3.1.44 \(\int \frac {x^4}{\text {ArcCos}(a x)} \, dx\) [44]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 41, 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 {\text {Si}(\text {ArcCos}(a x))}{8 a^5}-\frac {3 \text {Si}(3 \text {ArcCos}(a x))}{16 a^5}-\frac {\text {Si}(5 \text {ArcCos}(a x))}{16 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*SinIntegral[ArcCos[a*x]]/a^5 - (3*SinIntegral[3*ArcCos[a*x]])/(16*a^5) - SinIntegral[5*ArcCos[a*x]]/(16*a
^5)

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^4}{\cos ^{-1}(a x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}\\ &=-\frac {\text {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 {\text {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^5}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^5}-\frac {3 \text {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 = 0.06, size = 31, normalized size = 0.76 \begin {gather*} -\frac {2 \text {Si}(\text {ArcCos}(a x))+3 \text {Si}(3 \text {ArcCos}(a x))+\text {Si}(5 \text {ArcCos}(a x))}{16 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(2*SinIntegral[ArcCos[a*x]] + 3*SinIntegral[3*ArcCos[a*x]] + SinIntegral[5*ArcCos[a*x]])/a^5

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 31, normalized size = 0.76

method	result	size
derivativedivides	\(\frac {-\frac {3 \sinIntegral \left (3 \arccos \left (a x \right )\right )}{16}-\frac {\sinIntegral \left (5 \arccos \left (a x \right )\right )}{16}-\frac {\sinIntegral \left (\arccos \left (a x \right )\right )}{8}}{a^{5}}\)	\(31\)
default	\(\frac {-\frac {3 \sinIntegral \left (3 \arccos \left (a x \right )\right )}{16}-\frac {\sinIntegral \left (5 \arccos \left (a x \right )\right )}{16}-\frac {\sinIntegral \left (\arccos \left (a x \right )\right )}{8}}{a^{5}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccos(a*x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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^4/arccos(a*x),x, algorithm="maxima")

[Out]

integrate(x^4/arccos(a*x), x)

________________________________________________________________________________________

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^4/arccos(a*x),x, algorithm="fricas")

[Out]

integral(x^4/arccos(a*x), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\operatorname {acos}{\left (a x \right )}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.43, size = 35, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^4}{\mathrm {acos}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4/acos(a*x), x)

________________________________________________________________________________________