\(\int \frac {(c+a^2 c x^2) \arctan (a x)^2}{x^4} \, dx\) [265]

Optimal result

Integrand size = 20, antiderivative size = 135 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^4} \, dx=-\frac {a^2 c}{3 x}-\frac {1}{3} a^3 c \arctan (a x)-\frac {a c \arctan (a x)}{3 x^2}-\frac {2}{3} i a^3 c \arctan (a x)^2-\frac {c \arctan (a x)^2}{3 x^3}-\frac {a^2 c \arctan (a x)^2}{x}+\frac {4}{3} a^3 c \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {2}{3} i a^3 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5070, 4946, 5038, 331, 209, 5044, 4988, 2497} \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^4} \, dx=-\frac {2}{3} i a^3 c \arctan (a x)^2-\frac {1}{3} a^3 c \arctan (a x)+\frac {4}{3} a^3 c \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {2}{3} i a^3 c \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )-\frac {a^2 c \arctan (a x)^2}{x}-\frac {a^2 c}{3 x}-\frac {c \arctan (a x)^2}{3 x^3}-\frac {a c \arctan (a x)}{3 x^2} \]

[In]

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

[Out]

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

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]

Rule 5070

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

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.76 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^4} \, dx=\frac {c \left (-a^2 x^2+(1-2 i a x) (-i+a x)^2 \arctan (a x)^2+a x \arctan (a x) \left (-1-a^2 x^2+4 a^2 x^2 \log \left (1-e^{2 i \arctan (a x)}\right )\right )-2 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )}{3 x^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (117 ) = 234\).

Time = 0.75 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.01

method	result	size
derivativedivides	\(a^{3} \left (-\frac {c \arctan \left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {c \arctan \left (a x \right )^{2}}{a x}-\frac {2 c \left (\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}-2 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}\right )}{3}\right )\)	\(271\)
default	\(a^{3} \left (-\frac {c \arctan \left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {c \arctan \left (a x \right )^{2}}{a x}-\frac {2 c \left (\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}-2 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}\right )}{3}\right )\)	\(271\)
parts	\(-\frac {a^{2} c \arctan \left (a x \right )^{2}}{x}-\frac {c \arctan \left (a x \right )^{2}}{3 x^{3}}-\frac {2 c \left (a^{3} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {a \arctan \left (a x \right )}{2 x^{2}}-2 a^{3} \arctan \left (a x \right ) \ln \left (a x \right )+\frac {a^{3} \left (\frac {1}{a x}+\arctan \left (a x \right )-2 i \ln \left (a x \right ) \ln \left (i a x +1\right )+2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-2 i \operatorname {dilog}\left (i a x +1\right )+2 i \operatorname {dilog}\left (-i a x +1\right )+i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{2}\right )}{3}\)	\(271\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/48*(12*(a^3*c*arctan(a*x)^3 + 12*a^4*c*integrate(1/48*x^4*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) - 48*a^4*c*
integrate(1/48*x^4*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 96*a^3*c*integrate(1/48*x^3*arctan(a*x)/(a^2*x^6 + x
^4), x) + 288*a^2*c*integrate(1/48*x^2*arctan(a*x)^2/(a^2*x^6 + x^4), x) + 24*a^2*c*integrate(1/48*x^2*log(a^2
*x^2 + 1)^2/(a^2*x^6 + x^4), x) - 16*a^2*c*integrate(1/48*x^2*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 32*a*c*in
tegrate(1/48*x*arctan(a*x)/(a^2*x^6 + x^4), x) + 144*c*integrate(1/48*arctan(a*x)^2/(a^2*x^6 + x^4), x) + 12*c
*integrate(1/48*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x))*x^3 - 4*(3*a^2*c*x^2 + c)*arctan(a*x)^2 + (3*a^2*c*x^2
+ c)*log(a^2*x^2 + 1)^2)/x^3

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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