3.283 \(\int \frac{x^3 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx\)

Optimal. Leaf size=169 \[ \frac{\text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a^4 c}+\frac{i \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a^4 c}+\frac{\log \left (a^2 x^2+1\right )}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^3}{3 a^4 c}+\frac{\tan ^{-1}(a x)^2}{2 a^4 c}-\frac{x \tan ^{-1}(a x)}{a^3 c}+\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{a^4 c} \]

[Out]

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

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Rubi [A]  time = 0.293706, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {4916, 4852, 4846, 260, 4884, 4920, 4854, 4994, 6610} \[ \frac{\text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a^4 c}+\frac{i \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a^4 c}+\frac{\log \left (a^2 x^2+1\right )}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^3}{3 a^4 c}+\frac{\tan ^{-1}(a x)^2}{2 a^4 c}-\frac{x \tan ^{-1}(a x)}{a^3 c}+\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{a^4 c} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]

[Out]

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

Rule 4916

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(d*f^2)/e, 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] && GtQ[m, 1]

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 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 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 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 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[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{x^3 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx &=-\frac{\int \frac{x \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{a^2}+\frac{\int x \tan ^{-1}(a x)^2 \, dx}{a^2 c}\\ &=\frac{x^2 \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^3}{3 a^4 c}+\frac{\int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{a^3 c}-\frac{\int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a c}\\ &=\frac{x^2 \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^3}{3 a^4 c}+\frac{\tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a^4 c}-\frac{\int \tan ^{-1}(a x) \, dx}{a^3 c}+\frac{\int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^3 c}-\frac{2 \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c}\\ &=-\frac{x \tan ^{-1}(a x)}{a^3 c}+\frac{\tan ^{-1}(a x)^2}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^3}{3 a^4 c}+\frac{\tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a^4 c}+\frac{i \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^4 c}-\frac{i \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c}+\frac{\int \frac{x}{1+a^2 x^2} \, dx}{a^2 c}\\ &=-\frac{x \tan ^{-1}(a x)}{a^3 c}+\frac{\tan ^{-1}(a x)^2}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^3}{3 a^4 c}+\frac{\tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a^4 c}+\frac{\log \left (1+a^2 x^2\right )}{2 a^4 c}+\frac{i \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^4 c}+\frac{\text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a^4 c}\\ \end{align*}

Mathematica [A]  time = 0.112094, size = 123, normalized size = 0.73 \[ \frac{-i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-\log \left (\frac{1}{\sqrt{a^2 x^2+1}}\right )+\frac{1}{2} \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-\frac{1}{3} i \tan ^{-1}(a x)^3-a x \tan ^{-1}(a x)+\tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )}{a^4 c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]

[Out]

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

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Maple [C]  time = 0.727, size = 1695, normalized size = 10. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arctan(a*x)^2/(a^2*c*x^2+c),x)

[Out]

1/a^4/c*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-1/a^4/c*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+1/2/a^4/c*polylog(
3,-(1+I*a*x)^2/(a^2*x^2+1))-1/4*I/a^4/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*arctan(a*x)^2/a^4/c+1/8/a^3
/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)*x-1/4/a^3/c*ar
ctan(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*x-x*arctan(a*x)/a^3
/c+1/2*x^2*arctan(a*x)^2/a^2/c-1/8/a^3/c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^
2*x^2+1)+I)^3*x+1/8/a^3/c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*x+1/8*I/a^4/c*arctan(a*x)^2
*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^3+1/8*I/a^4/c*arctan(a*x)^2*Pi*csgn(I*((1+
I*a*x)^2/(a^2*x^2+1)+1)^2)^3-1/4*I/a^4/c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3-1/4*I/a^4/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)^3-1/4*I/a^4/c*arctan(a*x)^2*Pi*csgn(I
*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2+1/4*I/a^4/c*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+1/8*I/a^4/c*arctan(a*x)^2*Pi*cs
gn(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)-1/4*I/a^4/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-I/a^4/c*arctan(a*x)*polylog(2,-(1+I*a*x)^
2/(a^2*x^2+1))-1/2/a^4/c*arctan(a*x)^2*ln(a^2*x^2+1)+1/a^4/c*arctan(a*x)^2*ln(2)+I/a^4/c*arctan(a*x)-1/3*I/a^4
/c*arctan(a*x)^3+1/4/a^3/c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*
csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)*x-1/4*I/a^4/c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x
)^2/(a^2*x^2+1)+I)^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)+1/8*I/a^4/c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^
2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2-1/8/a^3/c*arctan(a*x)^2*Pi*csgn(I*(1
+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*x+1/4*I/a^4/c*arcta
n(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+1/2*I/a^4/c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*csgn(I*(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{x^{3} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctan(a*x)^2/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

integrate(x^3*arctan(a*x)^2/(a^2*c*x^2 + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctan(a*x)^2/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

integral(x^3*arctan(a*x)^2/(a^2*c*x^2 + c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{3} \operatorname{atan}^{2}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*atan(a*x)**2/(a**2*c*x**2+c),x)

[Out]

Integral(x**3*atan(a*x)**2/(a**2*x**2 + 1), x)/c

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctan(a*x)^2/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

integrate(x^3*arctan(a*x)^2/(a^2*c*x^2 + c), x)