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

Optimal result

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

[Out]

-1/4*Si(arccos(a*x))/a^3-1/4*Si(3*arccos(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 = {4732, 4491, 3380} \[ \int \frac {x^2}{\arccos (a x)} \, dx=-\frac {\text {Si}(\arccos (a x))}{4 a^3}-\frac {\text {Si}(3 \arccos (a x))}{4 a^3} \]

[In]

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

[Out]

-1/4*SinIntegral[ArcCos[a*x]]/a^3 - SinIntegral[3*ArcCos[a*x]]/(4*a^3)

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 4732

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(b*c^(m + 1))^(-1), Subst[Int[x^n*C
os[-a/b + x/b]^m*Sin[-a/b + x/b], x], x, a + b*ArcCos[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 ^2(x) \sin (x)}{x} \, dx,x,\arccos (a x)\right )}{a^3} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\arccos (a x)\right )}{a^3} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arccos (a x)\right )}{4 a^3}-\frac {\text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arccos (a x)\right )}{4 a^3} \\ & = -\frac {\text {Si}(\arccos (a x))}{4 a^3}-\frac {\text {Si}(3 \arccos (a x))}{4 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {x^2}{\arccos (a x)} \, dx=-\frac {\text {Si}(\arccos (a x))+\text {Si}(3 \arccos (a x))}{4 a^3} \]

[In]

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

[Out]

-1/4*(SinIntegral[ArcCos[a*x]] + SinIntegral[3*ArcCos[a*x]])/a^3

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81

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

[In]

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

[Out]

1/a^3*(-1/4*Si(3*arccos(a*x))-1/4*Si(arccos(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x^2/arccos(a*x), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/acos(a*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2/arccos(a*x), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\arccos (a x)} \, dx=-\frac {\operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{4 \, a^{3}} - \frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{4 \, a^{3}} \]

[In]

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

[Out]

-1/4*sin_integral(3*arccos(a*x))/a^3 - 1/4*sin_integral(arccos(a*x))/a^3

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2/acos(a*x), x)