\(\int \frac {\arctan (a x)}{x (c+a^2 c x^2)^3} \, dx\) [196]

Optimal result

Integrand size = 20, antiderivative size = 159 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^3} \, dx=-\frac {a x}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \arctan (a x)}{32 c^3}+\frac {\arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^2}{2 c^3}+\frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3} \]

[Out]

-1/16*a*x/c^3/(a^2*x^2+1)^2-11/32*a*x/c^3/(a^2*x^2+1)-11/32*arctan(a*x)/c^3+1/4*arctan(a*x)/c^3/(a^2*x^2+1)^2+
1/2*arctan(a*x)/c^3/(a^2*x^2+1)-1/2*I*arctan(a*x)^2/c^3+arctan(a*x)*ln(2-2/(1-I*a*x))/c^3-1/2*I*polylog(2,-1+2
/(1-I*a*x))/c^3

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5086, 5044, 4988, 2497, 5050, 205, 211} \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {\arctan (a x)}{2 c^3 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {11 a x}{32 c^3 \left (a^2 x^2+1\right )}-\frac {a x}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac {i \arctan (a x)^2}{2 c^3}-\frac {11 \arctan (a x)}{32 c^3}+\frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c^3} \]

[In]

Int[ArcTan[a*x]/(x*(c + a^2*c*x^2)^3),x]

[Out]

-1/16*(a*x)/(c^3*(1 + a^2*x^2)^2) - (11*a*x)/(32*c^3*(1 + a^2*x^2)) - (11*ArcTan[a*x])/(32*c^3) + ArcTan[a*x]/
(4*c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]/(2*c^3*(1 + a^2*x^2)) - ((I/2)*ArcTan[a*x]^2)/c^3 + (ArcTan[a*x]*Log[2 -
2/(1 - I*a*x)])/c^3 - ((I/2)*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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]

Rule 5050

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

Rule 5086

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int[
x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] &
& NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = \frac {\arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{4} a \int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^2 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = -\frac {a x}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^2}{2 c^3}+\frac {i \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^3}-\frac {(3 a) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-\frac {a \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c} \\ & = -\frac {a x}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x}{32 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^2}{2 c^3}+\frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-\frac {(3 a) \int \frac {1}{c+a^2 c x^2} \, dx}{32 c^2}-\frac {a \int \frac {1}{c+a^2 c x^2} \, dx}{4 c^2} \\ & = -\frac {a x}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \arctan (a x)}{32 c^3}+\frac {\arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^2}{2 c^3}+\frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.57 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^3} \, dx=-\frac {64 i \arctan (a x)^2-4 \arctan (a x) \left (12 \cos (2 \arctan (a x))+\cos (4 \arctan (a x))+32 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+64 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+24 \sin (2 \arctan (a x))+\sin (4 \arctan (a x))}{128 c^3} \]

[In]

Integrate[ArcTan[a*x]/(x*(c + a^2*c*x^2)^3),x]

[Out]

-1/128*((64*I)*ArcTan[a*x]^2 - 4*ArcTan[a*x]*(12*Cos[2*ArcTan[a*x]] + Cos[4*ArcTan[a*x]] + 32*Log[1 - E^((2*I)
*ArcTan[a*x])]) + (64*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + 24*Sin[2*ArcTan[a*x]] + Sin[4*ArcTan[a*x]])/c^3

Maple [C] (verified)

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

Time = 0.54 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.61

method	result	size
parts	\(\frac {\arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{3}}+\frac {\arctan \left (a x \right )}{2 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) \ln \left (x \right )}{c^{3}}-\frac {a \left (-\frac {i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{a}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 a^{2}}+\frac {\frac {11}{8} a^{2} x^{3}+\frac {13}{8} x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {11 \arctan \left (a x \right )}{16 a}\right )}{2 c^{3}}\)	\(256\)
derivativedivides	\(\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c^{3}}+\frac {\arctan \left (a x \right )}{2 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{3}}+\frac {\arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\frac {\frac {11}{8} a^{3} x^{3}+\frac {13}{8} a x}{\left (a^{2} x^{2}+1\right )^{2}}+\frac {11 \arctan \left (a x \right )}{8}-2 i \ln \left (a x \right ) \ln \left (i a x +1\right )+2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-2 i \operatorname {dilog}\left (i a x +1\right )+2 i \operatorname {dilog}\left (-i a x +1\right )+i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{4 c^{3}}\)	\(290\)
default	\(\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c^{3}}+\frac {\arctan \left (a x \right )}{2 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{3}}+\frac {\arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\frac {\frac {11}{8} a^{3} x^{3}+\frac {13}{8} a x}{\left (a^{2} x^{2}+1\right )^{2}}+\frac {11 \arctan \left (a x \right )}{8}-2 i \ln \left (a x \right ) \ln \left (i a x +1\right )+2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-2 i \operatorname {dilog}\left (i a x +1\right )+2 i \operatorname {dilog}\left (-i a x +1\right )+i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{4 c^{3}}\)	\(290\)
risch	\(-\frac {i}{64 c^{3} \left (i a x +1\right )^{2}}+\frac {i \operatorname {dilog}\left (i a x +1\right )}{2 c^{3}}+\frac {i \ln \left (i a x +1\right )^{2}}{8 c^{3}}-\frac {i}{64 c^{3} \left (i a x -1\right )}-\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c^{3}}-\frac {5 i}{32 c^{3} \left (i a x +1\right )}+\frac {5 i}{32 c^{3} \left (-i a x +1\right )}+\frac {i}{64 c^{3} \left (-i a x +1\right )^{2}}-\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2 c^{3}}-\frac {i \ln \left (-i a x +1\right )^{2}}{8 c^{3}}+\frac {i}{64 c^{3} \left (-i a x -1\right )}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c^{3}}-\frac {11 \arctan \left (a x \right )}{64 c^{3}}-\frac {5 i \ln \left (i a x +1\right )}{32 c^{3} \left (i a x +1\right )}-\frac {i \ln \left (i a x +1\right )}{32 c^{3} \left (i a x +1\right )^{2}}+\frac {5 i \ln \left (i a x +1\right )}{64 c^{3} \left (i a x -1\right )}-\frac {3 i \ln \left (i a x +1\right )}{128 c^{3} \left (i a x -1\right )^{2}}+\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c^{3}}+\frac {5 i \ln \left (-i a x +1\right )}{32 c^{3} \left (-i a x +1\right )}+\frac {i \ln \left (-i a x +1\right )}{32 c^{3} \left (-i a x +1\right )^{2}}-\frac {5 i \ln \left (-i a x +1\right )}{64 c^{3} \left (-i a x -1\right )}+\frac {3 i \ln \left (-i a x +1\right )}{128 c^{3} \left (-i a x -1\right )^{2}}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c^{3}}+\frac {\ln \left (-i a x +1\right ) a x}{64 c^{3} \left (-i a x -1\right )^{2}}-\frac {5 \ln \left (i a x +1\right ) a x}{64 c^{3} \left (i a x -1\right )}+\frac {\ln \left (i a x +1\right ) a x}{64 c^{3} \left (i a x -1\right )^{2}}-\frac {5 \ln \left (-i a x +1\right ) a x}{64 c^{3} \left (-i a x -1\right )}+\frac {i \ln \left (-i a x +1\right ) a^{2} x^{2}}{128 c^{3} \left (-i a x -1\right )^{2}}-\frac {i \ln \left (i a x +1\right ) a^{2} x^{2}}{128 c^{3} \left (i a x -1\right )^{2}}\)	\(571\)

[In]

int(arctan(a*x)/x/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/4*arctan(a*x)/c^3/(a^2*x^2+1)^2-1/2/c^3*arctan(a*x)*ln(a^2*x^2+1)+1/2*arctan(a*x)/c^3/(a^2*x^2+1)+1/c^3*arct
an(a*x)*ln(x)-1/2/c^3*a*(-I*ln(x)*(ln(1+I*a*x)-ln(1-I*a*x))/a-I*(dilog(1+I*a*x)-dilog(1-I*a*x))/a-1/4/a^2*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=RootOf(_Z^2*a^2+1))+1/2*(11/8*a^2*x^3+13/8*x)/(a^2*x^
2+1)^2+11/16/a*arctan(a*x))

Fricas [F]

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

[In]

integrate(arctan(a*x)/x/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

integral(arctan(a*x)/(a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x), x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RecursionError} \]

[In]

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

[Out]

Exception raised: RecursionError >> maximum recursion depth exceeded

Maxima [F]

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

[In]

integrate(arctan(a*x)/x/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(arctan(a*x)/x/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )}{x\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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