\(\int \frac {x \arctan (a x)^2}{c+a^2 c x^2} \, dx\) [285]

Optimal result

Integrand size = 20, antiderivative size = 102 \[ \int \frac {x \arctan (a x)^2}{c+a^2 c x^2} \, dx=-\frac {i \arctan (a x)^3}{3 a^2 c}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^2 c}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^2 c} \]

[Out]

-1/3*I*arctan(a*x)^3/a^2/c-arctan(a*x)^2*ln(2/(1+I*a*x))/a^2/c-I*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a^2/c-1/
2*polylog(3,1-2/(1+I*a*x))/a^2/c

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5040, 4964, 5004, 5114, 6745} \[ \int \frac {x \arctan (a x)^2}{c+a^2 c x^2} \, dx=-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 c}-\frac {i \arctan (a x)^3}{3 a^2 c}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a^2 c} \]

[In]

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

[Out]

((-1/3*I)*ArcTan[a*x]^3)/(a^2*c) - (ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^2*c) - (I*ArcTan[a*x]*PolyLog[2, 1 -
2/(1 + I*a*x)])/(a^2*c) - PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^2*c)

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 5004

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

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]

Rule 5114

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {i \arctan (a x)^3}{3 a^2 c}-\frac {\int \frac {\arctan (a x)^2}{i-a x} \, dx}{a c} \\ & = -\frac {i \arctan (a x)^3}{3 a^2 c}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}+\frac {2 \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a c} \\ & = -\frac {i \arctan (a x)^3}{3 a^2 c}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^2 c}+\frac {i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a c} \\ & = -\frac {i \arctan (a x)^3}{3 a^2 c}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^2 c}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^2 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90 \[ \int \frac {x \arctan (a x)^2}{c+a^2 c x^2} \, dx=-\frac {2 i \arctan (a x)^3+6 \arctan (a x)^2 \log \left (\frac {2 i}{i-a x}\right )+6 i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {i+a x}{-i+a x}\right )+3 \operatorname {PolyLog}\left (3,\frac {i+a x}{-i+a x}\right )}{6 a^2 c} \]

[In]

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

[Out]

-1/6*((2*I)*ArcTan[a*x]^3 + 6*ArcTan[a*x]^2*Log[(2*I)/(I - a*x)] + (6*I)*ArcTan[a*x]*PolyLog[2, (I + a*x)/(-I
+ a*x)] + 3*PolyLog[3, (I + a*x)/(-I + a*x)])/(a^2*c)

Maple [C] (warning: unable to verify)

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

Time = 6.35 (sec) , antiderivative size = 756, normalized size of antiderivative = 7.41

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {i \arctan \left (a x \right )^{3}}{3}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{2}}{4}-i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}}{c}}{a^{2}}\)	\(756\)
default	\(\frac {\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {i \arctan \left (a x \right )^{3}}{3}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{2}}{4}-i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}}{c}}{a^{2}}\)	\(756\)
parts	\(\frac {\ln \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{2}}{2 a^{2} c}-\frac {\frac {\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a}-\frac {i \arctan \left (a x \right )^{3}}{3 a}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{2}}{4 a}-\frac {i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{a}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a}}{a c}\)	\(773\)

[In]

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

[Out]

1/a^2*(1/2/c*arctan(a*x)^2*ln(a^2*x^2+1)-1/c*(arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-1/3*I*arctan(a*x)^
3+1/4*(-I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2
+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/(
(1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^
2)-2*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+I*Pi*csgn(I*((1+I*a*x)^2
/(a^2*x^2+1)+1)^2)^3-I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))+2*I*Pi*csgn(I*
(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+I*Pi*csg
n(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-I*Pi*csgn(I*(1+I*
a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+4*ln(2))*arctan(a*x)^2-I*arctan(a*x)*polylog(2,-(1+I*a*x)^
2/(a^2*x^2+1))+1/2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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