3.295 \(\int \frac{\tan ^{-1}(a x)^2}{x (c+a^2 c x^2)^2} \, dx\)
Optimal. Leaf size=170 \[ \frac{\text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^2}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{1}{4 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}-\frac{a x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac{i \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^2}{4 c^2}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^2} \]
[Out]
-1/(4*c^2*(1 + a^2*x^2)) - (a*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^2/(4*c^2) + ArcTan[a*x]^2/(2*
c^2*(1 + a^2*x^2)) - ((I/3)*ArcTan[a*x]^3)/c^2 + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 - (I*ArcTan[a*x]*P
olyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c^2)
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Rubi [A] time = 0.311218, antiderivative size = 170, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used =
{4966, 4924, 4868, 4884, 4992, 6610, 4930, 4892, 261} \[ \frac{\text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^2}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{1}{4 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}-\frac{a x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac{i \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^2}{4 c^2}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^2} \]
Antiderivative was successfully verified.
[In]
Int[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^2),x]
[Out]
-1/(4*c^2*(1 + a^2*x^2)) - (a*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^2/(4*c^2) + ArcTan[a*x]^2/(2*
c^2*(1 + a^2*x^2)) - ((I/3)*ArcTan[a*x]^3)/c^2 + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 - (I*ArcTan[a*x]*P
olyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c^2)
Rule 4966
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]
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 4884
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 4992
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a + b*ArcT
an[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u]
)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*I
)/(I + c*x))^2, 0]
Rule 6610
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
; !FalseQ[w]] /; FreeQ[n, x]
Rule 4930
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 4892
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*(a + b*ArcTan[c*x])
^p)/(2*d*(d + e*x^2)), x] + (-Dist[(b*c*p)/2, Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + Simp
[(a + b*ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p,
0]
Rule 261
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^2} \, dx &=-\left (a^2 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=\frac{\tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^3}{3 c^2}-a \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{i \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^2}\\ &=-\frac{a x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^2}{4 c^2}+\frac{\tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^3}{3 c^2}+\frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}+\frac{1}{2} a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx-\frac{(2 a) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\\ &=-\frac{1}{4 c^2 \left (1+a^2 x^2\right )}-\frac{a x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^2}{4 c^2}+\frac{\tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^3}{3 c^2}+\frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{i \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}+\frac{(i a) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\\ &=-\frac{1}{4 c^2 \left (1+a^2 x^2\right )}-\frac{a x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^2}{4 c^2}+\frac{\tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^3}{3 c^2}+\frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{i \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}+\frac{\text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.198638, size = 119, normalized size = 0.7 \[ \frac{24 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+12 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )+8 i \tan ^{-1}(a x)^3+24 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-6 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+6 \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )-3 \cos \left (2 \tan ^{-1}(a x)\right )-i \pi ^3}{24 c^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^2),x]
[Out]
((-I)*Pi^3 + (8*I)*ArcTan[a*x]^3 - 3*Cos[2*ArcTan[a*x]] + 6*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 24*ArcTan[a*x]^
2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 12*PolyLog[3, E^((
-2*I)*ArcTan[a*x])] - 6*ArcTan[a*x]*Sin[2*ArcTan[a*x]])/(24*c^2)
________________________________________________________________________________________
Maple [C] time = 0.53, size = 1936, normalized size = 11.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arctan(a*x)^2/x/(a^2*c*x^2+c)^2,x)
[Out]
-1/2/c^2*arctan(a*x)^2*ln(a^2*x^2+1)+1/c^2*arctan(a*x)^2*ln(2)-1/c^2*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-
1)+1/c^2*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/2*I/c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn
(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2+
1/2*arctan(a*x)^2/c^2/(a^2*x^2+1)-1/3*I*arctan(a*x)^3/c^2+1/c^2*arctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2)
)+1/c^2*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-1/c^2*arctan(a*x)/(8*a*x-8*I)-1/c^2*arctan(a*x)/(8*a*x+8
*I)+1/16*I/c^2/(a*x-I)-1/16*I/c^2/(a*x+I)+1/c^2*arctan(a*x)^2*ln(a*x)+I/c^2*arctan(a*x)/(8*a*x-8*I)*a*x-1/4*I/
c^2*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))-1/2*I/c^2*Pi*csgn(I
*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x
^2+1)+1))^2*arctan(a*x)^2+1/4*I/c^2*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x
^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-1/2*I/c^2*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a
^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2+1/2*I/c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+
I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2+1/2*I/c^2
*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2+1/4*I/c^2*arctan(a*x)^
2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-1/2*I/c^2*Pi*csgn(I*((1+I*a*x
)^2/(a^2*x^2+1)-1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2-1/2*I/c^2*
arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+1/4*I/c^2*arctan(
a*x)^2*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^
2-I/c^2*arctan(a*x)/(8*a*x+8*I)*a*x+2/c^2*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2/c^2*polylog(3,(1+I*a*x)/(a
^2*x^2+1)^(1/2))-1/4*arctan(a*x)^2/c^2-1/4*I/c^2*arctan(a*x)^2*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I
*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+1/16/c^2/(a*x+I)*a*x-2
*I/c^2*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I/c^2*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)
^(1/2))+1/2*I/c^2*Pi*arctan(a*x)^2+1/16/c^2/(a*x-I)*a*x-1/4*I/c^2*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2
+1))^3-1/2*I/c^2*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2+1/4*I/c^2*ar
ctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+1/2*I/c^2*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x
)^2/(a^2*x^2+1)+1))^3*arctan(a*x)^2-1/4*I/c^2*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^
2*x^2+1)+1)^2)^3+1/2*I/c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*arctan(a*x)^2
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctan(a*x)^2/x/(a^2*c*x^2+c)^2,x, algorithm="maxima")
[Out]
integrate(arctan(a*x)^2/((a^2*c*x^2 + c)^2*x), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arctan \left (a x\right )^{2}}{a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctan(a*x)^2/x/(a^2*c*x^2+c)^2,x, algorithm="fricas")
[Out]
integral(arctan(a*x)^2/(a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{a^{4} x^{5} + 2 a^{2} x^{3} + x}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(atan(a*x)**2/x/(a**2*c*x**2+c)**2,x)
[Out]
Integral(atan(a*x)**2/(a**4*x**5 + 2*a**2*x**3 + x), x)/c**2
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctan(a*x)^2/x/(a^2*c*x^2+c)^2,x, algorithm="giac")
[Out]
integrate(arctan(a*x)^2/((a^2*c*x^2 + c)^2*x), x)