\(\int \frac {x}{\arcsin (a x)} \, dx\) [47]

Optimal result

Integrand size = 8, antiderivative size = 14 \[ \int \frac {x}{\arcsin (a x)} \, dx=\frac {\text {Si}(2 \arcsin (a x))}{2 a^2} \]

[Out]

1/2*Si(2*arcsin(a*x))/a^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4731, 4491, 12, 3380} \[ \int \frac {x}{\arcsin (a x)} \, dx=\frac {\text {Si}(2 \arcsin (a x))}{2 a^2} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 (x)}{x} \, dx,x,\arcsin (a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\arcsin (a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\arcsin (a x)\right )}{2 a^2} \\ & = \frac {\text {Si}(2 \arcsin (a x))}{2 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\arcsin (a x)} \, dx=\frac {\text {Si}(2 \arcsin (a x))}{2 a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {\operatorname {Si}\left (2 \arcsin \left (a x \right )\right )}{2 a^{2}}\)	\(13\)
default	\(\frac {\operatorname {Si}\left (2 \arcsin \left (a x \right )\right )}{2 a^{2}}\)	\(13\)

[In]

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

[Out]

1/2*Si(2*arcsin(a*x))/a^2

Fricas [F]

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

[In]

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

[Out]

integral(x/arcsin(a*x), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x/asin(a*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x/arcsin(a*x), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {x}{\arcsin (a x)} \, dx=\frac {\operatorname {Si}\left (2 \, \arcsin \left (a x\right )\right )}{2 \, a^{2}} \]

[In]

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

[Out]

1/2*sin_integral(2*arcsin(a*x))/a^2

Mupad [F(-1)]

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

[In]

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

[Out]

int(x/asin(a*x), x)