3.37 \(\int \sin ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=69 \[ \frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^4-12 x \sin ^{-1}(a x)^2+24 x \]

[Out]

24*x-12*x*arcsin(a*x)^2+x*arcsin(a*x)^4-24*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a+4*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)
/a

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Rubi [A]  time = 0.12, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4619, 4677, 8} \[ \frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^4-12 x \sin ^{-1}(a x)^2+24 x \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^4,x]

[Out]

24*x - (24*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a - 12*x*ArcSin[a*x]^2 + (4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a + x*A
rcSin[a*x]^4

Rule 8

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

Rule 4619

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 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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]

Rubi steps

\begin {align*} \int \sin ^{-1}(a x)^4 \, dx &=x \sin ^{-1}(a x)^4-(4 a) \int \frac {x \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4-12 \int \sin ^{-1}(a x)^2 \, dx\\ &=-12 x \sin ^{-1}(a x)^2+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4+(24 a) \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}-12 x \sin ^{-1}(a x)^2+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4+24 \int 1 \, dx\\ &=24 x-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}-12 x \sin ^{-1}(a x)^2+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 69, normalized size = 1.00 \[ \frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^4-12 x \sin ^{-1}(a x)^2+24 x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^4,x]

[Out]

24*x - (24*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a - 12*x*ArcSin[a*x]^2 + (4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a + x*A
rcSin[a*x]^4

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fricas [A]  time = 0.41, size = 55, normalized size = 0.80 \[ \frac {a x \arcsin \left (a x\right )^{4} - 12 \, a x \arcsin \left (a x\right )^{2} + 24 \, a x + 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arcsin \left (a x\right )^{3} - 6 \, \arcsin \left (a x\right )\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^4,x, algorithm="fricas")

[Out]

(a*x*arcsin(a*x)^4 - 12*a*x*arcsin(a*x)^2 + 24*a*x + 4*sqrt(-a^2*x^2 + 1)*(arcsin(a*x)^3 - 6*arcsin(a*x)))/a

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giac [A]  time = 0.12, size = 65, normalized size = 0.94 \[ x \arcsin \left (a x\right )^{4} - 12 \, x \arcsin \left (a x\right )^{2} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a} + 24 \, x - \frac {24 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^4,x, algorithm="giac")

[Out]

x*arcsin(a*x)^4 - 12*x*arcsin(a*x)^2 + 4*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/a + 24*x - 24*sqrt(-a^2*x^2 + 1)*arc
sin(a*x)/a

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maple [A]  time = 0.04, size = 67, normalized size = 0.97 \[ \frac {a x \arcsin \left (a x \right )^{4}+4 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}-12 a x \arcsin \left (a x \right )^{2}+24 a x -24 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^4,x)

[Out]

1/a*(a*x*arcsin(a*x)^4+4*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)-12*a*x*arcsin(a*x)^2+24*a*x-24*arcsin(a*x)*(-a^2*x^2
+1)^(1/2))

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maxima [A]  time = 0.54, size = 75, normalized size = 1.09 \[ x \arcsin \left (a x\right )^{4} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a} - 12 \, {\left (\frac {x \arcsin \left (a x\right )^{2}}{a} - \frac {2 \, {\left (x - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^4,x, algorithm="maxima")

[Out]

x*arcsin(a*x)^4 + 4*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/a - 12*(x*arcsin(a*x)^2/a - 2*(x - sqrt(-a^2*x^2 + 1)*arc
sin(a*x)/a)/a)*a

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mupad [B]  time = 0.14, size = 48, normalized size = 0.70 \[ x\,\left ({\mathrm {asin}\left (a\,x\right )}^4-12\,{\mathrm {asin}\left (a\,x\right )}^2+24\right )+\frac {4\,\mathrm {asin}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}\,\left ({\mathrm {asin}\left (a\,x\right )}^2-6\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^4,x)

[Out]

x*(asin(a*x)^4 - 12*asin(a*x)^2 + 24) + (4*asin(a*x)*(1 - a^2*x^2)^(1/2)*(asin(a*x)^2 - 6))/a

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sympy [A]  time = 0.86, size = 65, normalized size = 0.94 \[ \begin {cases} x \operatorname {asin}^{4}{\left (a x \right )} - 12 x \operatorname {asin}^{2}{\left (a x \right )} + 24 x + \frac {4 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{a} - \frac {24 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**4,x)

[Out]

Piecewise((x*asin(a*x)**4 - 12*x*asin(a*x)**2 + 24*x + 4*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/a - 24*sqrt(-a**2*x
**2 + 1)*asin(a*x)/a, Ne(a, 0)), (0, True))

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