3.144 \(\int \frac{a+b \cos ^{-1}(c x)}{x} \, dx\)

Optimal. Leaf size=63 \[ -\frac{1}{2} i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\log \left (1+e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0716771, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4626, 3719, 2190, 2279, 2391} \[ -\frac{1}{2} i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\log \left (1+e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])/x,x]

[Out]

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

Rule 4626

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[(a + b*x)^n/Cot[x], x], x, ArcCos[c
*x]] /; FreeQ[{a, b, c}, 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{a+b \cos ^{-1}(c x)}{x} \, dx &=-\operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cos ^{-1}(c x)\right ) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-b \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cos ^{-1}(c x)\right ) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )+\frac{1}{2} (i b) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )\\ &=-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cos ^{-1}(c x)\right ) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac{1}{2} i b \text{Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.0175066, size = 58, normalized size = 0.92 \[ -\frac{1}{2} i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )+a \log (x)-\frac{1}{2} i b \cos ^{-1}(c x)^2+b \cos ^{-1}(c x) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCos[c*x])/x,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.06, size = 77, normalized size = 1.2 \begin{align*} a\ln \left ( cx \right ) -{\frac{i}{2}}b \left ( \arccos \left ( cx \right ) \right ) ^{2}+b\arccos \left ( cx \right ) \ln \left ( 1+ \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) -{\frac{i}{2}}b{\it polylog} \left ( 2,- \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))/x,x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x,x, algorithm="maxima")

[Out]

b*integrate(arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/x, x) + a*log(x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \arccos \left (c x\right ) + a}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arccos(c*x) + a)/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))/x,x)

[Out]

Integral((a + b*acos(c*x))/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccos(c*x) + a)/x, x)