) + 3*asin(a*x)**4/(32*a**5
) - 45*asin(a*x)**2/(128*a**5), Ne(a, 0)), (0, True))

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Giac [A]
time = 0.44, size = 192, normalized size = 1.01 \begin {gather*} \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{3}}{4 \, a^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{8 \, a^{4}} - \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{32 \, a^{4}} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{16 \, a^{5}} + \frac {3 \, \arcsin \left (a x\right )^{4}}{32 \, a^{5}} + \frac {51 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{64 \, a^{4}} + \frac {15 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{16 \, a^{5}} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2}}{128 \, a^{5}} + \frac {51 \, \arcsin \left (a x\right )^{2}}{128 \, a^{5}} - \frac {51 \, {\left (a^{2} x^{2} - 1\right )}}{128 \, a^{5}} - \frac {195}{1024 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/4*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^3/a^4 - 5/8*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^4 - 3/32*(-a^2*x^2 + 1
)^(3/2)*x*arcsin(a*x)/a^4 + 3/16*(a^2*x^2 - 1)^2*arcsin(a*x)^2/a^5 + 3/32*arcsin(a*x)^4/a^5 + 51/64*sqrt(-a^2*
x^2 + 1)*x*arcsin(a*x)/a^4 + 15/16*(a^2*x^2 - 1)*arcsin(a*x)^2/a^5 - 3/128*(a^2*x^2 - 1)^2/a^5 + 51/128*arcsin
(a*x)^2/a^5 - 51/128*(a^2*x^2 - 1)/a^5 - 195/1024/a^5

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,{\mathrm {asin}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*asin(a*x)^3)/(1 - a^2*x^2)^(1/2),x)

[Out]

int((x^4*asin(a*x)^3)/(1 - a^2*x^2)^(1/2), x)

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