3.368 \(\int \frac{(c+a^2 c x^2) \tan ^{-1}(a x)^3}{x^2} \, dx\)
Optimal. Leaf size=169 \[ \frac{3}{2} a c \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{3}{2} a c \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-3 i a c \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+3 i a c \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+a^2 c x \tan ^{-1}(a x)^3-\frac{c \tan ^{-1}(a x)^3}{x}+3 a c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]
[Out]
-((c*ArcTan[a*x]^3)/x) + a^2*c*x*ArcTan[a*x]^3 + 3*a*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] + 3*a*c*ArcTan[a*x]^2*
Log[2 - 2/(1 - I*a*x)] - (3*I)*a*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (3*I)*a*c*ArcTan[a*x]*PolyLog[
2, 1 - 2/(1 + I*a*x)] + (3*a*c*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (3*a*c*PolyLog[3, 1 - 2/(1 + I*a*x)])/2
________________________________________________________________________________________
Rubi [A] time = 0.39471, antiderivative size = 169, normalized size of antiderivative =
1., number of steps used = 11, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.55, Rules used = {4950, 4852, 4924, 4868, 4884, 4992, 6610, 4846, 4920, 4854, 4994}
\[ \frac{3}{2} a c \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{3}{2} a c \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-3 i a c \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+3 i a c \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+a^2 c x \tan ^{-1}(a x)^3-\frac{c \tan ^{-1}(a x)^3}{x}+3 a c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
[In]
Int[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^2,x]
[Out]
-((c*ArcTan[a*x]^3)/x) + a^2*c*x*ArcTan[a*x]^3 + 3*a*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] + 3*a*c*ArcTan[a*x]^2*
Log[2 - 2/(1 - I*a*x)] - (3*I)*a*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (3*I)*a*c*ArcTan[a*x]*PolyLog[
2, 1 - 2/(1 + I*a*x)] + (3*a*c*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (3*a*c*PolyLog[3, 1 - 2/(1 + I*a*x)])/2
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 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 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 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]
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3}{x^2} \, dx &=c \int \frac{\tan ^{-1}(a x)^3}{x^2} \, dx+\left (a^2 c\right ) \int \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c \tan ^{-1}(a x)^3}{x}+a^2 c x \tan ^{-1}(a x)^3+(3 a c) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (3 a^3 c\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^3}{x}+a^2 c x \tan ^{-1}(a x)^3+(3 i a c) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx+\left (3 a^2 c\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^3}{x}+a^2 c x \tan ^{-1}(a x)^3+3 a c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-\left (6 a^2 c\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (6 a^2 c\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^3}{x}+a^2 c x \tan ^{-1}(a x)^3+3 a c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+3 i a c \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\left (3 i a^2 c\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (3 i a^2 c\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^3}{x}+a^2 c x \tan ^{-1}(a x)^3+3 a c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+3 i a c \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{3}{2} a c \text{Li}_3\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.175158, size = 181, normalized size = 1.07 \[ a c \left (3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-\frac{\tan ^{-1}(a x)^3}{a x}+i \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-\frac{i \pi ^3}{8}\right )+a c \left (-3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )+a x \tan ^{-1}(a x)^3-i \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^2,x]
[Out]
a*c*((-I/8)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x) + 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (
3*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2) + a*c*((-I)*Ar
cTan[a*x]^3 + a*x*ArcTan[a*x]^3 + 3*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - (3*I)*ArcTan[a*x]*PolyLog[2
, -E^((2*I)*ArcTan[a*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/2)
________________________________________________________________________________________
Maple [C] time = 0.431, size = 1826, normalized size = 10.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a^2*c*x^2+c)*arctan(a*x)^3/x^2,x)
[Out]
3/2*I*a*c*Pi*arctan(a*x)^2*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))-3/2*I*a*c*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)*arctan(a*x)^2+3/2*a*c*
polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+6*a*c*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*a*c*polylog(3,(1+I*a*x)/(a
^2*x^2+1)^(1/2))+3*I*a*c*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*arctan(a*x)^
2+3/2*I*a*c*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*arctan(a*x)^2-3/2*I*a*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)^2-3/2*I*a*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-c*arctan(a*x)^3/x+a^2*c*x
*arctan(a*x)^3+6*a*c*arctan(a*x)^2*ln(2)-3*a*c*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-3*a*c*arctan(a*x)^2
*ln(a^2*x^2+1)+3*a*c*arctan(a*x)^2*ln(a*x)+6*a*c*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a*c*arctan(a*
x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a*c*arctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I*a*c*arctan(a*
x)^3-3/2*I*a*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)^2-3/2*I*a*c*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))*arc
tan(a*x)^2-3*I*a*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)^2)^2*arctan(a*x)^
2+3/2*I*a*c*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
*arctan(a*x)^2+3/2*I*a*c*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)*arctan
(a*x)^2+3/2*I*a*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+3/2*I*a*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+3/2*I*a*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)^2-3/2*I*a*c*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2-3/2*I*a*c*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*arctan(a*x)^2+3/2*I*a*c*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*
arctan(a*x)^2-3/2*I*a*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)^2-3*I*a
*c*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-6*I*a*c*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-
6*I*a*c*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3/2*I*a*c*Pi*arctan(a*x)^2
________________________________________________________________________________________
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)^3/x^2,x, algorithm="maxima")
[Out]
Timed out
________________________________________________________________________________________
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 )^{3}}{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)^3/x^2,x, algorithm="fricas")
[Out]
integral((a^2*c*x^2 + c)*arctan(a*x)^3/x^2, x)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int a^{2} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int \frac{\operatorname{atan}^{3}{\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)**3/x**2,x)
[Out]
c*(Integral(a**2*atan(a*x)**3, x) + Integral(atan(a*x)**3/x**2, x))
________________________________________________________________________________________
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 )^{3}}{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)^3/x^2,x, algorithm="giac")
[Out]
integrate((a^2*c*x^2 + c)*arctan(a*x)^3/x^2, x)