\(\int \frac {x^2}{\arcsin (a x)} \, dx\) [46]

Optimal result

Integrand size = 10, antiderivative size = 27 \[ \int \frac {x^2}{\arcsin (a x)} \, dx=\frac {\operatorname {CosIntegral}(\arcsin (a x))}{4 a^3}-\frac {\operatorname {CosIntegral}(3 \arcsin (a x))}{4 a^3} \]

[Out]

1/4*Ci(arcsin(a*x))/a^3-1/4*Ci(3*arcsin(a*x))/a^3

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 27, 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, 3383} \[ \int \frac {x^2}{\arcsin (a x)} \, dx=\frac {\operatorname {CosIntegral}(\arcsin (a x))}{4 a^3}-\frac {\operatorname {CosIntegral}(3 \arcsin (a x))}{4 a^3} \]

[In]

Int[x^2/ArcSin[a*x],x]

[Out]

CosIntegral[ArcSin[a*x]]/(4*a^3) - CosIntegral[3*ArcSin[a*x]]/(4*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 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 ^2(x)}{x} \, dx,x,\arcsin (a x)\right )}{a^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\arcsin (a x)\right )}{a^3} \\ & = \frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a x)\right )}{4 a^3}-\frac {\text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{4 a^3} \\ & = \frac {\operatorname {CosIntegral}(\arcsin (a x))}{4 a^3}-\frac {\operatorname {CosIntegral}(3 \arcsin (a x))}{4 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{\arcsin (a x)} \, dx=\frac {\operatorname {CosIntegral}(\arcsin (a x))-\operatorname {CosIntegral}(3 \arcsin (a x))}{4 a^3} \]

[In]

Integrate[x^2/ArcSin[a*x],x]

[Out]

(CosIntegral[ArcSin[a*x]] - CosIntegral[3*ArcSin[a*x]])/(4*a^3)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{4}-\frac {\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{4}}{a^{3}}\)	\(22\)
default	\(\frac {\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{4}-\frac {\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{4}}{a^{3}}\)	\(22\)

[In]

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

[Out]

1/a^3*(1/4*Ci(arcsin(a*x))-1/4*Ci(3*arcsin(a*x)))

Fricas [F]

\[ \int \frac {x^2}{\arcsin (a x)} \, dx=\int { \frac {x^{2}}{\arcsin \left (a x\right )} \,d x } \]

[In]

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

[Out]

integral(x^2/arcsin(a*x), x)

Sympy [F]

\[ \int \frac {x^2}{\arcsin (a x)} \, dx=\int \frac {x^{2}}{\operatorname {asin}{\left (a x \right )}}\, dx \]

[In]

integrate(x**2/asin(a*x),x)

[Out]

Integral(x**2/asin(a*x), x)

Maxima [F]

\[ \int \frac {x^2}{\arcsin (a x)} \, dx=\int { \frac {x^{2}}{\arcsin \left (a x\right )} \,d x } \]

[In]

integrate(x^2/arcsin(a*x),x, algorithm="maxima")

[Out]

integrate(x^2/arcsin(a*x), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\arcsin (a x)} \, dx=-\frac {\operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a^{3}} + \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{4 \, a^{3}} \]

[In]

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

[Out]

-1/4*cos_integral(3*arcsin(a*x))/a^3 + 1/4*cos_integral(arcsin(a*x))/a^3

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\arcsin (a x)} \, dx=\int \frac {x^2}{\mathrm {asin}\left (a\,x\right )} \,d x \]

[In]

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

[Out]

int(x^2/asin(a*x), x)