3.287 \(\int \frac{\tan ^{-1}(a x)^2}{x (c+a^2 c x^2)} \, dx\)
Optimal. Leaf size=91 \[ \frac{\text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c}-\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c} \]
[Out]
((-I/3)*ArcTan[a*x]^3)/c + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*
a*x)])/c + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c)
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Rubi [A] time = 0.179331, antiderivative size = 91, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used =
{4924, 4868, 4884, 4992, 6610} \[ \frac{\text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c}-\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c} \]
Antiderivative was successfully verified.
[In]
Int[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)),x]
[Out]
((-I/3)*ArcTan[a*x]^3)/c + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*
a*x)])/c + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c)
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]
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )} \, dx &=-\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{i \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c}\\ &=-\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c}-\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}\\ &=-\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c}-\frac{i \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c}+\frac{(i a) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}\\ &=-\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c}-\frac{i \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c}+\frac{\text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c}\\ \end{align*}
Mathematica [B] time = 0.0496027, size = 243, normalized size = 2.67 \[ \frac{\text{PolyLog}\left (3,\frac{-a x-i}{a x-i}\right )}{2 c}+\frac{\text{PolyLog}\left (3,-\frac{a x+i}{-a x+i}\right )}{2 c}-\frac{\text{PolyLog}\left (3,\frac{a x+i}{a x-i}\right )}{2 c}+\frac{i \tan ^{-1}(a x) \text{PolyLog}\left (2,\frac{-a x-i}{a x-i}\right )}{c}+\frac{i \tan ^{-1}(a x) \text{PolyLog}\left (2,-\frac{a x+i}{-a x+i}\right )}{c}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left (2,\frac{a x+i}{a x-i}\right )}{c}+\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{\log \left (\frac{2 i}{-a x+i}\right ) \tan ^{-1}(a x)^2}{c}+\frac{2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2 i}{-a x+i}\right )}{c} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)),x]
[Out]
((I/3)*ArcTan[a*x]^3)/c + (2*ArcTan[a*x]^2*ArcTanh[1 - (2*I)/(I - a*x)])/c + (ArcTan[a*x]^2*Log[(2*I)/(I - a*x
)])/c + (I*ArcTan[a*x]*PolyLog[2, (-I - a*x)/(-I + a*x)])/c + (I*ArcTan[a*x]*PolyLog[2, -((I + a*x)/(I - a*x))
])/c - (I*ArcTan[a*x]*PolyLog[2, (I + a*x)/(-I + a*x)])/c + PolyLog[3, (-I - a*x)/(-I + a*x)]/(2*c) + PolyLog[
3, -((I + a*x)/(I - a*x))]/(2*c) - PolyLog[3, (I + a*x)/(-I + a*x)]/(2*c)
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Maple [C] time = 0.283, size = 1767, normalized size = 19.4 \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),x)
[Out]
1/c*arctan(a*x)^2*ln(2)-1/c*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)+1/c*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*
x^2+1)^(1/2))+1/c*arctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/c*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(
1/2))-1/2/c*arctan(a*x)^2*ln(a^2*x^2+1)+1/c*arctan(a*x)^2*ln(a*x)+1/2*I/c*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*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*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+1/4*I/c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+1/2*I/c*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*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*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*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*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)^2-1/4*I/c*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*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*arc
tan(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*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*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)^2-1/4*I/c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2
+1))^3-1/2*I/c*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)^2-1/3*I*arctan(a
*x)^3/c+1/2*I/c*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)^2-1/4*I/c*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)+1/2*
I/c*Pi*arctan(a*x)^2-2*I/c*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I/c*arctan(a*x)*polylog(2,(1+
I*a*x)/(a^2*x^2+1)^(1/2))+2/c*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2/c*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2
))
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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 )} 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),x, algorithm="maxima")
[Out]
integrate(arctan(a*x)^2/((a^2*c*x^2 + c)*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^{2} c x^{3} + c 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),x, algorithm="fricas")
[Out]
integral(arctan(a*x)^2/(a^2*c*x^3 + c*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^{2} x^{3} + x}\, dx}{c} \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),x)
[Out]
Integral(atan(a*x)**2/(a**2*x**3 + x), x)/c
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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 )} 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),x, algorithm="giac")
[Out]
integrate(arctan(a*x)^2/((a^2*c*x^2 + c)*x), x)