3.43 \(\int \frac{x^5}{\cos ^{-1}(a x)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{5 \text{Si}\left (2 \cos ^{-1}(a x)\right )}{32 a^6}-\frac{\text{Si}\left (4 \cos ^{-1}(a x)\right )}{8 a^6}-\frac{\text{Si}\left (6 \cos ^{-1}(a x)\right )}{32 a^6} \]

[Out]

(-5*SinIntegral[2*ArcCos[a*x]])/(32*a^6) - SinIntegral[4*ArcCos[a*x]]/(8*a^6) - SinIntegral[6*ArcCos[a*x]]/(32
*a^6)

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Rubi [A]  time = 0.0788047, antiderivative size = 43, 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} \[ -\frac{5 \text{Si}\left (2 \cos ^{-1}(a x)\right )}{32 a^6}-\frac{\text{Si}\left (4 \cos ^{-1}(a x)\right )}{8 a^6}-\frac{\text{Si}\left (6 \cos ^{-1}(a x)\right )}{32 a^6} \]

Antiderivative was successfully verified.

[In]

Int[x^5/ArcCos[a*x],x]

[Out]

(-5*SinIntegral[2*ArcCos[a*x]])/(32*a^6) - SinIntegral[4*ArcCos[a*x]]/(8*a^6) - SinIntegral[6*ArcCos[a*x]]/(32
*a^6)

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^5}{\cos ^{-1}(a x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^5(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^6}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{5 \sin (2 x)}{32 x}+\frac{\sin (4 x)}{8 x}+\frac{\sin (6 x)}{32 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^6}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (6 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^6}-\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^6}-\frac{5 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^6}\\ &=-\frac{5 \text{Si}\left (2 \cos ^{-1}(a x)\right )}{32 a^6}-\frac{\text{Si}\left (4 \cos ^{-1}(a x)\right )}{8 a^6}-\frac{\text{Si}\left (6 \cos ^{-1}(a x)\right )}{32 a^6}\\ \end{align*}

Mathematica [A]  time = 0.087031, size = 33, normalized size = 0.77 \[ -\frac{5 \text{Si}\left (2 \cos ^{-1}(a x)\right )+4 \text{Si}\left (4 \cos ^{-1}(a x)\right )+\text{Si}\left (6 \cos ^{-1}(a x)\right )}{32 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/ArcCos[a*x],x]

[Out]

-(5*SinIntegral[2*ArcCos[a*x]] + 4*SinIntegral[4*ArcCos[a*x]] + SinIntegral[6*ArcCos[a*x]])/(32*a^6)

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Maple [A]  time = 0.055, size = 33, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{6}} \left ( -{\frac{5\,{\it Si} \left ( 2\,\arccos \left ( ax \right ) \right ) }{32}}-{\frac{{\it Si} \left ( 4\,\arccos \left ( ax \right ) \right ) }{8}}-{\frac{{\it Si} \left ( 6\,\arccos \left ( ax \right ) \right ) }{32}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/arccos(a*x),x)

[Out]

1/a^6*(-5/32*Si(2*arccos(a*x))-1/8*Si(4*arccos(a*x))-1/32*Si(6*arccos(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\arccos \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arccos(a*x),x, algorithm="maxima")

[Out]

integrate(x^5/arccos(a*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{5}}{\arccos \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arccos(a*x),x, algorithm="fricas")

[Out]

integral(x^5/arccos(a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/acos(a*x),x)

[Out]

Integral(x**5/acos(a*x), x)

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Giac [A]  time = 1.15649, size = 50, normalized size = 1.16 \begin{align*} -\frac{\operatorname{Si}\left (6 \, \arccos \left (a x\right )\right )}{32 \, a^{6}} - \frac{\operatorname{Si}\left (4 \, \arccos \left (a x\right )\right )}{8 \, a^{6}} - \frac{5 \, \operatorname{Si}\left (2 \, \arccos \left (a x\right )\right )}{32 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arccos(a*x),x, algorithm="giac")

[Out]

-1/32*sin_integral(6*arccos(a*x))/a^6 - 1/8*sin_integral(4*arccos(a*x))/a^6 - 5/32*sin_integral(2*arccos(a*x))
/a^6