3.178 \(\int \frac{\tan ^{-1}(a x)}{x (c+a^2 c x^2)} \, dx\)

Optimal. Leaf size=64 \[ -\frac{i \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c}-\frac{i \tan ^{-1}(a x)^2}{2 c}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c} \]

[Out]

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

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Rubi [A]  time = 0.0996789, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4924, 4868, 2447} \[ -\frac{i \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c}-\frac{i \tan ^{-1}(a x)^2}{2 c}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]/(x*(c + a^2*c*x^2)),x]

[Out]

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

Rule 4924

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*d*(p + 1)), x] + Dist[I/d, Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]

Rule 4868

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTan[c*x]
)^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2 - 2/(1 + (e*x)/d)
])/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

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

Mathematica [A]  time = 0.0243861, size = 103, normalized size = 1.61 \[ \frac{i \text{PolyLog}(2,-i a x)}{2 c}-\frac{i \text{PolyLog}(2,i a x)}{2 c}+\frac{i \text{PolyLog}\left (2,-\frac{a x+i}{-a x+i}\right )}{2 c}+\frac{i \tan ^{-1}(a x)^2}{2 c}+\frac{\log \left (\frac{2 i}{-a x+i}\right ) \tan ^{-1}(a x)}{c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTan[a*x]/(x*(c + a^2*c*x^2)),x]

[Out]

((I/2)*ArcTan[a*x]^2)/c + (ArcTan[a*x]*Log[(2*I)/(I - a*x)])/c + ((I/2)*PolyLog[2, (-I)*a*x])/c - ((I/2)*PolyL
og[2, I*a*x])/c + ((I/2)*PolyLog[2, -((I + a*x)/(I - a*x))])/c

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Maple [B]  time = 0.092, size = 251, normalized size = 3.9 \begin{align*} -{\frac{\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,c}}+{\frac{\arctan \left ( ax \right ) \ln \left ( ax \right ) }{c}}+{\frac{{\frac{i}{2}}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) }{c}}-{\frac{{\frac{i}{2}}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) }{c}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+iax \right ) }{c}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( 1-iax \right ) }{c}}-{\frac{{\frac{i}{4}}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{c}}+{\frac{{\frac{i}{8}} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{c}}+{\frac{{\frac{i}{4}}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{c}}+{\frac{{\frac{i}{4}}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{c}}+{\frac{{\frac{i}{4}}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{c}}-{\frac{{\frac{i}{8}} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{c}}-{\frac{{\frac{i}{4}}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{c}}-{\frac{{\frac{i}{4}}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)/x/(a^2*c*x^2+c),x)

[Out]

-1/2/c*arctan(a*x)*ln(a^2*x^2+1)+1/c*arctan(a*x)*ln(a*x)+1/2*I/c*ln(a*x)*ln(1+I*a*x)-1/2*I/c*ln(a*x)*ln(1-I*a*
x)+1/2*I/c*dilog(1+I*a*x)-1/2*I/c*dilog(1-I*a*x)-1/4*I/c*ln(a^2*x^2+1)*ln(a*x-I)+1/8*I/c*ln(a*x-I)^2+1/4*I/c*l
n(a*x-I)*ln(-1/2*I*(a*x+I))+1/4*I/c*dilog(-1/2*I*(a*x+I))+1/4*I/c*ln(a^2*x^2+1)*ln(a*x+I)-1/8*I/c*ln(a*x+I)^2-
1/4*I/c*ln(a*x+I)*ln(1/2*I*(a*x-I))-1/4*I/c*dilog(1/2*I*(a*x-I))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

integrate(arctan(a*x)/((a^2*c*x^2 + c)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arctan \left (a x\right )}{a^{2} c x^{3} + c x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

integral(arctan(a*x)/(a^2*c*x^3 + c*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{atan}{\left (a x \right )}}{a^{2} x^{3} + x}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)/x/(a**2*c*x**2+c),x)

[Out]

Integral(atan(a*x)/(a**2*x**3 + x), x)/c

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

integrate(arctan(a*x)/((a^2*c*x^2 + c)*x), x)