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

Optimal. Leaf size=29 \[ \frac {\text {Si}\left (2 \sin ^{-1}(a x)\right )}{4 a^4}-\frac {\text {Si}\left (4 \sin ^{-1}(a x)\right )}{8 a^4} \]

[Out]

1/4*Si(2*arcsin(a*x))/a^4-1/8*Si(4*arcsin(a*x))/a^4

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Rubi [A]  time = 0.06, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4635, 4406, 3299} \[ \frac {\text {Si}\left (2 \sin ^{-1}(a x)\right )}{4 a^4}-\frac {\text {Si}\left (4 \sin ^{-1}(a x)\right )}{8 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinIntegral[2*ArcSin[a*x]]/(4*a^4) - SinIntegral[4*ArcSin[a*x]]/(8*a^4)

Rule 3299

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

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^3}{\sin ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos (x) \sin ^3(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}-\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}+\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^4}\\ &=\frac {\text {Si}\left (2 \sin ^{-1}(a x)\right )}{4 a^4}-\frac {\text {Si}\left (4 \sin ^{-1}(a x)\right )}{8 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 24, normalized size = 0.83 \[ -\frac {\text {Si}\left (4 \sin ^{-1}(a x)\right )-2 \text {Si}\left (2 \sin ^{-1}(a x)\right )}{8 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(-2*SinIntegral[2*ArcSin[a*x]] + SinIntegral[4*ArcSin[a*x]])/a^4

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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\arcsin \left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3/arcsin(a*x), x)

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giac [A]  time = 0.15, size = 25, normalized size = 0.86 \[ -\frac {\operatorname {Si}\left (4 \, \arcsin \left (a x\right )\right )}{8 \, a^{4}} + \frac {\operatorname {Si}\left (2 \, \arcsin \left (a x\right )\right )}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sin_integral(4*arcsin(a*x))/a^4 + 1/4*sin_integral(2*arcsin(a*x))/a^4

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maple [A]  time = 0.03, size = 24, normalized size = 0.83 \[ \frac {\frac {\Si \left (2 \arcsin \left (a x \right )\right )}{4}-\frac {\Si \left (4 \arcsin \left (a x \right )\right )}{8}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^4*(1/4*Si(2*arcsin(a*x))-1/8*Si(4*arcsin(a*x)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\arcsin \left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3/arcsin(a*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^3}{\mathrm {asin}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3/asin(a*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\operatorname {asin}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/asin(a*x), x)

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