3.100 \(\int \frac{a+b \tan ^{-1}(c x^3)}{x} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{6} i b \text{PolyLog}\left (2,-i c x^3\right )-\frac{1}{6} i b \text{PolyLog}\left (2,i c x^3\right )+a \log (x) \]

[Out]

a*Log[x] + (I/6)*b*PolyLog[2, (-I)*c*x^3] - (I/6)*b*PolyLog[2, I*c*x^3]

________________________________________________________________________________________

Rubi [A]  time = 0.0503424, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5031, 4848, 2391} \[ \frac{1}{6} i b \text{PolyLog}\left (2,-i c x^3\right )-\frac{1}{6} i b \text{PolyLog}\left (2,i c x^3\right )+a \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^3])/x,x]

[Out]

a*Log[x] + (I/6)*b*PolyLog[2, (-I)*c*x^3] - (I/6)*b*PolyLog[2, I*c*x^3]

Rule 5031

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*ArcTan[c*x])^p
/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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 \tan ^{-1}\left (c x^3\right )}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{a+b \tan ^{-1}(c x)}{x} \, dx,x,x^3\right )\\ &=a \log (x)+\frac{1}{6} (i b) \operatorname{Subst}\left (\int \frac{\log (1-i c x)}{x} \, dx,x,x^3\right )-\frac{1}{6} (i b) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x} \, dx,x,x^3\right )\\ &=a \log (x)+\frac{1}{6} i b \text{Li}_2\left (-i c x^3\right )-\frac{1}{6} i b \text{Li}_2\left (i c x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0047405, size = 39, normalized size = 1. \[ \frac{1}{6} i b \text{PolyLog}\left (2,-i c x^3\right )-\frac{1}{6} i b \text{PolyLog}\left (2,i c x^3\right )+a \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTan[c*x^3])/x,x]

[Out]

a*Log[x] + (I/6)*b*PolyLog[2, (-I)*c*x^3] - (I/6)*b*PolyLog[2, I*c*x^3]

________________________________________________________________________________________

Maple [C]  time = 0.086, size = 63, normalized size = 1.6 \begin{align*} a\ln \left ( x \right ) +b\ln \left ( x \right ) \arctan \left ( c{x}^{3} \right ) -{\frac{b}{2\,c}\sum _{{\it \_R1}={\it RootOf} \left ({c}^{2}{{\it \_Z}}^{6}+1 \right ) }{\frac{1}{{{\it \_R1}}^{3}} \left ( \ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x^3))/x,x)

[Out]

a*ln(x)+b*ln(x)*arctan(c*x^3)-1/2*b/c*sum(1/_R1^3*(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1)),_R1=RootOf(_Z^6*c
^2+1))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{\arctan \left (c x^{3}\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*arctan(c*x^3))/x,x, algorithm="maxima")

[Out]

b*integrate(arctan(c*x^3)/x, x) + a*log(x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \arctan \left (c x^{3}\right ) + a}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))/x,x, algorithm="fricas")

[Out]

integral((b*arctan(c*x^3) + a)/x, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{atan}{\left (c x^{3} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x**3))/x,x)

[Out]

Integral((a + b*atan(c*x**3))/x, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arctan \left (c x^{3}\right ) + a}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))/x,x, algorithm="giac")

[Out]

integrate((b*arctan(c*x^3) + a)/x, x)