\(\int \frac {x \arctan (a x)}{c+a^2 c x^2} \, dx\) [176]

Optimal result

Integrand size = 18, antiderivative size = 72 \[ \int \frac {x \arctan (a x)}{c+a^2 c x^2} \, dx=-\frac {i \arctan (a x)^2}{2 a^2 c}-\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5040, 4964, 2449, 2352} \[ \int \frac {x \arctan (a x)}{c+a^2 c x^2} \, dx=-\frac {i \arctan (a x)^2}{2 a^2 c}-\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a^2 c} \]

[In]

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

[Out]

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[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 5040

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*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07 \[ \int \frac {x \arctan (a x)}{c+a^2 c x^2} \, dx=-\frac {i \arctan (a x)^2}{2 a^2 c}-\frac {\arctan (a x) \log \left (\frac {2 i}{i-a x}\right )}{a^2 c}-\frac {i \operatorname {PolyLog}\left (2,\frac {i+a x}{-i+a x}\right )}{2 a^2 c} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.65

method	result	size
parts	\(\frac {\ln \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )}{2 a^{2} c}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }}{8 a^{3} c}\)	\(119\)
risch	\(\frac {i \ln \left (-i a x +1\right )^{2}}{8 c \,a^{2}}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c \,a^{2}}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c \,a^{2}}-\frac {i \ln \left (i a x +1\right )^{2}}{8 c \,a^{2}}-\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c \,a^{2}}+\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c \,a^{2}}\)	\(124\)
derivativedivides	\(\frac {\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c}}{a^{2}}\)	\(159\)
default	\(\frac {\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c}}{a^{2}}\)	\(159\)

[In]

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

[Out]

1/2/a^2/c*ln(a^2*x^2+1)*arctan(a*x)-1/8/a^3/c*sum(1/_alpha*(2*ln(x-_alpha)*ln(a^2*x^2+1)-a^2*(1/a^2/_alpha*ln(
x-_alpha)^2+2*_alpha*ln(x-_alpha)*ln(1/2*(x+_alpha)/_alpha)+2*_alpha*dilog(1/2*(x+_alpha)/_alpha))),_alpha=Roo
tOf(_Z^2*a^2+1))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x \arctan (a x)}{c+a^2 c x^2} \, dx=\frac {\int \frac {x \operatorname {atan}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

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

[Out]

Integral(x*atan(a*x)/(a**2*x**2 + 1), x)/c

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int((x*atan(a*x))/(c + a^2*c*x^2),x)

[Out]

int((x*atan(a*x))/(c + a^2*c*x^2), x)