3.4.44 \(\int \frac {x^2}{\sqrt {1-a^2 x^2} \text {ArcSin}(a x)} \, dx\) [344]

Optimal. Leaf size=27 \[ -\frac {\text {CosIntegral}(2 \text {ArcSin}(a x))}{2 a^3}+\frac {\log (\text {ArcSin}(a x))}{2 a^3} \]

[Out]

-1/2*Ci(2*arcsin(a*x))/a^3+1/2*ln(arcsin(a*x))/a^3

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Rubi [A]
time = 0.10, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4809, 3393, 3383} \begin {gather*} \frac {\log (\text {ArcSin}(a x))}{2 a^3}-\frac {\text {CosIntegral}(2 \text {ArcSin}(a x))}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*CosIntegral[2*ArcSin[a*x]]/a^3 + Log[ArcSin[a*x]]/(2*a^3)

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c
^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x],
x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac {\log \left (\sin ^{-1}(a x)\right )}{2 a^3}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {\text {Ci}\left (2 \sin ^{-1}(a x)\right )}{2 a^3}+\frac {\log \left (\sin ^{-1}(a x)\right )}{2 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.81 \begin {gather*} \frac {-\text {CosIntegral}(2 \text {ArcSin}(a x))+\log (\text {ArcSin}(a x))}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-CosIntegral[2*ArcSin[a*x]] + Log[ArcSin[a*x]])/(2*a^3)

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Maple [A]
time = 0.00, size = 21, normalized size = 0.78

method	result	size
default	\(\frac {\ln \left (\arcsin \left (a x \right )\right )-\cosineIntegral \left (2 \arcsin \left (a x \right )\right )}{2 a^{3}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(ln(arcsin(a*x))-Ci(2*arcsin(a*x)))/a^3

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*x^2/((a^2*x^2 - 1)*arcsin(a*x)), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(sqrt(-(a*x - 1)*(a*x + 1))*asin(a*x)), x)

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Giac [A]
time = 0.43, size = 23, normalized size = 0.85 \begin {gather*} -\frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{2 \, a^{3}} + \frac {\log \left (\arcsin \left (a x\right )\right )}{2 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cos_integral(2*arcsin(a*x))/a^3 + 1/2*log(arcsin(a*x))/a^3

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^2}{\mathrm {asin}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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