3.263 \(\int \frac{(c+a^2 c x^2) \tan ^{-1}(a x)^2}{x^2} \, dx\)

Optimal. Leaf size=113 \[ -i a c \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+i a c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+a^2 c x \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^2}{x}+2 a c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 a c \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.222704, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4950, 4852, 4924, 4868, 2447, 4846, 4920, 4854, 2402, 2315} \[ -i a c \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+i a c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+a^2 c x \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^2}{x}+2 a c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 a c \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4950

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]))

Rule 4852

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

Rule 4924

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4868

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 2447

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 4846

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

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2}{x^2} \, dx &=c \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx+\left (a^2 c\right ) \int \tan ^{-1}(a x)^2 \, dx\\ &=-\frac{c \tan ^{-1}(a x)^2}{x}+a^2 c x \tan ^{-1}(a x)^2+(2 a c) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx-\left (2 a^3 c\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^2}{x}+a^2 c x \tan ^{-1}(a x)^2+(2 i a c) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx+\left (2 a^2 c\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^2}{x}+a^2 c x \tan ^{-1}(a x)^2+2 a c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\left (2 a^2 c\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 a^2 c\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^2}{x}+a^2 c x \tan ^{-1}(a x)^2+2 a c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a c \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+(2 i a c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )\\ &=-\frac{c \tan ^{-1}(a x)^2}{x}+a^2 c x \tan ^{-1}(a x)^2+2 a c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a c \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+i a c \text{Li}_2\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.148467, size = 123, normalized size = 1.09 \[ a c \left (-i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+a x \tan ^{-1}(a x)^2-i \tan ^{-1}(a x)^2+2 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )+a c \left (-i \left (\tan ^{-1}(a x)^2+\text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )\right )-\frac{\tan ^{-1}(a x)^2}{a x}+2 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.092, size = 262, normalized size = 2.3 \begin{align*}{a}^{2}cx \left ( \arctan \left ( ax \right ) \right ) ^{2}-{\frac{c \left ( \arctan \left ( ax \right ) \right ) ^{2}}{x}}-2\,ac\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) +2\,ac\arctan \left ( ax \right ) \ln \left ( ax \right ) -iac\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) -iac\ln \left ( ax \right ) \ln \left ( 1-iax \right ) -iac{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) +iac\ln \left ( ax \right ) \ln \left ( 1+iax \right ) +iac{\it dilog} \left ( 1+iax \right ) +{\frac{i}{2}}ac \left ( \ln \left ( ax-i \right ) \right ) ^{2}+iac\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) -iac{\it dilog} \left ( 1-iax \right ) +iac\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -iac\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) -{\frac{i}{2}}ac \left ( \ln \left ( ax+i \right ) \right ) ^{2}+iac{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int a^{2} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x^{2}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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