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

Optimal. Leaf size=310 \[ -\frac {3}{2} i a^2 c \text {Li}_2\left (\frac {2}{1-i a x}-1\right )+\frac {3}{4} i a^2 c \text {Li}_4\left (1-\frac {2}{i a x+1}\right )-\frac {3}{4} i a^2 c \text {Li}_4\left (\frac {2}{i a x+1}-1\right )-\frac {3}{2} i a^2 c \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)^2+\frac {3}{2} i a^2 c \text {Li}_2\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)^2-\frac {3}{2} a^2 c \text {Li}_3\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)+\frac {3}{2} a^2 c \text {Li}_3\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)-\frac {1}{2} a^2 c \tan ^{-1}(a x)^3-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2+3 a^2 c \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {c \tan ^{-1}(a x)^3}{2 x^2}-\frac {3 a c \tan ^{-1}(a x)^2}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.56, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4950, 4852, 4918, 4924, 4868, 2447, 4884, 4850, 4988, 4994, 4998, 6610} \[ -\frac {3}{2} i a^2 c \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {3}{4} i a^2 c \text {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i a^2 c \text {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c \tan ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c \tan ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} a^2 c \tan ^{-1}(a x)^3-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2+3 a^2 c \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {c \tan ^{-1}(a x)^3}{2 x^2}-\frac {3 a c \tan ^{-1}(a x)^2}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4850

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTan[c*x])^p*ArcTanh[1 - 2/(1 +
I*c*x)], x] - Dist[2*b*c*p, Int[((a + b*ArcTan[c*x])^(p - 1)*ArcTanh[1 - 2/(1 + I*c*x)])/(1 + c^2*x^2), x], x
] /; FreeQ[{a, b, c}, x] && IGtQ[p, 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 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 4918

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 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 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 4988

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[(
Log[1 + u]*(a + b*ArcTan[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[1 - u]*(a + b*ArcTan[c*x])^p)/(d +
e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I - c*x))^
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 4998

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a
+ b*ArcTan[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[k
+ 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[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 {\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3}{x^3} \, dx &=c \int \frac {\tan ^{-1}(a x)^3}{x^3} \, dx+\left (a^2 c\right ) \int \frac {\tan ^{-1}(a x)^3}{x} \, dx\\ &=-\frac {c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} (3 a c) \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (6 a^3 c\right ) \int \frac {\tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} (3 a c) \int \frac {\tan ^{-1}(a x)^2}{x^2} \, dx-\frac {1}{2} \left (3 a^3 c\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (3 a^3 c\right ) \int \frac {\tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c\right ) \int \frac {\tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {3 a c \tan ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+\left (3 a^2 c\right ) \int \frac {\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx+\left (3 i a^3 c\right ) \int \frac {\tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 i a^3 c\right ) \int \frac {\tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2-\frac {3 a c \tan ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\left (3 i a^2 c\right ) \int \frac {\tan ^{-1}(a x)}{x (i+a x)} \, dx+\frac {1}{2} \left (3 a^3 c\right ) \int \frac {\text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2-\frac {3 a c \tan ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+3 a^2 c \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i a^2 c \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i a^2 c \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )-\left (3 a^3 c\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2-\frac {3 a c \tan ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+3 a^2 c \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i a^2 c \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i a^2 c \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i a^2 c \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 337, normalized size = 1.09 \[ -\frac {3}{4} i a^2 c \text {Li}_4\left (\frac {-a x-i}{a x-i}\right )+\frac {3}{4} i a^2 c \text {Li}_4\left (\frac {a x+i}{a x-i}\right )+\frac {3}{2} i a^2 c \text {Li}_2\left (\frac {-a x-i}{a x-i}\right ) \tan ^{-1}(a x)^2-\frac {3}{2} i a^2 c \text {Li}_2\left (\frac {a x+i}{a x-i}\right ) \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c \text {Li}_3\left (\frac {-a x-i}{a x-i}\right ) \tan ^{-1}(a x)-\frac {3}{2} a^2 c \text {Li}_3\left (\frac {a x+i}{a x-i}\right ) \tan ^{-1}(a x)+\frac {3}{2} a^2 c \left (-i \text {Li}_2\left (e^{2 i \tan ^{-1}(a x)}\right )-\frac {1}{3} \tan ^{-1}(a x) \left (\left (\tan ^{-1}(a x)+3 i\right ) \tan ^{-1}(a x)+\frac {3 \tan ^{-1}(a x)}{a x}-6 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )\right )\right )+\frac {c \left (-a^2 x^2-1\right ) \tan ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} a^2 c \tan ^{-1}(a x)^3+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2 i}{-a x+i}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*c*ArcTan[a*x]^3)/2 + (c*(-1 - a^2*x^2)*ArcTan[a*x]^3)/(2*x^2) + 2*a^2*c*ArcTan[a*x]^3*ArcTanh[1 - (2*I)/(
I - a*x)] + (3*a^2*c*(-1/3*(ArcTan[a*x]*((3*ArcTan[a*x])/(a*x) + ArcTan[a*x]*(3*I + ArcTan[a*x]) - 6*Log[1 - E
^((2*I)*ArcTan[a*x])])) - I*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/2 + ((3*I)/2)*a^2*c*ArcTan[a*x]^2*PolyLog[2, (
-I - a*x)/(-I + a*x)] - ((3*I)/2)*a^2*c*ArcTan[a*x]^2*PolyLog[2, (I + a*x)/(-I + a*x)] + (3*a^2*c*ArcTan[a*x]*
PolyLog[3, (-I - a*x)/(-I + a*x)])/2 - (3*a^2*c*ArcTan[a*x]*PolyLog[3, (I + a*x)/(-I + a*x)])/2 - ((3*I)/4)*a^
2*c*PolyLog[4, (-I - a*x)/(-I + a*x)] + ((3*I)/4)*a^2*c*PolyLog[4, (I + a*x)/(-I + a*x)]

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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 10.36, size = 568, normalized size = 1.83 \[ -\frac {a^{2} c \arctan \left (a x \right )^{3}}{2}-3 i a^{2} c \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 a c \arctan \left (a x \right )^{2}}{2 x}-\frac {c \arctan \left (a x \right )^{3}}{2 x^{2}}-3 i a^{2} c \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+a^{2} c \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i a^{2} c \arctan \left (a x \right )^{2}}{2}+6 a^{2} c \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i a^{2} c \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {3 a^{2} c \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+6 i a^{2} c \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 a^{2} c \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i a^{2} c \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-a^{2} c \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+6 i a^{2} c \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+a^{2} c \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i a^{2} c \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 a^{2} c \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 a^{2} c \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i a^{2} c \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*c*arctan(a*x)^3-3/4*I*a^2*c*polylog(4,-(1+I*a*x)^2/(a^2*x^2+1))-3/2*a*c*arctan(a*x)^2/x-1/2*c*arctan(
a*x)^3/x^2-3/2*I*a^2*c*arctan(a*x)^2+a^2*c*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*a^2*c*polylog(4
,(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*a^2*c*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*a^2*c*arctan(a*x)
^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3/2*a^2*c*arctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-3*I*a^2*c*
polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a^2*c*arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*a^2*c*arctan
(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-a^2*c*arctan(a*x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+3/2*I*a^2*c*
arctan(a*x)^2*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+a^2*c*arctan(a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*a^
2*c*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a^2*c*arctan(a*x)*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*a^2*c*arcta
n(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*a^2*c*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {4 \, c \arctan \left (a x\right )^{3} - 3 \, c \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + x^{2} \int \frac {12 \, a^{2} c x^{2} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) - 12 \, a c x \arctan \left (a x\right )^{2} - 56 \, {\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3} + 3 \, {\left (a c x - 2 \, {\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{a^{2} x^{5} + x^{3}}\,{d x}}{64 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(4*c*arctan(a*x)^3 - 3*c*arctan(a*x)*log(a^2*x^2 + 1)^2 - 64*x^2*integrate(-1/64*(12*a^2*c*x^2*arctan(a*
x)*log(a^2*x^2 + 1) - 12*a*c*x*arctan(a*x)^2 - 56*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x)^3 + 3*(a*c*x - 2*(
a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^5 + x^3), x))/x^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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