3.1.43 \(\int \frac {x^5}{\text {ArcCos}(a x)} \, dx\) [43]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4732, 4491, 3380} \begin {gather*} -\frac {5 \text {Si}(2 \text {ArcCos}(a x))}{32 a^6}-\frac {\text {Si}(4 \text {ArcCos}(a x))}{8 a^6}-\frac {\text {Si}(6 \text {ArcCos}(a x))}{32 a^6} \end {gather*}

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

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Mathematica [A]
time = 0.07, size = 33, normalized size = 0.77 \begin {gather*} -\frac {5 \text {Si}(2 \text {ArcCos}(a x))+4 \text {Si}(4 \text {ArcCos}(a x))+\text {Si}(6 \text {ArcCos}(a x))}{32 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/32*(5*SinIntegral[2*ArcCos[a*x]] + 4*SinIntegral[4*ArcCos[a*x]] + SinIntegral[6*ArcCos[a*x]])/a^6

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Maple [A]
time = 0.14, size = 33, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {-\frac {5 \sinIntegral \left (2 \arccos \left (a x \right )\right )}{32}-\frac {\sinIntegral \left (4 \arccos \left (a x \right )\right )}{8}-\frac {\sinIntegral \left (6 \arccos \left (a x \right )\right )}{32}}{a^{6}}\)	\(33\)
default	\(\frac {-\frac {5 \sinIntegral \left (2 \arccos \left (a x \right )\right )}{32}-\frac {\sinIntegral \left (4 \arccos \left (a x \right )\right )}{8}-\frac {\sinIntegral \left (6 \arccos \left (a x \right )\right )}{32}}{a^{6}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/arccos(a*x),x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\operatorname {acos}{\left (a x \right )}}\, dx \end {gather*}

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 = 0.46, size = 37, normalized size = 0.86 \begin {gather*} -\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 {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^5}{\mathrm {acos}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^5/acos(a*x), x)

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