3.23 \(\int x^3 \sin ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=167 \[ -\frac {3 \sin ^{-1}(a x)^3}{32 a^4}+\frac {45 \sin ^{-1}(a x)}{256 a^4}-\frac {9 x^2 \sin ^{-1}(a x)}{32 a^2}-\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{16 a}-\frac {45 x \sqrt {1-a^2 x^2}}{256 a^3}+\frac {9 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{32 a^3}+\frac {1}{4} x^4 \sin ^{-1}(a x)^3-\frac {3}{32} x^4 \sin ^{-1}(a x) \]

[Out]

45/256*arcsin(a*x)/a^4-9/32*x^2*arcsin(a*x)/a^2-3/32*x^4*arcsin(a*x)-3/32*arcsin(a*x)^3/a^4+1/4*x^4*arcsin(a*x
)^3-45/256*x*(-a^2*x^2+1)^(1/2)/a^3-3/128*x^3*(-a^2*x^2+1)^(1/2)/a+9/32*x*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a^3
+3/16*x^3*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a

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Rubi [A]  time = 0.30, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4627, 4707, 4641, 321, 216} \[ -\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}-\frac {45 x \sqrt {1-a^2 x^2}}{256 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{16 a}-\frac {9 x^2 \sin ^{-1}(a x)}{32 a^2}+\frac {9 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{32 a^3}-\frac {3 \sin ^{-1}(a x)^3}{32 a^4}+\frac {45 \sin ^{-1}(a x)}{256 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^3-\frac {3}{32} x^4 \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcSin[a*x]^3,x]

[Out]

(-45*x*Sqrt[1 - a^2*x^2])/(256*a^3) - (3*x^3*Sqrt[1 - a^2*x^2])/(128*a) + (45*ArcSin[a*x])/(256*a^4) - (9*x^2*
ArcSin[a*x])/(32*a^2) - (3*x^4*ArcSin[a*x])/32 + (9*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(32*a^3) + (3*x^3*Sqrt[
1 - a^2*x^2]*ArcSin[a*x]^2)/(16*a) - (3*ArcSin[a*x]^3)/(32*a^4) + (x^4*ArcSin[a*x]^3)/4

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 x^3 \sin ^{-1}(a x)^3 \, dx &=\frac {1}{4} x^4 \sin ^{-1}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^4 \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{16 a}+\frac {1}{4} x^4 \sin ^{-1}(a x)^3-\frac {3}{8} \int x^3 \sin ^{-1}(a x) \, dx-\frac {9 \int \frac {x^2 \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac {3}{32} x^4 \sin ^{-1}(a x)+\frac {9 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{32 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{16 a}+\frac {1}{4} x^4 \sin ^{-1}(a x)^3-\frac {9 \int \frac {\sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{32 a^3}-\frac {9 \int x \sin ^{-1}(a x) \, dx}{16 a^2}+\frac {1}{32} (3 a) \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}-\frac {9 x^2 \sin ^{-1}(a x)}{32 a^2}-\frac {3}{32} x^4 \sin ^{-1}(a x)+\frac {9 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{32 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{16 a}-\frac {3 \sin ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^3+\frac {9 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{128 a}+\frac {9 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{32 a}\\ &=-\frac {45 x \sqrt {1-a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}-\frac {9 x^2 \sin ^{-1}(a x)}{32 a^2}-\frac {3}{32} x^4 \sin ^{-1}(a x)+\frac {9 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{32 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{16 a}-\frac {3 \sin ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^3+\frac {9 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{256 a^3}+\frac {9 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{64 a^3}\\ &=-\frac {45 x \sqrt {1-a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}+\frac {45 \sin ^{-1}(a x)}{256 a^4}-\frac {9 x^2 \sin ^{-1}(a x)}{32 a^2}-\frac {3}{32} x^4 \sin ^{-1}(a x)+\frac {9 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{32 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{16 a}-\frac {3 \sin ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^3\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 112, normalized size = 0.67 \[ \frac {8 \left (8 a^4 x^4-3\right ) \sin ^{-1}(a x)^3-3 a x \sqrt {1-a^2 x^2} \left (2 a^2 x^2+15\right )+24 a x \sqrt {1-a^2 x^2} \left (2 a^2 x^2+3\right ) \sin ^{-1}(a x)^2-3 \left (8 a^4 x^4+24 a^2 x^2-15\right ) \sin ^{-1}(a x)}{256 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcSin[a*x]^3,x]

[Out]

(-3*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2) - 3*(-15 + 24*a^2*x^2 + 8*a^4*x^4)*ArcSin[a*x] + 24*a*x*Sqrt[1 - a^
2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x]^2 + 8*(-3 + 8*a^4*x^4)*ArcSin[a*x]^3)/(256*a^4)

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fricas [A]  time = 0.53, size = 96, normalized size = 0.57 \[ \frac {8 \, {\left (8 \, a^{4} x^{4} - 3\right )} \arcsin \left (a x\right )^{3} - 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right ) - 3 \, {\left (2 \, a^{3} x^{3} - 8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{2} + 15 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{256 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)^3,x, algorithm="fricas")

[Out]

1/256*(8*(8*a^4*x^4 - 3)*arcsin(a*x)^3 - 3*(8*a^4*x^4 + 24*a^2*x^2 - 15)*arcsin(a*x) - 3*(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^4

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giac [A]  time = 0.20, size = 185, normalized size = 1.11 \[ -\frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{2}}{16 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{3}}{4 \, a^{4}} + \frac {15 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{32 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3}}{2 \, a^{4}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{128 \, a^{3}} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )}{32 \, a^{4}} + \frac {5 \, \arcsin \left (a x\right )^{3}}{32 \, a^{4}} - \frac {51 \, \sqrt {-a^{2} x^{2} + 1} x}{256 \, a^{3}} - \frac {15 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{32 \, a^{4}} - \frac {51 \, \arcsin \left (a x\right )}{256 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)^3,x, algorithm="giac")

[Out]

-3/16*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^2/a^3 + 1/4*(a^2*x^2 - 1)^2*arcsin(a*x)^3/a^4 + 15/32*sqrt(-a^2*x^2 +
1)*x*arcsin(a*x)^2/a^3 + 1/2*(a^2*x^2 - 1)*arcsin(a*x)^3/a^4 + 3/128*(-a^2*x^2 + 1)^(3/2)*x/a^3 - 3/32*(a^2*x
^2 - 1)^2*arcsin(a*x)/a^4 + 5/32*arcsin(a*x)^3/a^4 - 51/256*sqrt(-a^2*x^2 + 1)*x/a^3 - 15/32*(a^2*x^2 - 1)*arc
sin(a*x)/a^4 - 51/256*arcsin(a*x)/a^4

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maple [A]  time = 0.09, size = 154, normalized size = 0.92 \[ \frac {\frac {a^{4} x^{4} \arcsin \left (a x \right )^{3}}{4}-\frac {3 \arcsin \left (a x \right )^{2} \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{32}-\frac {3 a^{4} x^{4} \arcsin \left (a x \right )}{32}-\frac {3 a x \left (2 a^{2} x^{2}+3\right ) \sqrt {-a^{2} x^{2}+1}}{256}-\frac {27 \arcsin \left (a x \right )}{256}-\frac {9 \left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )}{32}-\frac {9 a x \sqrt {-a^{2} x^{2}+1}}{64}+\frac {3 \arcsin \left (a x \right )^{3}}{16}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsin(a*x)^3,x)

[Out]

1/a^4*(1/4*a^4*x^4*arcsin(a*x)^3-3/32*arcsin(a*x)^2*(-2*a^3*x^3*(-a^2*x^2+1)^(1/2)-3*a*x*(-a^2*x^2+1)^(1/2)+3*
arcsin(a*x))-3/32*a^4*x^4*arcsin(a*x)-3/256*a*x*(2*a^2*x^2+3)*(-a^2*x^2+1)^(1/2)-27/256*arcsin(a*x)-9/32*(a^2*
x^2-1)*arcsin(a*x)-9/64*a*x*(-a^2*x^2+1)^(1/2)+3/16*arcsin(a*x)^3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, x^{4} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3} + 3 \, a \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} x^{4} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2}}{4 \, {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)^3,x, algorithm="maxima")

[Out]

1/4*x^4*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3 + 3*a*integrate(1/4*sqrt(a*x + 1)*sqrt(-a*x + 1)*x^4*arct
an2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2/(a^2*x^2 - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\mathrm {asin}\left (a\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*asin(a*x)^3,x)

[Out]

int(x^3*asin(a*x)^3, x)

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sympy [A]  time = 3.34, size = 160, normalized size = 0.96 \[ \begin {cases} \frac {x^{4} \operatorname {asin}^{3}{\left (a x \right )}}{4} - \frac {3 x^{4} \operatorname {asin}{\left (a x \right )}}{32} + \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1}}{128 a} - \frac {9 x^{2} \operatorname {asin}{\left (a x \right )}}{32 a^{2}} + \frac {9 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{32 a^{3}} - \frac {45 x \sqrt {- a^{2} x^{2} + 1}}{256 a^{3}} - \frac {3 \operatorname {asin}^{3}{\left (a x \right )}}{32 a^{4}} + \frac {45 \operatorname {asin}{\left (a x \right )}}{256 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*asin(a*x)**3,x)

[Out]

Piecewise((x**4*asin(a*x)**3/4 - 3*x**4*asin(a*x)/32 + 3*x**3*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(16*a) - 3*x**
3*sqrt(-a**2*x**2 + 1)/(128*a) - 9*x**2*asin(a*x)/(32*a**2) + 9*x*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(32*a**3)
- 45*x*sqrt(-a**2*x**2 + 1)/(256*a**3) - 3*asin(a*x)**3/(32*a**4) + 45*asin(a*x)/(256*a**4), Ne(a, 0)), (0, Tr
ue))

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