3.1.6 \(\int \frac {\text {ArcCos}(a x)}{x} \, dx\) [6]

Optimal. Leaf size=51 \[ -\frac {1}{2} i \text {ArcCos}(a x)^2+\text {ArcCos}(a x) \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-\frac {1}{2} i \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right ) \]

[Out]

-1/2*I*arccos(a*x)^2+arccos(a*x)*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)-1/2*I*polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2)
)^2)

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Rubi [A]
time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4722, 3800, 2221, 2317, 2438} \begin {gather*} -\frac {1}{2} i \text {Li}_2\left (-e^{2 i \text {ArcCos}(a x)}\right )-\frac {1}{2} i \text {ArcCos}(a x)^2+\text {ArcCos}(a x) \log \left (1+e^{2 i \text {ArcCos}(a x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/2*I)*ArcCos[a*x]^2 + ArcCos[a*x]*Log[1 + E^((2*I)*ArcCos[a*x])] - (I/2)*PolyLog[2, -E^((2*I)*ArcCos[a*x])]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 4722

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcCos[c
*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\cos ^{-1}(a x)}{x} \, dx &=-\text {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {1}{2} i \cos ^{-1}(a x)^2+2 i \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )+\frac {1}{2} i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}(a x)}\right )\\ &=-\frac {1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\frac {1}{2} i \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 51, normalized size = 1.00 \begin {gather*} -\frac {1}{2} i \text {ArcCos}(a x)^2+\text {ArcCos}(a x) \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-\frac {1}{2} i \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/2*I)*ArcCos[a*x]^2 + ArcCos[a*x]*Log[1 + E^((2*I)*ArcCos[a*x])] - (I/2)*PolyLog[2, -E^((2*I)*ArcCos[a*x])]

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Maple [A]
time = 0.31, size = 68, normalized size = 1.33

method	result	size
derivativedivides	\(-\frac {i \arccos \left (a x \right )^{2}}{2}+\arccos \left (a x \right ) \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {i \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(68\)
default	\(-\frac {i \arccos \left (a x \right )^{2}}{2}+\arccos \left (a x \right ) \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {i \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*arccos(a*x)^2+arccos(a*x)*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)-1/2*I*polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2)
)^2)

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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(arccos(a*x)/x,x, algorithm="maxima")

[Out]

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

[Out]

integral(arccos(a*x)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)/x, x)

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

[Out]

integrate(arccos(a*x)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acos(a*x)/x, x)

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