\(\int \frac {(a+b \arctan (c x))^3}{x} \, dx\) [30]

Optimal result

Integrand size = 14, antiderivative size = 206 \[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=2 (a+b \arctan (c x))^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {3}{2} b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )+\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i c x}\right ) \]

[Out]

-2*(a+b*arctan(c*x))^3*arctanh(-1+2/(1+I*c*x))-3/2*I*b*(a+b*arctan(c*x))^2*polylog(2,1-2/(1+I*c*x))+3/2*I*b*(a
+b*arctan(c*x))^2*polylog(2,-1+2/(1+I*c*x))-3/2*b^2*(a+b*arctan(c*x))*polylog(3,1-2/(1+I*c*x))+3/2*b^2*(a+b*ar
ctan(c*x))*polylog(3,-1+2/(1+I*c*x))+3/4*I*b^3*polylog(4,1-2/(1+I*c*x))-3/4*I*b^3*polylog(4,-1+2/(1+I*c*x))

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4942, 5108, 5004, 5114, 5118, 6745} \[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3-\frac {3}{2} b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))+\frac {3}{2} b^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))-\frac {3}{2} i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2+\frac {3}{2} i b \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))^2+\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{i c x+1}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,\frac {2}{i c x+1}-1\right ) \]

[In]

Int[(a + b*ArcTan[c*x])^3/x,x]

[Out]

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

Rule 4942

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 5004

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 5108

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[L
og[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 5114

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[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 5118

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 6745

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*} \text {integral}& = 2 (a+b \arctan (c x))^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-(6 b c) \int \frac {(a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = 2 (a+b \arctan (c x))^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+(3 b c) \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-(3 b c) \int \frac {(a+b \arctan (c x))^2 \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = 2 (a+b \arctan (c x))^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )+\left (3 i b^2 c\right ) \int \frac {(a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (3 i b^2 c\right ) \int \frac {(a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = 2 (a+b \arctan (c x))^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {3}{2} b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )+\frac {1}{2} \left (3 b^3 c\right ) \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\frac {1}{2} \left (3 b^3 c\right ) \int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = 2 (a+b \arctan (c x))^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {3}{2} b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )+\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i c x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.79 \[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=a^3 \log (c x)+\frac {3}{2} i a^2 b (\operatorname {PolyLog}(2,-i c x)-\operatorname {PolyLog}(2,i c x))+3 a b^2 \left (-\frac {i \pi ^3}{24}+\frac {2}{3} i \arctan (c x)^3+\arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-\arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )-\frac {1}{64} i b^3 \left (\pi ^4-32 \arctan (c x)^4+64 i \arctan (c x)^3 \log \left (1-e^{-2 i \arctan (c x)}\right )-64 i \arctan (c x)^3 \log \left (1+e^{2 i \arctan (c x)}\right )-96 \arctan (c x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-96 \arctan (c x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+96 i \arctan (c x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-96 i \arctan (c x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (c x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (c x)}\right )\right ) \]

[In]

Integrate[(a + b*ArcTan[c*x])^3/x,x]

[Out]

a^3*Log[c*x] + ((3*I)/2)*a^2*b*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + 3*a*b^2*((-1/24*I)*Pi^3 + ((2*I)/3
)*ArcTan[c*x]^3 + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])]
+ I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + PolyL
og[3, E^((-2*I)*ArcTan[c*x])]/2 - PolyLog[3, -E^((2*I)*ArcTan[c*x])]/2) - (I/64)*b^3*(Pi^4 - 32*ArcTan[c*x]^4
+ (64*I)*ArcTan[c*x]^3*Log[1 - E^((-2*I)*ArcTan[c*x])] - (64*I)*ArcTan[c*x]^3*Log[1 + E^((2*I)*ArcTan[c*x])] -
96*ArcTan[c*x]^2*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 96*ArcTan[c*x]^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (
96*I)*ArcTan[c*x]*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - (96*I)*ArcTan[c*x]*PolyLog[3, -E^((2*I)*ArcTan[c*x])] +
48*PolyLog[4, E^((-2*I)*ArcTan[c*x])] + 48*PolyLog[4, -E^((2*I)*ArcTan[c*x])])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.58 (sec) , antiderivative size = 2026, normalized size of antiderivative = 9.83

\[\text {Expression too large to display}\]

[In]

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

[Out]

a^3*ln(c*x)+b^3*(ln(c*x)*arctan(c*x)^3-arctan(c*x)^3*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+arctan(c*x)^3*ln(1-(1+I*c*x
)/(c^2*x^2+1)^(1/2))-3*I*arctan(c*x)^2*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*arctan(c*x)*polylog(3,(1+I*c*x
)/(c^2*x^2+1)^(1/2))+6*I*polylog(4,(1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c*x)^3*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2)
)-3*I*arctan(c*x)^2*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*arctan(c*x)*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/
2))+6*I*polylog(4,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*Pi*(csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c
^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))-csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+
(1+I*c*x)^2/(c^2*x^2+1)))^2+csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*((1
+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))-csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/
(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2-csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*((1+I*c*x)^2/(c^2*x^2
+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2+csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3-csgn(I
*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2
*x^2+1)))^2+csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3+1)*arctan(c*x)^3+3/2*I*arctan(c*x)
^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-3/2*arctan(c*x)*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-3/4*I*polylog(4,-(1
+I*c*x)^2/(c^2*x^2+1)))+3*a*b^2*(ln(c*x)*arctan(c*x)^2+I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-1/2*p
olylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+arctan(c*x)^2*ln(1-(1+I*c*x)/(c
^2*x^2+1)^(1/2))-2*I*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2
))+arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*p
olylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*Pi*(csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)
))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))-csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)
^2/(c^2*x^2+1)))^2+csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*((1+I*c*x)^2
/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))-csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+
1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2-csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1
+(1+I*c*x)^2/(c^2*x^2+1)))^2+csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3-csgn(I*((1+I*c*
x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))
^2+csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3+1)*arctan(c*x)^2)+3*a^2*b*(ln(c*x)*arctan(c
*x)+1/2*I*ln(c*x)*ln(1+I*c*x)-1/2*I*ln(c*x)*ln(1-I*c*x)+1/2*I*dilog(1+I*c*x)-1/2*I*dilog(1-I*c*x))

Fricas [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}}{x}\, dx \]

[In]

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

[Out]

Integral((a + b*atan(c*x))**3/x, x)

Maxima [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{x} \,d x } \]

[In]

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

[Out]

a^3*log(x) + 1/32*integrate((28*b^3*arctan(c*x)^3 + 3*b^3*arctan(c*x)*log(c^2*x^2 + 1)^2 + 96*a*b^2*arctan(c*x
)^2 + 96*a^2*b*arctan(c*x))/x, x)

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{x} \,d x \]

[In]

int((a + b*atan(c*x))^3/x,x)

[Out]

int((a + b*atan(c*x))^3/x, x)