3.46 \(\int \frac {x^2}{\cos ^{-1}(a x)} \, dx\)

Optimal. Leaf size=27 \[ -\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{4 a^3}-\frac {\text {Si}\left (3 \cos ^{-1}(a x)\right )}{4 a^3} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 27, 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 = {4636, 4406, 3299} \[ -\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{4 a^3}-\frac {\text {Si}\left (3 \cos ^{-1}(a x)\right )}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4636

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

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 20, normalized size = 0.74 \[ -\frac {\text {Si}\left (\cos ^{-1}(a x)\right )+\text {Si}\left (3 \cos ^{-1}(a x)\right )}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\arccos \left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2/arccos(a*x), x)

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giac [A]  time = 1.93, size = 23, normalized size = 0.85 \[ -\frac {\operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{4 \, a^{3}} - \frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.04, size = 22, normalized size = 0.81 \[ \frac {-\frac {\Si \left (3 \arccos \left (a x \right )\right )}{4}-\frac {\Si \left (\arccos \left (a x \right )\right )}{4}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\arccos \left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arccos(a*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^2}{\mathrm {acos}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/acos(a*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {acos}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/acos(a*x), x)

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