3.1.47 \(\int \frac {x}{\text {ArcCos}(a x)} \, dx\) [47]

Optimal. Leaf size=14 \[ -\frac {\text {Si}(2 \text {ArcCos}(a x))}{2 a^2} \]

[Out]

-1/2*Si(2*arccos(a*x))/a^2

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4732, 4491, 12, 3380} \begin {gather*} -\frac {\text {Si}(2 \text {ArcCos}(a x))}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*SinIntegral[2*ArcCos[a*x]]/a^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\text {Si}(2 \text {ArcCos}(a x))}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*SinIntegral[2*ArcCos[a*x]]/a^2

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 13, normalized size = 0.93

method	result	size
derivativedivides	\(-\frac {\sinIntegral \left (2 \arccos \left (a x \right )\right )}{2 a^{2}}\)	\(13\)
default	\(-\frac {\sinIntegral \left (2 \arccos \left (a x \right )\right )}{2 a^{2}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*Si(2*arccos(a*x))/a^2

________________________________________________________________________________________

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/arccos(a*x),x, algorithm="maxima")

[Out]

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

[Out]

integral(x/arccos(a*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/acos(a*x), x)

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 12, normalized size = 0.86 \begin {gather*} -\frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{2 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sin_integral(2*arccos(a*x))/a^2

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {x}{\mathrm {acos}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/acos(a*x), x)

________________________________________________________________________________________