3.44 \(\int \frac{x^4}{\cos ^{-1}(a x)} \, dx\)

Optimal. Leaf size=41 \[ -\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{16 a^5}-\frac{\text{Si}\left (5 \cos ^{-1}(a x)\right )}{16 a^5} \]

[Out]

-SinIntegral[ArcCos[a*x]]/(8*a^5) - (3*SinIntegral[3*ArcCos[a*x]])/(16*a^5) - SinIntegral[5*ArcCos[a*x]]/(16*a
^5)

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Rubi [A]  time = 0.0748271, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4636, 4406, 3299} \[ -\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{16 a^5}-\frac{\text{Si}\left (5 \cos ^{-1}(a x)\right )}{16 a^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCos[a*x],x]

[Out]

-SinIntegral[ArcCos[a*x]]/(8*a^5) - (3*SinIntegral[3*ArcCos[a*x]])/(16*a^5) - SinIntegral[5*ArcCos[a*x]]/(16*a
^5)

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]

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 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]

Rubi steps

\begin{align*} \int \frac{x^4}{\cos ^{-1}(a x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^4(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{\sin (x)}{8 x}+\frac{3 \sin (3 x)}{16 x}+\frac{\sin (5 x)}{16 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^5}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (5 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{16 a^5}-\frac{\text{Si}\left (5 \cos ^{-1}(a x)\right )}{16 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0741457, size = 31, normalized size = 0.76 \[ -\frac{2 \text{Si}\left (\cos ^{-1}(a x)\right )+3 \text{Si}\left (3 \cos ^{-1}(a x)\right )+\text{Si}\left (5 \cos ^{-1}(a x)\right )}{16 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcCos[a*x],x]

[Out]

-(2*SinIntegral[ArcCos[a*x]] + 3*SinIntegral[3*ArcCos[a*x]] + SinIntegral[5*ArcCos[a*x]])/(16*a^5)

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Maple [A]  time = 0.047, size = 31, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{5}} \left ( -{\frac{3\,{\it Si} \left ( 3\,\arccos \left ( ax \right ) \right ) }{16}}-{\frac{{\it Si} \left ( 5\,\arccos \left ( ax \right ) \right ) }{16}}-{\frac{{\it Si} \left ( \arccos \left ( ax \right ) \right ) }{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccos(a*x),x)

[Out]

1/a^5*(-3/16*Si(3*arccos(a*x))-1/16*Si(5*arccos(a*x))-1/8*Si(arccos(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\arccos \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x),x, algorithm="maxima")

[Out]

integrate(x^4/arccos(a*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4}}{\arccos \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x),x, algorithm="fricas")

[Out]

integral(x^4/arccos(a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acos(a*x),x)

[Out]

Integral(x**4/acos(a*x), x)

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Giac [A]  time = 1.12244, size = 47, normalized size = 1.15 \begin{align*} -\frac{\operatorname{Si}\left (5 \, \arccos \left (a x\right )\right )}{16 \, a^{5}} - \frac{3 \, \operatorname{Si}\left (3 \, \arccos \left (a x\right )\right )}{16 \, a^{5}} - \frac{\operatorname{Si}\left (\arccos \left (a x\right )\right )}{8 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x),x, algorithm="giac")

[Out]

-1/16*sin_integral(5*arccos(a*x))/a^5 - 3/16*sin_integral(3*arccos(a*x))/a^5 - 1/8*sin_integral(arccos(a*x))/a
^5