3.69 \(\int \frac{\cos ^{-1}(a x^n)}{x} \, dx\)

Optimal. Leaf size=68 \[ -\frac{i \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac{i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.0614399, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4831, 3719, 2190, 2279, 2391} \[ -\frac{i \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac{i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4831

Int[ArcCos[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> -Dist[p^(-1), Subst[Int[x^n*Tan[x], x], x, ArcCos[a*x^p]]
, x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]

Rule 3719

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

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

Mathematica [B]  time = 0.137089, size = 141, normalized size = 2.07 \[ \frac{a \left (\text{PolyLog}\left (2,e^{-2 \sinh ^{-1}\left (\sqrt{-a^2} x^n\right )}\right )+2 n \log (x) \log \left (\sqrt{-a^2} x^n+\sqrt{1-a^2 x^{2 n}}\right )-\sinh ^{-1}\left (\sqrt{-a^2} x^n\right )^2-2 \sinh ^{-1}\left (\sqrt{-a^2} x^n\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (\sqrt{-a^2} x^n\right )}\right )\right )}{2 \sqrt{-a^2} n}+\log (x) \cos ^{-1}\left (a x^n\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 89, normalized size = 1.3 \begin{align*}{\frac{-{\frac{i}{2}} \left ( \arccos \left ( a{x}^{n} \right ) \right ) ^{2}}{n}}+{\frac{\arccos \left ( a{x}^{n} \right ) }{n}\ln \left ( 1+ \left ( a{x}^{n}+i\sqrt{1-{a}^{2} \left ({x}^{n} \right ) ^{2}} \right ) ^{2} \right ) }-{\frac{{\frac{i}{2}}}{n}{\it polylog} \left ( 2,- \left ( a{x}^{n}+i\sqrt{1-{a}^{2} \left ({x}^{n} \right ) ^{2}} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a n \int \frac{\sqrt{a x^{n} + 1} \sqrt{-a x^{n} + 1} x^{n} \log \left (x\right )}{a^{2} x x^{2 \, n} - x}\,{d x} + \arctan \left (\sqrt{a x^{n} + 1} \sqrt{-a x^{n} + 1}, a x^{n}\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*n*integrate(sqrt(a*x^n + 1)*sqrt(-a*x^n + 1)*x^n*log(x)/(a^2*x*x^(2*n) - x), x) + arctan2(sqrt(a*x^n + 1)*s
qrt(-a*x^n + 1), a*x^n)*log(x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x**n)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(a*x^n)/x, x)