3.4.4 \(\int \frac {x^3 \text {ArcSin}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [304]

Optimal. Leaf size=157 \[ -\frac {40 x}{9 a^3}-\frac {2 x^3}{27 a}+\frac {40 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{9 a^2}+\frac {2 x \text {ArcSin}(a x)^2}{a^3}+\frac {x^3 \text {ArcSin}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{3 a^2} \]

[Out]

-40/9*x/a^3-2/27*x^3/a+2*x*arcsin(a*x)^2/a^3+1/3*x^3*arcsin(a*x)^2/a+40/9*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^4+2
/9*x^2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^2-2/3*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/a^4-1/3*x^2*arcsin(a*x)^3*(-a^2
*x^2+1)^(1/2)/a^2

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Rubi [A]
time = 0.23, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4795, 4767, 4715, 8, 4723, 30} \begin {gather*} \frac {2 x \text {ArcSin}(a x)^2}{a^3}-\frac {40 x}{9 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{3 a^2}+\frac {2 x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{9 a^2}-\frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{3 a^4}+\frac {40 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{9 a^4}+\frac {x^3 \text {ArcSin}(a x)^2}{3 a}-\frac {2 x^3}{27 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-40*x)/(9*a^3) - (2*x^3)/(27*a) + (40*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(9*a^4) + (2*x^2*Sqrt[1 - a^2*x^2]*ArcSi
n[a*x])/(9*a^2) + (2*x*ArcSin[a*x]^2)/a^3 + (x^3*ArcSin[a*x]^2)/(3*a) - (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3
*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*a^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

Rule 4715

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

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 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -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^3 \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}+\frac {2 \int \frac {x \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{3 a^2}+\frac {\int x^2 \sin ^{-1}(a x)^2 \, dx}{a}\\ &=\frac {x^3 \sin ^{-1}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}-\frac {2}{3} \int \frac {x^3 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx+\frac {2 \int \sin ^{-1}(a x)^2 \, dx}{a^3}\\ &=\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}+\frac {2 x \sin ^{-1}(a x)^2}{a^3}+\frac {x^3 \sin ^{-1}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}-\frac {4 \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{9 a^2}-\frac {4 \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{a^2}-\frac {2 \int x^2 \, dx}{9 a}\\ &=-\frac {2 x^3}{27 a}+\frac {40 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}+\frac {2 x \sin ^{-1}(a x)^2}{a^3}+\frac {x^3 \sin ^{-1}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}-\frac {4 \int 1 \, dx}{9 a^3}-\frac {4 \int 1 \, dx}{a^3}\\ &=-\frac {40 x}{9 a^3}-\frac {2 x^3}{27 a}+\frac {40 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}+\frac {2 x \sin ^{-1}(a x)^2}{a^3}+\frac {x^3 \sin ^{-1}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 100, normalized size = 0.64 \begin {gather*} \frac {-2 a x \left (60+a^2 x^2\right )+6 \sqrt {1-a^2 x^2} \left (20+a^2 x^2\right ) \text {ArcSin}(a x)+9 a x \left (6+a^2 x^2\right ) \text {ArcSin}(a x)^2-9 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \text {ArcSin}(a x)^3}{27 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*x*(60 + a^2*x^2) + 6*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2)*ArcSin[a*x] + 9*a*x*(6 + a^2*x^2)*ArcSin[a*x]^2 -
9*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcSin[a*x]^3)/(27*a^4)

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Maple [A]
time = 0.12, size = 180, normalized size = 1.15

method	result	size
default	\(-\frac {\left (9 a^{4} x^{4} \arcsin \left (a x \right )^{3}+9 \arcsin \left (a x \right )^{3} a^{2} x^{2}+9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-6 a^{4} x^{4} \arcsin \left (a x \right )-114 a^{2} x^{2} \arcsin \left (a x \right )-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-18 \arcsin \left (a x \right )^{3}+54 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +120 \arcsin \left (a x \right )-120 a x \sqrt {-a^{2} x^{2}+1}\right ) \sqrt {-a^{2} x^{2}+1}}{27 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27/a^4*(9*a^4*x^4*arcsin(a*x)^3+9*arcsin(a*x)^3*a^2*x^2+9*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^3*x^3-6*a^4*x^
4*arcsin(a*x)-114*a^2*x^2*arcsin(a*x)-2*a^3*x^3*(-a^2*x^2+1)^(1/2)-18*arcsin(a*x)^3+54*arcsin(a*x)^2*(-a^2*x^2
+1)^(1/2)*a*x+120*arcsin(a*x)-120*a*x*(-a^2*x^2+1)^(1/2))*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)

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Maxima [A]
time = 0.48, size = 131, normalized size = 0.83 \begin {gather*} -\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arcsin \left (a x\right )^{3} + \frac {2}{27} \, a {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arcsin \left (a x\right )}{a^{3}} - \frac {a^{2} x^{3} + 60 \, x}{a^{4}}\right )} + \frac {{\left (a^{2} x^{3} + 6 \, x\right )} \arcsin \left (a x\right )^{2}}{3 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arcsin(a*x)^3 + 2/27*a*(3*(sqrt(-a^2*x^2 + 1)*x^2
+ 20*sqrt(-a^2*x^2 + 1)/a^2)*arcsin(a*x)/a^3 - (a^2*x^3 + 60*x)/a^4) + 1/3*(a^2*x^3 + 6*x)*arcsin(a*x)^2/a^3

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Fricas [A]
time = 1.52, size = 85, normalized size = 0.54 \begin {gather*} -\frac {2 \, a^{3} x^{3} - 9 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arcsin \left (a x\right )^{2} + 120 \, a x + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (3 \, {\left (a^{2} x^{2} + 2\right )} \arcsin \left (a x\right )^{3} - 2 \, {\left (a^{2} x^{2} + 20\right )} \arcsin \left (a x\right )\right )}}{27 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(2*a^3*x^3 - 9*(a^3*x^3 + 6*a*x)*arcsin(a*x)^2 + 120*a*x + 3*sqrt(-a^2*x^2 + 1)*(3*(a^2*x^2 + 2)*arcsin(
a*x)^3 - 2*(a^2*x^2 + 20)*arcsin(a*x)))/a^4

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Sympy [A]
time = 0.63, size = 148, normalized size = 0.94 \begin {gather*} \begin {cases} \frac {x^{3} \operatorname {asin}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{3}}{27 a} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{9 a^{2}} + \frac {2 x \operatorname {asin}^{2}{\left (a x \right )}}{a^{3}} - \frac {40 x}{9 a^{3}} - \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{3 a^{4}} + \frac {40 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{9 a^{4}} & \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**3*asin(a*x)**3/(-a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((x**3*asin(a*x)**2/(3*a) - 2*x**3/(27*a) - x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(3*a**2) + 2*x**2*
sqrt(-a**2*x**2 + 1)*asin(a*x)/(9*a**2) + 2*x*asin(a*x)**2/a**3 - 40*x/(9*a**3) - 2*sqrt(-a**2*x**2 + 1)*asin(
a*x)**3/(3*a**4) + 40*sqrt(-a**2*x**2 + 1)*asin(a*x)/(9*a**4), Ne(a, 0)), (0, True))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,{\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^3*asin(a*x)^3)/(1 - a^2*x^2)^(1/2),x)

[Out]

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

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