\(\int \frac {x^3}{\arcsin (a x)} \, dx\) [45]

Optimal result

Integrand size = 10, antiderivative size = 29 \[ \int \frac {x^3}{\arcsin (a x)} \, dx=\frac {\text {Si}(2 \arcsin (a x))}{4 a^4}-\frac {\text {Si}(4 \arcsin (a x))}{8 a^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4731, 4491, 3380} \[ \int \frac {x^3}{\arcsin (a x)} \, dx=\frac {\text {Si}(2 \arcsin (a x))}{4 a^4}-\frac {\text {Si}(4 \arcsin (a x))}{8 a^4} \]

[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 3380

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 4491

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 4731

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{\arcsin (a x)} \, dx=-\frac {-2 \text {Si}(2 \arcsin (a x))+\text {Si}(4 \arcsin (a x))}{8 a^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {Si}\left (2 \arcsin \left (a x \right )\right )}{4}-\frac {\operatorname {Si}\left (4 \arcsin \left (a x \right )\right )}{8}}{a^{4}}\)	\(24\)
default	\(\frac {\frac {\operatorname {Si}\left (2 \arcsin \left (a x \right )\right )}{4}-\frac {\operatorname {Si}\left (4 \arcsin \left (a x \right )\right )}{8}}{a^{4}}\)	\(24\)

[In]

int(x^3/arcsin(a*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {x^3}{\arcsin (a x)} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )} \,d x } \]

[In]

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

[Out]

integral(x^3/arcsin(a*x), x)

Sympy [F]

\[ \int \frac {x^3}{\arcsin (a x)} \, dx=\int \frac {x^{3}}{\operatorname {asin}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral(x**3/asin(a*x), x)

Maxima [F]

\[ \int \frac {x^3}{\arcsin (a x)} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )} \,d x } \]

[In]

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

[Out]

integrate(x^3/arcsin(a*x), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {x^3}{\arcsin (a x)} \, dx=-\frac {\operatorname {Si}\left (4 \, \arcsin \left (a x\right )\right )}{8 \, a^{4}} + \frac {\operatorname {Si}\left (2 \, \arcsin \left (a x\right )\right )}{4 \, a^{4}} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\arcsin (a x)} \, dx=\int \frac {x^3}{\mathrm {asin}\left (a\,x\right )} \,d x \]

[In]

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

[Out]

int(x^3/asin(a*x), x)