\(\int \frac {\arctan (a x)^2}{x^4 (c+a^2 c x^2)} \, dx\) [290]

Optimal result

Integrand size = 22, antiderivative size = 166 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=-\frac {a^2}{3 c x}-\frac {a^3 \arctan (a x)}{3 c}-\frac {a \arctan (a x)}{3 c x^2}+\frac {4 i a^3 \arctan (a x)^2}{3 c}-\frac {\arctan (a x)^2}{3 c x^3}+\frac {a^2 \arctan (a x)^2}{c x}+\frac {a^3 \arctan (a x)^3}{3 c}-\frac {8 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c}+\frac {4 i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c} \]

[Out]

-1/3*a^2/c/x-1/3*a^3*arctan(a*x)/c-1/3*a*arctan(a*x)/c/x^2+4/3*I*a^3*arctan(a*x)^2/c-1/3*arctan(a*x)^2/c/x^3+a
^2*arctan(a*x)^2/c/x+1/3*a^3*arctan(a*x)^3/c-8/3*a^3*arctan(a*x)*ln(2-2/(1-I*a*x))/c+4/3*I*a^3*polylog(2,-1+2/
(1-I*a*x))/c

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5038, 4946, 331, 209, 5044, 4988, 2497, 5004} \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {a^3 \arctan (a x)^3}{3 c}+\frac {4 i a^3 \arctan (a x)^2}{3 c}-\frac {a^3 \arctan (a x)}{3 c}-\frac {8 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c}+\frac {4 i a^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{3 c}+\frac {a^2 \arctan (a x)^2}{c x}-\frac {a^2}{3 c x}-\frac {\arctan (a x)^2}{3 c x^3}-\frac {a \arctan (a x)}{3 c x^2} \]

[In]

Int[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)),x]

[Out]

-1/3*a^2/(c*x) - (a^3*ArcTan[a*x])/(3*c) - (a*ArcTan[a*x])/(3*c*x^2) + (((4*I)/3)*a^3*ArcTan[a*x]^2)/c - ArcTa
n[a*x]^2/(3*c*x^3) + (a^2*ArcTan[a*x]^2)/(c*x) + (a^3*ArcTan[a*x]^3)/(3*c) - (8*a^3*ArcTan[a*x]*Log[2 - 2/(1 -
I*a*x)])/(3*c) + (((4*I)/3)*a^3*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 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 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)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\arctan (a x)^2}{x^4} \, dx}{c} \\ & = -\frac {\arctan (a x)^2}{3 c x^3}+a^4 \int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx+\frac {(2 a) \int \frac {\arctan (a x)}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2} \, dx}{c} \\ & = -\frac {\arctan (a x)^2}{3 c x^3}+\frac {a^2 \arctan (a x)^2}{c x}+\frac {a^3 \arctan (a x)^3}{3 c}+\frac {(2 a) \int \frac {\arctan (a x)}{x^3} \, dx}{3 c}-\frac {\left (2 a^3\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac {\left (2 a^3\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{c} \\ & = -\frac {a \arctan (a x)}{3 c x^2}+\frac {4 i a^3 \arctan (a x)^2}{3 c}-\frac {\arctan (a x)^2}{3 c x^3}+\frac {a^2 \arctan (a x)^2}{c x}+\frac {a^3 \arctan (a x)^3}{3 c}+\frac {a^2 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac {\left (2 i a^3\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{3 c}-\frac {\left (2 i a^3\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c} \\ & = -\frac {a^2}{3 c x}-\frac {a \arctan (a x)}{3 c x^2}+\frac {4 i a^3 \arctan (a x)^2}{3 c}-\frac {\arctan (a x)^2}{3 c x^3}+\frac {a^2 \arctan (a x)^2}{c x}+\frac {a^3 \arctan (a x)^3}{3 c}-\frac {8 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c}-\frac {a^4 \int \frac {1}{1+a^2 x^2} \, dx}{3 c}+\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{3 c}+\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = -\frac {a^2}{3 c x}-\frac {a^3 \arctan (a x)}{3 c}-\frac {a \arctan (a x)}{3 c x^2}+\frac {4 i a^3 \arctan (a x)^2}{3 c}-\frac {\arctan (a x)^2}{3 c x^3}+\frac {a^2 \arctan (a x)^2}{c x}+\frac {a^3 \arctan (a x)^3}{3 c}-\frac {8 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c}+\frac {4 i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.72 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {a^3 \left (-\frac {1-4 \arctan (a x)^2+\frac {\left (1+a^2 x^2\right ) \arctan (a x)^2}{a^2 x^2}}{a x}+\arctan (a x) \left (-\frac {1+a^2 x^2}{a^2 x^2}+\arctan (a x) (4 i+\arctan (a x))-8 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+4 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )}{3 c} \]

[In]

Integrate[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)),x]

[Out]

(a^3*(-((1 - 4*ArcTan[a*x]^2 + ((1 + a^2*x^2)*ArcTan[a*x]^2)/(a^2*x^2))/(a*x)) + ArcTan[a*x]*(-((1 + a^2*x^2)/
(a^2*x^2)) + ArcTan[a*x]*(4*I + ArcTan[a*x]) - 8*Log[1 - E^((2*I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, E^((2*I)*A
rcTan[a*x])]))/(3*c)

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.77

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )^{2}}{c a x}+\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}+4 \arctan \left (a x \right ) \ln \left (a x \right )-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}+2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\arctan \left (a x \right )^{3}\right )}{3 c}\right )\)	\(293\)
default	\(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )^{2}}{c a x}+\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}+4 \arctan \left (a x \right ) \ln \left (a x \right )-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}+2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\arctan \left (a x \right )^{3}\right )}{3 c}\right )\)	\(293\)
parts	\(\frac {a^{3} \arctan \left (a x \right )^{3}}{c}-\frac {\arctan \left (a x \right )^{2}}{3 c \,x^{3}}+\frac {a^{2} \arctan \left (a x \right )^{2}}{c x}-\frac {2 \left (a^{3} \left (\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}+4 \arctan \left (a x \right ) \ln \left (a x \right )-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}+2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )+a^{3} \arctan \left (a x \right )^{3}\right )}{3 c}\)	\(298\)

[In]

int(arctan(a*x)^2/x^4/(a^2*c*x^2+c),x,method=_RETURNVERBOSE)

[Out]

a^3*(-1/3/c*arctan(a*x)^2/a^3/x^3+1/c*arctan(a*x)^2/a/x+1/c*arctan(a*x)^3-2/3/c*(1/2*arctan(a*x)/a^2/x^2+4*arc
tan(a*x)*ln(a*x)-2*arctan(a*x)*ln(a^2*x^2+1)+1/2*arctan(a*x)+1/2/a/x+2*I*ln(a*x)*ln(1+I*a*x)-2*I*ln(a*x)*ln(1-
I*a*x)+2*I*dilog(1+I*a*x)-2*I*dilog(1-I*a*x)-I*(ln(a*x-I)*ln(a^2*x^2+1)-dilog(-1/2*I*(I+a*x))-ln(a*x-I)*ln(-1/
2*I*(I+a*x))-1/2*ln(a*x-I)^2)+I*(ln(I+a*x)*ln(a^2*x^2+1)-dilog(1/2*I*(a*x-I))-ln(I+a*x)*ln(1/2*I*(a*x-I))-1/2*
ln(I+a*x)^2)+arctan(a*x)^3))

Fricas [F]

\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]

[In]

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{6} + x^{4}}\, dx}{c} \]

[In]

integrate(atan(a*x)**2/x**4/(a**2*c*x**2+c),x)

[Out]

Integral(atan(a*x)**2/(a**2*x**6 + x**4), x)/c

Maxima [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\text {Timed out} \]

[In]

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]

[In]

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

int(atan(a*x)^2/(x^4*(c + a^2*c*x^2)),x)

[Out]

int(atan(a*x)^2/(x^4*(c + a^2*c*x^2)), x)