\(\int \frac {\arctan (a x)}{x^3 (c+a^2 c x^2)} \, dx\) [180]

Optimal result

Integrand size = 20, antiderivative size = 113 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )} \, dx=-\frac {a}{2 c x}-\frac {a^2 \arctan (a x)}{2 c}-\frac {\arctan (a x)}{2 c x^2}+\frac {i a^2 \arctan (a x)^2}{2 c}-\frac {a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5038, 4946, 331, 209, 5044, 4988, 2497} \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {i a^2 \arctan (a x)^2}{2 c}-\frac {a^2 \arctan (a x)}{2 c}-\frac {a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {i a^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c}-\frac {\arctan (a x)}{2 c x^2}-\frac {a}{2 c x} \]

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

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 5038

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.98 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )} \, dx=-\frac {\frac {\arctan (a x)}{x^2}+\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-a^2 x^2\right )}{x}+i a^2 \left (\arctan (a x)^2-2 i \arctan (a x) \log \left (\frac {2 i}{i-a x}\right )+\operatorname {PolyLog}(2,-i a x)-\operatorname {PolyLog}(2,i a x)+\operatorname {PolyLog}\left (2,\frac {i+a x}{-i+a x}\right )\right )}{2 c} \]

[In]

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

[Out]

-1/2*(ArcTan[a*x]/x^2 + (a*Hypergeometric2F1[-1/2, 1, 1/2, -(a^2*x^2)])/x + I*a^2*(ArcTan[a*x]^2 - (2*I)*ArcTa
n[a*x]*Log[(2*I)/(I - a*x)] + PolyLog[2, (-I)*a*x] - PolyLog[2, I*a*x] + PolyLog[2, (I + a*x)/(-I + a*x)]))/c

Maple [C] (verified)

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

Time = 0.39 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.88

method	result	size
parts	\(\frac {\arctan \left (a x \right ) a^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {\arctan \left (a x \right )}{2 c \,x^{2}}-\frac {\arctan \left (a x \right ) a^{2} \ln \left (x \right )}{c}-\frac {a \left (\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{4}+\frac {1}{x}+a \arctan \left (a x \right )-2 a^{2} \left (-\frac {i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{2 a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{2 a}\right )\right )}{2 c}\)	\(213\)
derivativedivides	\(a^{2} \left (\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {\arctan \left (a x \right )}{2 c \,a^{2} x^{2}}-\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c}-\frac {-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}+\frac {1}{a x}+\arctan \left (a x \right )+i \ln \left (a x \right ) \ln \left (i a x +1\right )-i \ln \left (a x \right ) \ln \left (-i a x +1\right )+i \operatorname {dilog}\left (i a x +1\right )-i \operatorname {dilog}\left (-i a x +1\right )}{2 c}\right )\)	\(250\)
default	\(a^{2} \left (\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {\arctan \left (a x \right )}{2 c \,a^{2} x^{2}}-\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c}-\frac {-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}+\frac {1}{a x}+\arctan \left (a x \right )+i \ln \left (a x \right ) \ln \left (i a x +1\right )-i \ln \left (a x \right ) \ln \left (-i a x +1\right )+i \operatorname {dilog}\left (i a x +1\right )-i \operatorname {dilog}\left (-i a x +1\right )}{2 c}\right )\)	\(250\)
risch	\(-\frac {i a^{2} \ln \left (i a x +1\right )^{2}}{8 c}+\frac {i a^{2} \operatorname {dilog}\left (-i a x +1\right )}{2 c}-\frac {i a^{2} \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c}-\frac {a}{2 c x}+\frac {i a^{2} \ln \left (i a x +1\right )}{4 c}-\frac {i a^{2} \ln \left (i a x \right )}{4 c}-\frac {i a^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c}+\frac {i a^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c}+\frac {i \ln \left (i a x +1\right )}{4 c \,x^{2}}-\frac {i a^{2} \operatorname {dilog}\left (i a x +1\right )}{2 c}+\frac {i a^{2} \ln \left (-i a x \right )}{4 c}+\frac {i a^{2} \ln \left (-i a x +1\right )^{2}}{8 c}-\frac {i \ln \left (-i a x +1\right )}{4 c \,x^{2}}+\frac {i a^{2} \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c}-\frac {i a^{2} \ln \left (-i a x +1\right )}{4 c}\)	\(265\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}{\left (a x \right )}}{a^{2} x^{5} + x^{3}}\, dx}{c} \]

[In]

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

[Out]

Integral(atan(a*x)/(a**2*x**5 + x**3), x)/c

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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