\(\int \frac {\arctan (a x)}{x (c+a^2 c x^2)} \, dx\) [178]

Optimal result

Integrand size = 20, antiderivative size = 64 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx=-\frac {i \arctan (a x)^2}{2 c}+\frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {i \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5044, 4988, 2497} \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx=-\frac {i \arctan (a x)^2}{2 c}+\frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {i \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c} \]

[In]

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

[Out]

((-1/2*I)*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 2497

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]]

Rule 4988

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 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {i \arctan (a x)^2}{2 c}+\frac {i \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c} \\ & = -\frac {i \arctan (a x)^2}{2 c}+\frac {\arctan (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 \arctan (a x)^2}{2 c}+\frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {i \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.61 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx=\frac {i \arctan (a x)^2}{2 c}+\frac {\arctan (a x) \log \left (\frac {2 i}{i-a x}\right )}{c}+\frac {i \operatorname {PolyLog}(2,-i a x)}{2 c}-\frac {i \operatorname {PolyLog}(2,i a x)}{2 c}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i+a x}{i-a x}\right )}{2 c} \]

[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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (56 ) = 112\).

Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.09

method	result	size
risch	\(-\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2 c}-\frac {i \ln \left (-i a x +1\right )^{2}}{8 c}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c}+\frac {i \operatorname {dilog}\left (i a x +1\right )}{2 c}+\frac {i \ln \left (i a x +1\right )^{2}}{8 c}+\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c}-\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c}\)	\(134\)
parts	\(-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}+\frac {\arctan \left (a x \right ) \ln \left (x \right )}{c}-\frac {a \left (-\frac {i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{a}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 a^{2}}\right )}{2 c}\)	\(182\)
derivativedivides	\(\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c}\)	\(219\)
default	\(\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c}\)	\(219\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}{\left (a x \right )}}{a^{2} x^{3} + x}\, dx}{c} \]

[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

Maxima [F]

\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )}{x\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

int(atan(a*x)/(x*(c + a^2*c*x^2)),x)

[Out]

int(atan(a*x)/(x*(c + a^2*c*x^2)), x)